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# Research news

edited March 3 in Strategy

I've been writing weekly progress reports to my grad students. It may make sense to copy them here. Let me give it a try.

«1

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1.

17 November 2016:

Hi -

Daniel Cicala points out the math conference December 3rd and 4th at Union College in upstate NY. There will be talks on category theory organized by Susan Niefeld - she does this every year, and I've spoken there once. If you can go, do it!

(If you read the category theory mailing list you can stay up to date on such conferences.)

Here is this week's progress:

1) Blake and I went to San Diego and learned about Metron's "ExAMS" software for designing complex systems. Then John Foley came up and, with help from Joseph Moeller, we figured out a bunch of stuff.

This software raises lots of interesting questions. I believe it's based on "timed hierarchical colored Petri nets with guards". I would like to make sure this is true, and understand this kind of network category-theoretically. In case anyone wants to help me, here's an intro:

When we met, Tom Mifflin at Metron seemed pretty eager for our work to go in this direction.

2) I went to the Mathematical Association of America conference and gave a talk on The answer to the ultimate question of life, the universe and everything. Brandon and Daniel also went there.

3) I finally blogged about Brendan's thesis:

As you can see, it's a lazy blog article - yet it still manages to give a detailed introduction to his work! If you haven't yet learned everything that Brendan is doing, this is a good place to start.

Comment Source:17 November 2016: Hi - Daniel Cicala points out the [math conference December 3rd and 4th at Union College](http://www.math.union.edu/%7Etoddg/ucc/) in upstate NY. There will be talks on category theory organized by Susan Niefeld - she does this every year, and I've spoken there once. If you can go, do it! (If you read the category theory mailing list you can stay up to date on such conferences.) Here is this week's progress: 1) Blake and I went to San Diego and learned about Metron's "ExAMS" software for designing complex systems. Then John Foley came up and, with help from Joseph Moeller, we figured out a bunch of stuff. I blogged about ExAMS here: * [Complex Adaptive System Design (Part 2)](https://johncarlosbaez.wordpress.com/2016/10/18/complex-adaptive-system-design-part-2/) This software raises lots of interesting questions. I believe it's based on "timed hierarchical colored Petri nets with guards". I would like to make sure this is true, and understand this kind of network category-theoretically. In case anyone wants to help me, here's an intro: * Wil M. P. van der Aalst, Christian Stahl, and Michael Westergaard, [Strategies for modeling complex processes using colored Petri nets](http://wwwis.win.tue.nl/%7Ewvdaalst/publications/p710.pdf). When we met, Tom Mifflin at Metron seemed pretty eager for our work to go in this direction. 2) I went to the Mathematical Association of America conference and gave a talk on [The answer to the ultimate question of life, the universe and everything](http://math.ucr.edu/home/baez/42/). Brandon and Daniel also went there. 3) I finally blogged about Brendan's thesis: * [Open and interconnected systems](https://johncarlosbaez.wordpress.com/2016/10/23/open-and-interconnected-systems/). As you can see, it's a lazy blog article - yet it still manages to give a detailed introduction to his work! If you haven't yet learned everything that Brendan is doing, this is a good place to start.
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2.

23 November 2016:

Lots of progress this week:

1) Jason uploaded his thesis to the arXiv! That's great, because it establishes priority - now he can prove he did this stuff before anyone else, even if it takes a while for him to publish a paper.

2) Blake gave a talk about open chemical reaction networks at the Santa Fe Institute workshop on Statistical Mechanics, Information Processing and Biology:

It was a real hit. Susanne Still said it was "great" - she works on Markov processes and the information bottleneck method for predictive inference. Jim Crutchfield liked it so much he invited Blake to give a talk up at U. C. Davis! He's a real bigshot: the Wikipedia article on him says

Over the last three decades Prof. Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas.

If Blake can strike up a relationship with Crutchfield and maybe work on a project, that'll be excellent.

3) Joshua Tan, a grad student at Oxford (and friend of Brendan), invited me to join a bunch of people in writing a grant proposal.

It's for an NSF grant called "Smart & Connected Communities", and part of the plan would be to model cities as composable, open systems using category theory. Here are the other people involved in writing the proposal:

• Dennis Frenchman is a professor at MIT and an expert in building digital tools for cities. He is the likely PI.

• Sokwoo Rhee is a director at NIST managing 100+ smart cities projects and will be collaborating directly with us, but he is a silent partner due to federal rules.

• Stephen Walter is a program director at the City of Boston, Mayor's Office of New Urban Mechanics.

• Matthew Claudel is a student of Dennis' who has been working with me to write the proposal. His research is in urban innovation.

• Possibly: Eric Gordon, a professor at the Engagement Lab at Emerson College, who works on civic participation.

• Other personnel attached to the project include Elizabeth Christoforetti (Harvard, MIT urban planner) and Nissia Sabri (startup, hardware specialist).

This would be a great step toward my ultimate goal: using network theory for studying complex systems like biological systems and designing systems to deal with climate change.

Comment Source:23 November 2016: Lots of progress this week: 1) Jason [uploaded his thesis to the arXiv](https://arxiv.org/abs/1611.07591)! That's great, because it establishes priority - now he can prove he did this stuff before anyone else, even if it takes a while for him to publish a paper. 2) Blake gave a talk about open chemical reaction networks at the Santa Fe Institute workshop on [Statistical Mechanics, Information Processing and Biology](http://www.santafe.edu/gevent/detail/science/2452/): * [Compositional frameworks for open systems](https://johncarlosbaez.wordpress.com/2016/11/27/compositional-frameworks-for-open-systems/). It was a real hit. [Susanne Still](http://www2.hawaii.edu/%7Esstill/) said it was "great" - she works on Markov processes and the information bottleneck method for predictive inference. [Jim Crutchfield](http://csc.ucdavis.edu/%7Echaos/) liked it so much he invited Blake to give a talk up at U. C. Davis! He's a real bigshot: the Wikipedia article on him says > Over the last three decades Prof. Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas. If Blake can strike up a relationship with Crutchfield and maybe work on a project, that'll be excellent. 3) [Joshua Tan](http://www.joshuatan.com/research/), a grad student at Oxford (and friend of Brendan), invited me to join a bunch of people in writing a grant proposal. It's for an NSF grant called "[Smart & Connected Communities](https://www.nsf.gov/pubs/2016/nsf16610/nsf16610.htm#pgm_desc_txt)", and part of the plan would be to model cities as composable, open systems using category theory. Here are the other people involved in writing the proposal: * Dennis Frenchman is a professor at MIT and an expert in building digital tools for cities. He is the likely PI. * Sokwoo Rhee is a director at NIST managing 100+ smart cities projects and will be collaborating directly with us, but he is a silent partner due to federal rules. * Stephen Walter is a program director at the City of Boston, Mayor's Office of New Urban Mechanics. * Matthew Claudel is a student of Dennis' who has been working with me to write the proposal. His research is in urban innovation. * Possibly: Eric Gordon, a professor at the Engagement Lab at Emerson College, who works on civic participation. * Other personnel attached to the project include Elizabeth Christoforetti (Harvard, MIT urban planner) and Nissia Sabri (startup, hardware specialist). This would be a great step toward my ultimate goal: using network theory for studying complex systems like biological systems and designing systems to deal with climate change.
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3.
edited December 2016

28 November 2016:

1) Daniel Cicala passed his oral exam today! He spoke about this paper that he put on the arXiv last week:

Abstract. We introduce the notion of a span of cospans and define, for them, horizontal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.

2) Tobias Fritz is visiting us! He'll be speaking in the network theory seminar tomorrow and also joining our group meeting on Wednesday at 11 am. Here's his talk:

Abstract. The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some hypothetical Bayesian network structure. In the presence of hidden nodes (unobserved variables), this is a challenging problem for which no exact methods are known. The inflation technique of http://arxiv.org/abs/1609.00672 provides a new practical tool for approaching this problem. It has the potential to be generalized to other kinds of networks, in particular those that live in semicartesian monoidal categories.

3) My former grad student Chris Rogers will be giving a special seminar on symplectic stuff on Thursday 3:40-5:00, either in the Undergraduate Study Room or in some better room like room 284 or 268 - it's not exactly clear, but I'll try to inform you when I find out.

It will be very good for Brandon and Adam to attend this, since they're doing symplectic stuff. However, Chris will blow them out of the water with his erudition.

• From Hamiltonian mechanics to homotopy Lie theory

Abstract. In Hamiltonian mechanics, physicists model the phase space of a physical system using symplectic geometry, and they use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces a new Lie algebra called a "central extension''. This central extension plays a crucial role, especially in quantum mechanics. The famous Heisenberg algebra, for example, arises precisely in this way.

In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of "homotopy Lie algebras''. A homotopy Lie algebra is a topologist's version of a Lie algebra: a chain complex equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation theory. The homotopy Lie algebras produced from our construction turn out to have interesting relationships with the theory of loop groups and what are called "string structures'' in algebraic topology

Comment Source:28 November 2016: 1) Daniel Cicala passed his oral exam today! He spoke about this paper that he put on the arXiv last week: * [Spans of cospans](https://arxiv.org/abs/1611.07886). > **Abstract.** We introduce the notion of a span of cospans and define, for them, horizontal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting. 2) Tobias Fritz is visiting us! He'll be speaking in the network theory seminar tomorrow and also joining our group meeting on Wednesday at 11 am. Here's his talk: * [Inferring hidden network structure: the case of causal inference](https://simons.berkeley.edu/talks/tobias-fritz-12-06-2016). > **Abstract.** The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some hypothetical Bayesian network structure. In the presence of hidden nodes (unobserved variables), this is a challenging problem for which no exact methods are known. The inflation technique of [http://arxiv.org/abs/1609.00672](http://arxiv.org/abs/1609.00672) provides a new practical tool for approaching this problem. It has the potential to be generalized to other kinds of networks, in particular those that live in semicartesian monoidal categories. 3) My former grad student Chris Rogers will be giving a special seminar on symplectic stuff on Thursday 3:40-5:00, either in the Undergraduate Study Room or in some better room like room 284 or 268 - it's not exactly clear, but I'll try to inform you when I find out. It will be very good for Brandon and Adam to attend this, since they're doing symplectic stuff. However, Chris will blow them out of the water with his erudition. * From Hamiltonian mechanics to homotopy Lie theory > **Abstract.** In Hamiltonian mechanics, physicists model the phase space of a physical system using symplectic geometry, and they use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces a new Lie algebra called a "central extension''. This central extension plays a crucial role, especially in quantum mechanics. The famous Heisenberg algebra, for example, arises precisely in this way. > In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of "homotopy Lie algebras''. A homotopy Lie algebra is a topologist's version of a Lie algebra: a chain complex equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation theory. The homotopy Lie algebras produced from our construction turn out to have interesting relationships with the theory of loop groups and what are called "string structures'' in algebraic topology
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4.

10 December 2016:

Some very good news this time:

1) Brendan Fong has accepted a postdoc at MIT working with David Spivak. Having seen them discuss math together, I think we can expect great things!

2) I'm 99% sure that Daniel Cicala has been accepted to the Kan Extension Seminar, a high-powered online course on category theory. This time it'll be about functorial semantics - you can see the papers they'll discuss by clicking the link.

3) I gave a talk on Compositionality in network theory at this week's workshop on Compositionality at the Simons Institute for the Theory of Computing. You can see a video by clicking the link. I explained Brendan's theory of decorated cospans, illustrating it with a paper that Blake and I are writing about Petri nets.

4) Brendan gave a talk on Modelling interconnected systems with decorated corelations at the same workshop. This goes further, introducing decorated corelations, which are a generalization of decorated cospans. Again you can see a video by clicking the link.

These talks seem to have gone over well, along with other closely connected talks by David Spivak, Ross Duncan, Pawel Sobocinski and others. I was invited by Michael Mislove, who edits a column on semantics at the journal Logic, Semantics and Theory of Programming, to contribute a column. The whole lot of us were invited to participate more in various conferences on logic and computer science, since what we're doing seems to fit into that heading.

5) Blake's work on Markov processes was cited in at least two talks, and Prakash Panagaden gave me a draft of his paper on a bicategory of Markov processes, which I append here - Blake, Kenny and Daniel should read it!

I think we can and should do better, but we'll have to avoid stepping on Prakash's toes. For one thing, we can build a symmetric monoidal bicategory. For another thing, they are doing discrete-time Markov processes, with 2-morphisms being maps called 'simulations'. We can do something else. Daniel's work on perfect measure spaces should come into this, as well as what Kenny has been doing on bicategories with coarse-grainings as 2-morphisms.

6) Some negative news: the grant proposal I mentioned recently, engineered by Joshua Tan, has fallen through. I'm not too upset.

Comment Source:10 December 2016: Some very good news this time: 1) Brendan Fong has accepted a postdoc at MIT working with David Spivak. Having seen them discuss math together, I think we can expect great things! 2) I'm 99% sure that Daniel Cicala has been accepted to the [Kan Extension Seminar](https://golem.ph.utexas.edu/category/2016/10/the_kan_extension_seminar_retu.html), a high-powered online course on category theory. This time it'll be about functorial semantics - you can see the papers they'll discuss by clicking the link. 3) I gave a talk on [Compositionality in network theory](https://johncarlosbaez.wordpress.com/2016/11/29/compositionality-in-network-theory/) at this week's workshop on Compositionality at the Simons Institute for the Theory of Computing. You can see a video by clicking the link. I explained Brendan's theory of decorated cospans, illustrating it with a paper that Blake and I are writing about Petri nets. 4) Brendan gave a talk on [Modelling interconnected systems with decorated corelations](https://johncarlosbaez.wordpress.com/2016/12/09/modelling-interconnected-systems-with-decorated-corelations/) at the same workshop. This goes further, introducing decorated corelations, which are a generalization of decorated cospans. Again you can see a video by clicking the link. These talks seem to have gone over well, along with other closely connected talks by David Spivak, Ross Duncan, Pawel Sobocinski and others. I was invited by Michael Mislove, who edits a column on semantics at the journal Logic, Semantics and Theory of Programming, to contribute a column. The whole lot of us were invited to participate more in various conferences on logic and computer science, since what we're doing seems to fit into that heading. 5) Blake's work on Markov processes was cited in at least two talks, and Prakash Panagaden gave me a draft of his paper on a bicategory of Markov processes, which I append here - Blake, Kenny and Daniel should read it! I think we can and should do better, but we'll have to avoid stepping on Prakash's toes. For one thing, we can build a symmetric monoidal bicategory. For another thing, they are doing discrete-time Markov processes, with 2-morphisms being maps called 'simulations'. We can do something else. Daniel's work on perfect measure spaces should come into this, as well as what Kenny has been doing on bicategories with coarse-grainings as 2-morphisms. 6) Some negative news: the grant proposal I mentioned recently, engineered by Joshua Tan, has fallen through. I'm not too upset.
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5.
edited December 2016

Finally catching up to today, 13 December 2016:

1) Kenny Courser's paper A bicategory of decorated cospans was accepted by Theory and Applications of Categories!

The referee wants him to say more about examples. That makes a lot of sense. I'd also like to deal with this issue: in our favorite examples, the 2-morphisms in Kenny's bicategory are a bit too restrictive.

For example, in the bicategory of cospans of finite sets where the apex is decorated by a graph, the 2-morphism do not allow arbitrary graph morphisms, only those that are "cocartesian lifts" of maps between finite sets.

2) On December 14th, Brendan is giving a talk called "All hypergraph categories are decorated corelation categories" at Macquarie University in Australia.

Brendan: make sure to say hi to Ross Street and my old friend James Dolan!

3) At Berkeley, it became clear that the stuff we do fits into "theoretical computer science", which is a very broad subject by now.

All of us were invited to submit papers to CALCO 2017, a conference on algebra and coalgebra in computer science. Daniel reminded me of this, saying:

Actually, a few of the gang could probably submit, since their interests include:

• String Diagrams and Network Theory

- Combinatorial approaches

- Theory of PROPs and operads

- Rewriting problems and higher-dimensional approaches

- Automated reasoning with string diagrams

- Applications of string diagrams

- Connections with Control Theory, Engineering and Concurrency


So, think of submitting papers here! Daniel has a plan to do this.

Comment Source:Finally catching up to today, 13 December 2016: 1) Kenny Courser's paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100) was accepted by _Theory and Applications of Categories_! The referee wants him to say more about examples. That makes a lot of sense. I'd also like to deal with this issue: in our favorite examples, the 2-morphisms in Kenny's bicategory are a bit too restrictive. For example, in the bicategory of cospans of finite sets where the apex is decorated by a graph, the 2-morphism do not allow arbitrary graph morphisms, only those that are "cocartesian lifts" of maps between finite sets. 2) On December 14th, Brendan is giving a talk called "All hypergraph categories are decorated corelation categories" at Macquarie University in Australia. Brendan: make sure to say hi to Ross Street and my old friend James Dolan! 3) At Berkeley, it became clear that the stuff we do fits into "theoretical computer science", which is a very broad subject by now. All of us were invited to submit papers to [CALCO 2017](http://coalg.org/calco17/index.html), a conference on algebra and coalgebra in computer science. Daniel reminded me of this, saying: > Actually, a few of the gang could probably submit, since their interests include: > * String Diagrams and Network Theory > - Combinatorial approaches > - Theory of PROPs and operads > - Rewriting problems and higher-dimensional approaches > - Automated reasoning with string diagrams > - Applications of string diagrams > - Connections with Control Theory, Engineering and Concurrency So, think of submitting papers here! Daniel has a plan to do this.
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6.

For some reason dates aren't showing up in Azimuth Forum entries, so it's good I included the dates in the entries here! I have some new entries...

Comment Source:For some reason dates aren't showing up in Azimuth Forum entries, so it's good I included the dates in the entries here! I have some new entries... 
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7.

22 December 2016:

Here are two things that happened this week:

1) Brandon and Brendan's paper Corelations are the prop for extraspecial commutative Frobenius monoids has been accepted for publication by Theory and Applications of Categories subject to making some small corrections.

2) It doesn't really count as mathematics, but I've started the Azimuth Backup Project to help back up climate data before Trump becomes president - because almost all his big hires are people who deny the importance of global warming.

Other teams are doing this too, and you can get the basic idea in this article of mine:

You can see our team's progress here:

We've got a great team, including a guy who used to drive a Mars rover for NASA, and so far we've backed up about a terabyte of data! In a couple of days I'll start a Kickstarter campaign to raise funds to store the data. We'll try to store it at least until larger institutions accept this responsibility.

Comment Source:22 December 2016: Here are two things that happened this week: 1) Brandon and Brendan's paper [Corelations are the prop for extraspecial commutative Frobenius monoids](https://arxiv.org/abs/1601.02307) has been accepted for publication by _Theory and Applications of Categories_ subject to making some small corrections. 2) It doesn't really count as mathematics, but I've started the Azimuth Backup Project to help back up climate data before Trump becomes president - because almost all his big hires are people who deny the importance of global warming. Other teams are doing this too, and you can get the basic idea in this article of mine: * [Saving Climate Data (Part 1)](https://johncarlosbaez.wordpress.com/2016/12/13/saving-climate-data/). You can see our team's progress here: * [Azimuth Backup Project (Part 1)](https://johncarlosbaez.wordpress.com/2016/12/16/azimuth-backup-project/). * [Azimuth Backup Project (Part 2)](https://johncarlosbaez.wordpress.com/2016/12/20/azimuth-backup-project-part-2/). We've got a great team, including a guy who used to drive a Mars rover for NASA, and so far we've backed up about a terabyte of data! In a couple of days I'll start a Kickstarter campaign to raise funds to store the data. We'll try to store it at least until larger institutions accept this responsibility.
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8.
edited January 2017

13 January 2017:

We had a hugely productive meeting on Wednesday and I'm really excited about the new ideas:

• The connection between Brandon Coya's work on bond graphs and Ross Street's work on weak bimonoids, noticed by Brendan, is really fascinating - it implies that there's a "quantum groupoid" associated to electrical circuits, and it implies that our conjectured list of axioms characterizing the category of bond graphs was missing some highly nonobvious relations.

• Daniel Cicala is revisiting Franciscus Rebro's work on the bicategory of cobordisms and will prove it's a symmetric monoidal bicategory.

• Kenny Courser had the smart idea of revisiting Jeffrey Morton and Jamie Vicary's work on Khovanov's categorified Heisenberg algebra and making it rigorous using our new ability to get ahold of symmetric monoidal bicategories, and I realized we can actually do this.

• Adam Yassine seems to have proved that there's a bicategory of symplectic manifolds and cospans whose legs are Poisson fibrations — good for the study of open systems in classical mechanics.

And that's not all! In our Metron project,

• Blake Pollard and John Foley are developing a new framework for search and rescue operations (and many other distributed optimization problems).

• Joseph Moeller created a new algebraic structure generalizing the "operad for communication networks", and I think we can prove this new structure has an elegant category-theoretic description.

It's all great stuff. But these weekly reports are supposed to be about things that have been completed, just to focus your attention on getting things finished. So here are two things like that:

• The math department at U.C. Riverside is hosting the Fall Meeting of the AMS Western Section on Saturday and Sunday, Nov. 4 and 5, 2017. My proposal for a special session on Applied Category Theory has been accepted! I hope you submit proposals for talks — if you're able to come despite the fact that, as usual for such meetings, we have no money. I'll say in a while how you can propose a talk: there will be a webpage where you can do this.

• U. C. Riverside has agreed to be a repository of climate data collected by the Azimuth Backup Project. This means we don't have to figure out how to hold this data permanently. We've raised over $10,000 by now, so we're fine in the short term. Comment Source:13 January 2017: We had a hugely productive meeting on Wednesday and I'm really excited about the new ideas: * The connection between Brandon Coya's work on bond graphs and Ross Street's work on weak bimonoids, noticed by Brendan, is really fascinating - it implies that there's a "quantum groupoid" associated to electrical circuits, and it implies that our conjectured list of axioms characterizing the category of bond graphs was missing some highly nonobvious relations. * Daniel Cicala is revisiting Franciscus Rebro's work on the bicategory of cobordisms and will prove it's a symmetric monoidal bicategory. * Kenny Courser had the smart idea of revisiting Jeffrey Morton and Jamie Vicary's work on Khovanov's categorified Heisenberg algebra and making it rigorous using our new ability to get ahold of symmetric monoidal bicategories, and I realized we can actually do this. * Adam Yassine seems to have proved that there's a bicategory of symplectic manifolds and cospans whose legs are Poisson fibrations &mdash; good for the study of open systems in classical mechanics. And that's not all! In our Metron project, * Blake Pollard and John Foley are developing a new framework for search and rescue operations (and many other distributed optimization problems). * Joseph Moeller created a new algebraic structure generalizing the "operad for communication networks", and I think we can prove this new structure has an elegant category-theoretic description. It's all great stuff. But these weekly reports are supposed to be about things that have been completed, just to focus your attention on getting things finished. So here are two things like that: * The math department at U.C. Riverside is hosting the Fall Meeting of the AMS Western Section on Saturday and Sunday, Nov. 4 and 5, 2017. My proposal for a special session on **Applied Category Theory** has been accepted! I hope you submit proposals for talks &mdash; if you're able to come despite the fact that, as usual for such meetings, we have no money. I'll say in a while how you can propose a talk: there will be a webpage where you can do this. * U. C. Riverside has agreed to be a repository of climate data collected by the Azimuth Backup Project. This means we don't have to figure out how to hold this data permanently. We've raised over$10,000 by now, so we're fine in the short term.
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9.
edited February 2017

21 January 2017:

1) Kenny Courser dealt with the referee's comments on his paper A bicategory of decorated cospans, resubmitted it to Theory and Applications of Categories, and put the new improved version on the arXiv.

2) Daniel Cicala has submitted his paper Spans of cospans to Theory and Applications of Categories. Is that true, Daniel? If not, make it so. Also, it probably makes sense to update the version on the arXiv.

(Kenny and Daniel are now teaming up to write a wonderful paper on bicategories of spans and cospans, including the "maps-of-spans" case already dealt with more generally by Mike, but more importantly the "spans-of-spans" case and "spans-of-cospans" case.)

3) Joseph Moeller has written up the statement of a theorem on generalizations of the operad of communication networks, suitable for inclusion in the paper that Blake is working on for DARPA. This paper is due the day after tomorrow!

4) I've been invited to join another project run by the US Defense Department!

I got an email from Michael A. Smith, a microbiologist who's the director of the Defense Biological Product Assurance Office. These are the people who check whether mysterious white powders showing up in people's mail are actually anthrax... and they also keep track of diseases that suddenly spring up, like Ebola or MIRS. They used to be called the Critical Reagents Program.

Smith is interested in whether modifications of the network of hospitals and laboratories in Africa could help them better keep track of new epidemics that show up on this continent. He read my book with Jacob Biamonte on Petri nets, and he thinks I might able to help. I'll start by listening in to a phone conversation he's having with Gary Kobinger, who helped develop a vaccine for Ebola, and who

developed and pioneered use of small mobile laboratories — a lab in a suitcase, essentially — that have changed the way testing is done during Ebola outbreaks.

I'm already way too busy, but this sounds interesting! Michael Smith said that his higher-ups mainly evaluate him on whether he spends money fast enough. I could certainly help with that... but it would also be cool if network ideas could help combat disease.

5) I arranged to visit the Institute for Scientific Interchange in Turin, Italy during the last week of June. This center is very interested in the mathematical foundations of complex systems, including networks.

I ran a workshop there in May 2015 and the director, Mario Rasetti, said I was welcome to visit any time. I decided to see if this offer was real: I'm visiting the category theorist Marco Grandis in nearby Genoa in mid-June, so I emailed Rasetti and asked if I could visit after that.

It turns out he's been reading our papers with interest! He'd actually like me to visit for a sabbatical or maybe even get some long-term affiliation with this center. I'll find out more in June.

Comment Source:21 January 2017: 1) Kenny Courser dealt with the referee's comments on his paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100), resubmitted it to _Theory and Applications of Categories_, and put the new improved version on the arXiv. 2) Daniel Cicala has submitted his paper [Spans of cospans](https://arxiv.org/abs/1611.07886) to _Theory and Applications of Categories_. Is that true, Daniel? If not, make it so. Also, it probably makes sense to update the version on the arXiv. (Kenny and Daniel are now teaming up to write a wonderful paper on bicategories of spans and cospans, including the "maps-of-spans" case already dealt with more generally by Mike, but more importantly the "spans-of-spans" case and "spans-of-cospans" case.) 3) Joseph Moeller has written up the statement of a theorem on generalizations of the operad of communication networks, suitable for inclusion in the paper that Blake is working on for DARPA. This paper is due the day after tomorrow! 4) I've been invited to join another project run by the US Defense Department! I got an email from [Michael A. Smith](https://www.linkedin.com/in/michael-smith-b6468844), a microbiologist who's the director of the [Defense Biological Product Assurance Office](https://globalbiodefense.com/2017/01/02/dod-seeks/). These are the people who check whether mysterious white powders showing up in people's mail are actually anthrax... and they also keep track of diseases that suddenly spring up, like Ebola or MIRS. They used to be called the Critical Reagents Program. Smith is interested in whether modifications of the network of hospitals and laboratories in Africa could help them better keep track of new epidemics that show up on this continent. He read [my book with Jacob Biamonte on Petri nets](http://math.ucr.edu/home/baez/stoch_stable.pdf), and he thinks I might able to help. I'll start by listening in to a phone conversation he's having with Gary Kobinger, who helped develop a vaccine for Ebola, and who > developed and pioneered use of small mobile laboratories — a lab in a suitcase, essentially — that have changed the way testing is done during Ebola outbreaks. I'm already way too busy, but this sounds interesting! Michael Smith said that his higher-ups mainly evaluate him on whether he spends money fast enough. I could certainly help with that... but it would also be cool if network ideas could help combat disease. 5) I arranged to visit the [Institute for Scientific Interchange](https://en.wikipedia.org/wiki/Institute_for_Scientific_Interchange) in Turin, Italy during the last week of June. This center is very interested in the [mathematical foundations of complex systems](http://www.isi.it/en/research/mathematics-foundation-of-complex-systems), including networks. I ran a workshop there in May 2015 and the director, Mario Rasetti, said I was welcome to visit any time. I decided to see if this offer was real: I'm visiting the category theorist Marco Grandis in nearby Genoa in mid-June, so I emailed Rasetti and asked if I could visit after that. It turns out he's been reading our papers with interest! He'd actually like me to visit for a sabbatical or maybe even get some long-term affiliation with this center. I'll find out more in June. 
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10.

14 February 2017:

1) I gave a talk called Biology as Information Dynamics at a workshop called Biological Complexity: Can it be Quantified? at the Beyond Center at Arizona State University. The last result in my talk is new: it's an improved version of Fisher's fundamental theorem of natural selection, which is more general than the usual version, and phrased in terms of information geometry.

I talked about gene regulatory networks with Sara Walker, who works there. A gene regulatory network is a very simple thing: a finite set of 'genes' with two kinds of arrows from some genes to some other genes: 'express' and 'repress'. It would be nice to study 'open' gene regulatory networks using the decorated cospan formalism. There seems to be a fairly simple semantics for these networks where time goes in integer steps and at each time each gene is either on (expressed) or off (repressed).

I also had some conversations with Jim Crutchfield, who wants Blake Pollard to visit him up at U.C. Davis.

2) Tom Mifflin of Metron said that DARPA likes our work so much that they're starting two new programs related to these ideas. One will be at the Strategic Technology Office, and it will be big. Tom suggested I could get more money for grad students. The other is a small "seedling" program, just $500,000, for Metron to help develop autonomous vehicles for the Navy. 3) Blake noticed that Jason Erbele's paper Categories in control was cited by Dominique Luzeaux in a paper on the category-theoretic foundations of systems engineering. The main interesting thing about this paper is that Lzueaux is the Deputy Director of the Joint Directorate for Networks, Infrastructure and Information Systems, part of the French defense department. It may seem weird that the military is interested in category theory and operads, but it makes sense. They have a lot of money, they're willing to experiment to stay ahead of other countries, and they have huge organizational/strategic problems that involve complex networked systems. Thus, they've traditionally been at the forefront of "systems of systems engineering". Comment Source:14 February 2017: 1) I gave a talk called [Biology as Information Dynamics](http://math.ucr.edu/home/baez/bio_asu/) at a workshop called [Biological Complexity: Can it be Quantified?](https://beyond.asu.edu/workshop/biological-complexity-can-it-be-quantified) at the [Beyond Center](https://beyond.asu.edu/) at Arizona State University. The last result in my talk is new: it's an improved version of Fisher's fundamental theorem of natural selection, which is more general than the usual version, and phrased in terms of information geometry. I talked about gene regulatory networks with Sara Walker, who works there. A gene regulatory network is a very simple thing: a finite set of 'genes' with two kinds of arrows from some genes to some other genes: 'express' and 'repress'. It would be nice to study 'open' gene regulatory networks using the decorated cospan formalism. There seems to be a fairly simple semantics for these networks where time goes in integer steps and at each time each gene is either on (expressed) or off (repressed). I also had some conversations with Jim Crutchfield, who wants Blake Pollard to visit him up at U.C. Davis. 2) Tom Mifflin of Metron said that DARPA likes our work so much that they're starting two new programs related to these ideas. One will be at the Strategic Technology Office, and it will be big. Tom suggested I could get more money for grad students. The other is a small "seedling" program, just$500,000, for Metron to help develop autonomous vehicles for the Navy. 3) Blake noticed that Jason Erbele's paper [Categories in control](http://www.tac.mta.ca/tac/volumes/30/24/30-24abs.html) was cited by [Dominique Luzeaux](https://scholar.google.com/citations?user=8G2BKfEAAAAJ&hl=en) in a paper on the category-theoretic foundations of systems engineering. The main interesting thing about this paper is that Lzueaux is the Deputy Director of the Joint Directorate for Networks, Infrastructure and Information Systems, part of the French defense department. It may seem weird that the military is interested in category theory and operads, but it makes sense. They have a lot of money, they're willing to experiment to stay ahead of other countries, and they have huge organizational/strategic problems that involve complex networked systems. Thus, they've traditionally been at the forefront of "[systems of systems engineering](https://en.wikipedia.org/wiki/System_of_systems_engineering)".
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11.

27 February 2017:

Four pieces of news:

1) Brendan and Brandon made all the required changes to their paper "Corelations are the prop for extraspecial commutative Frobenius monoids", so this paper has now been published by Theory and Applications of Categories. You can see it here:

http://www.tac.mta.ca/tac/index.html#vol32

It's in the same issue as two papers by Voevodsky, inventor of homotopy type theory, along with papers by Clemens Berger, Michael Batanin and Jiri Adamek — all famous category theorists! So, congratulations!

2) Blake applied to give a talk on his work at the American Physical Society March Meeting in New Orleans on March 13-17. They asked him to present a poster. He applied to the Grad Student Association for funding to go on this trip, but it looks like he'll have to foot some of the bill himself. (Sadly, while the Metron project he and I are working on pays him enough money so he doesn't have to be a teaching assistant, it doesn't give us any travel money, and the Grad Student Association gives rather limited funds.)

3) Blake has also gotten an invitation from the famous information theorist Jim Crutchfield to speak about our work at U. C. Davis. I mentioned this earlier, but now it's really come true! I forget when Blake is going up there.

4) Not exactly academic, but: the Azimuth Climate Data Backup Project's Kickstarter campaign has succeeded! Our original goal was $5000. We got$20,427 of donations, and after Kickstarter took its cut we received $18,590.96. I wrote thank-you notes to all 627 contributors, and compiled a list thanking everyone who was okay with having their name made public. We are close to reaching our goal of backing up 40 terabytes of data. The next step will be to put copies in several secure locations. 3 places have volunteered to hold copies: the Princeton math department, a UNESCO center in France, and the U. C. Riverside department of computing and communications. It will take a while to accomplish these transfers. It looks like I'll be interviewed about this tomorrow by Amy Harmon of the New York Times. But reporters are often busy and distracted, so I'll believe it when it happens. Comment Source:27 February 2017: Four pieces of news: 1) Brendan and Brandon made all the required changes to their paper "Corelations are the prop for extraspecial commutative Frobenius monoids", so this paper has now been published by Theory and Applications of Categories. You can see it here: [http://www.tac.mta.ca/tac/index.html#vol32](http://www.tac.mta.ca/tac/index.html#vol32) It's in the same issue as two papers by Voevodsky, inventor of homotopy type theory, along with papers by Clemens Berger, Michael Batanin and Jiri Adamek &mdash; all famous category theorists! So, congratulations! 2) Blake applied to give a talk on his work at the American Physical Society March Meeting in New Orleans on March 13-17. They asked him to present a poster. He applied to the Grad Student Association for funding to go on this trip, but it looks like he'll have to foot some of the bill himself. (Sadly, while the Metron project he and I are working on pays him enough money so he doesn't have to be a teaching assistant, it doesn't give us any travel money, and the Grad Student Association gives rather limited funds.) 3) Blake has also gotten an invitation from the famous information theorist [Jim Crutchfield](http://csc.ucdavis.edu/~chaos/) to speak about our work at U. C. Davis. I mentioned this earlier, but now it's really come true! I forget when Blake is going up there. 4) Not exactly academic, but: the Azimuth Climate Data Backup Project's [Kickstarter campaign](https://www.kickstarter.com/projects/592742410/azimuth-climate-data-backup-project) has succeeded! Our original goal was$5000. We got $20,427 of donations, and after Kickstarter took its cut we received$18,590.96. I wrote thank-you notes to all 627 contributors, and compiled [a list thanking everyone who was okay with having their name made public](https://johncarlosbaez.wordpress.com/2017/02/18/azimuth-backup-project-part-4/). We are close to reaching our goal of backing up 40 terabytes of data. The next step will be to put copies in several secure locations. 3 places have volunteered to hold copies: the Princeton math department, a UNESCO center in France, and the U. C. Riverside department of computing and communications. It will take a while to accomplish these transfers. It looks like I'll be interviewed about this tomorrow by [Amy Harmon](https://www.nytimes.com/by/amy-harmon) of the _New York Times_. But reporters are often busy and distracted, so I'll believe it when it happens.
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12.

John, Congratulations on such a great response at Kickstarter and all the other good news!

Comment Source:John, Congratulations on such a great response at Kickstarter and all the other good news!
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13.

Thanks!

Comment Source:Thanks!
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14.

8 March 2017:

1) Jason Erbele and Daniel Cicala have applied to attend the AMS workshop on homotopy type theory in the “breathtaking mountain setting” of Snowbird Resort in Utah, June 4th to 10th. It's being run by my friends Dan Christensen, Mike Shulman and Emily Riehl along with Chris Kapulkin and Dan Licata (who I don't know so well).

2) Next week Blake Pollard is going to the American Physical Society meeting in New Orleans and presenting a poster on his work on open systems.

3) My book with Jacob Biamonte, Quantum Techniques for Stochastic Processes, seems to have been accepted for publication by World Scientific Press.

4) Less significant, but fun: the Australian science fiction writer Greg Egan and I proved an interesting equation expressing pi in terms of the golden ratio:

Comment Source:8 March 2017: 1) Jason Erbele and Daniel Cicala have applied to attend the AMS workshop on homotopy type theory in the “breathtaking mountain setting” of Snowbird Resort in Utah, June 4th to 10th. It's being run by my friends Dan Christensen, Mike Shulman and Emily Riehl along with Chris Kapulkin and Dan Licata (who I don't know so well). 2) Next week Blake Pollard is going to the American Physical Society meeting in New Orleans and presenting a poster on his work on open systems. 3) My book with Jacob Biamonte, _[Quantum Techniques for Stochastic Processes](http://math.ucr.edu/home/baez/stoch_stable.pdf)_, seems to have been accepted for publication by World Scientific Press. 4) Less significant, but fun: the Australian science fiction writer Greg Egan and I proved an interesting equation expressing pi in terms of the golden ratio: * [Pi and the golden ratio](https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/), _Azimuth_, 7 March 2017. 
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15.
edited March 2017

27 March 2017:

1) On Friday, Blake and John Foley and I finished off two essays for our Complex Adaptive Composition and Design Environment project.

The first is called "Compositional Tasking: an operad-based approach to adaptive behaviors for distributed systems-of-systems and planning under uncertainty." This is a general overview of our plan to use operads to design and "task" (boss around) networks while moving up and down levels of abstraction as desired. I find this really exciting! With luck the folks at Metron will create some software to illustrate these ideas

The second, called "Compositional Tasking", has more of the mathematical details. Joseph played a key role in this by checking that the Grothendieck construction can create the operads we need. I hope we continue to improve this essay and publish it in one or more papers, which I hope to write this summer.

2) Prakash Panangaden, an expert in categories and computer science at McGill University in Montreal, has come out with a paper which takes the category-theoretic characterization of relative entropy that Tobias Fritz and I found and extends it from finite sets to more general measurable spaces (sets with a sigma-algebra of subsets):

Abstract. The inspiration for the present work comes from two recent developments. The first is the beginning of a categorical understanding of Bayesian inversion and learning, the second is a categorical reconstruction of relative entropy. The present paper provides a categorical treatment of entropy in the spirit of Baez and Fritz in the setting of Polish spaces, thus setting the stage to explore the role of entropy in learning.

"Polish spaces" are a nice class of measurable spaces, loved by the analysts who lived in Poland before the Nazis invaded.

3) Prakash has also written a paper about a bicategory where the morphisms are open Markov processes. This is based on a paper that Brendan, Blake and I wrote:

Abstract. We construct bicategories of Markov processes where the objects are input and output sets, the morphisms (one-cells) are Markov processes and the two-cells are simulations. This builds on the work of Baez, Fong and Pollard, who showed that a certain kinds of finite-space continuous-time Markov chain (CTMC) satisfying a detailed-balance condition can be viewed as morphisms in a category. This view allows a compositional description of their CTMCs. Our contribution is to develop a notion of simulation between processes and construct a bicategory where the two-cells are simulation morphisms. Our version is for processes that are essentially probabilistic transition systems with discrete time steps and which do not satisfy a detailed balance condition. We have also extended the theory to continuous space processes.

In short, Prakash is moving in and starting to offer competition in our field of work. A lot of people in computer science respect his work, so they will start to read our stuff and explore similar ideas. This means we can't laze around when it comes to publishing ideas we have... but it's basically good, because it means more people will be inclined to hire my grad students!

Comment Source:27 March 2017: 1) On Friday, Blake and John Foley and I finished off two essays for our [Complex Adaptive Composition and Design Environment](https://johncarlosbaez.wordpress.com/2016/10/02/complex-adaptive-system-design-part-1/) project. The first is called "Compositional Tasking: an operad-based approach to adaptive behaviors for distributed systems-of-systems and planning under uncertainty." This is a general overview of our plan to use operads to design and "task" (boss around) networks while moving up and down levels of abstraction as desired. I find this really exciting! With luck the folks at Metron will create some software to illustrate these ideas The second, called "Compositional Tasking", has more of the mathematical details. Joseph played a key role in this by checking that the Grothendieck construction can create the operads we need. I hope we continue to improve this essay and publish it in one or more papers, which I hope to write this summer. 2) Prakash Panangaden, an expert in categories and computer science at McGill University in Montreal, has come out with a paper which takes the category-theoretic characterization of relative entropy that Tobias Fritz and I found and extends it from finite sets to more general measurable spaces (sets with a sigma-algebra of subsets): > **Abstract**. The inspiration for the present work comes from two recent developments. The first is the beginning of a categorical understanding of Bayesian inversion and learning, the second is a categorical reconstruction of relative entropy. The present paper provides a categorical treatment of entropy in the spirit of Baez and Fritz in the setting of Polish spaces, thus setting the stage to explore the role of entropy in learning. "Polish spaces" are a nice class of measurable spaces, loved by the analysts who lived in Poland before the Nazis invaded. 3) Prakash has also written a paper about a bicategory where the morphisms are open Markov processes. This is based on a paper that Brendan, Blake and I wrote: > **Abstract**. We construct bicategories of Markov processes where the objects are input and output sets, the morphisms (one-cells) are Markov processes and the two-cells are simulations. This builds on the work of Baez, Fong and Pollard, who showed that a certain kinds of finite-space continuous-time Markov chain (CTMC) satisfying a detailed-balance condition can be viewed as morphisms in a category. This view allows a compositional description of their CTMCs. Our contribution is to develop a notion of simulation between processes and construct a bicategory where the two-cells are simulation morphisms. Our version is for processes that are essentially probabilistic transition systems with discrete time steps and which do not satisfy a detailed balance condition. We have also extended the theory to continuous space processes. In short, Prakash is moving in and starting to offer competition in our field of work. A lot of people in computer science respect his work, so they will start to read our stuff and explore similar ideas. This means we can't laze around when it comes to publishing ideas we have... but it's basically good, because it means more people will be inclined to hire my grad students! 
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16.

John, congratulations to you and your colleagues/students for inspiring others with your work!

Comment Source:John, congratulations to you and your colleagues/students for inspiring others with your work!
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17.

Thanks!

Comment Source:Thanks!
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18.
edited May 2017

5 April 2017:

1) Gheorghe Craciun visited UCR and gave a talk today on his proof of the Global Attractor Conjecture, which until recently was one of the biggest open questions in mathematical chemistry.

2) Blake Pollard and I put our paper A compositional framework for reaction networks on the arXiv today. The conclusions summarize a lot of the work our group has done so far, and fits it into a big commutative diagram. Craciun and I have already used these ideas to construct a large new class of reaction networks with nice stability properties.

Abstract. Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned "rate constants", any reaction network gives rise to a nonlinear dynamical system called its "rate equation". Here we generalize these ideas to "open" reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks are morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding "open dynamical system". This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a "black-boxing" functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes.

3) My former student Brendan Fong, who developed the "decorated cospan" and "decorated corelation" approach to network theory in this thesis, put related two papers onto the arXiv: Decorated corelations and A universal construction for (co)relations. I need to blog about these!

4) Brendan also gave an expository talk about "The mathematics of system composition" at BAE Systems, a British defense company.

5) My former student Mike Stay wrote two papers with Greg Meredith on the use of categories in computer science: Name-free combinators for concurrency and Representing operational semantics with enriched Lawvere theories.

6) My student Daniel Cicala got invited to the American Mathematical Society conference on Homotopy Type Theory that will take place in Snowbird, Utah on June 4-10.

Comment Source:5 April 2017: 1) Gheorghe Craciun visited UCR and gave a talk today on his proof of the [Global Attractor Conjecture](https://sinews.siam.org/Details-Page/discussing-the-proof-of-the-global-attractor-conjecture-1), which until recently was one of the biggest open questions in mathematical chemistry. 2) Blake Pollard and I put our paper [A compositional framework for reaction networks](https://arxiv.org/abs/1704.02051) on the arXiv today. The conclusions summarize a lot of the work our group has done so far, and fits it into a big commutative diagram. Craciun and I have already used these ideas to construct a large new class of reaction networks with nice stability properties. > **Abstract.** Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned "rate constants", any reaction network gives rise to a nonlinear dynamical system called its "rate equation". Here we generalize these ideas to "open" reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks are morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding "open dynamical system". This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a "black-boxing" functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes. 3) My former student Brendan Fong, who developed the "decorated cospan" and "decorated corelation" approach to network theory in this thesis, put related two papers onto the arXiv: [Decorated corelations](https://arxiv.org/abs/1703.09888) and [A universal construction for (co)relations](https://arxiv.org/abs/1703.08247). I need to blog about these! 4) Brendan also gave an expository talk about "The mathematics of system composition" at [BAE Systems](https://en.wikipedia.org/wiki/BAE_Systems), a British defense company. 5) My former student Mike Stay wrote two papers with Greg Meredith on the use of categories in computer science: [Name-free combinators for concurrency](https://arxiv.org/abs/1703.07054) and [Representing operational semantics with enriched Lawvere theories](https://arxiv.org/abs/1704.03080). 6) My student Daniel Cicala got invited to the American Mathematical Society conference on [Homotopy Type Theory](https://homotopytypetheory.org/2016/10/04/hott-mrc/) that will take place in Snowbird, Utah on June 4-10. 
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19.

24 April 2017:

1) This week Blake and I visited Metron Scientific Solutions. I only visited for a few hours, but Blake stayed for several days, including a hackathon where the Metron team tried to solve a search and rescue problem posed by DARPA!

One thing I did during my short stay was try to sell people on the vision of using operads to design systems 'a bit at a time'. Here 'a bit at a time' is meant in two ways. The basic idea is that a 'system' is an element of some algebra $A$ of some operad $O$. Then:

a) We can design small systems and stick them together using the operations of $O$ to get bigger systems, and

b) We can design systems at a high level of abstraction (meaning: not much detail) and then move to a lower level of abstraction (meaning: add extra details) with the help of a map of operads $O' \to O$. We do this by starting with an element of some algebra $A$ of $O$, and then choosing a way to 'lift' it to an element of some algebra $A'$ of $O'$. Here the primed guys are the ones with more detail.

The second point here is subtler than the first and I'm not really trying to explain it here, so don't feel bad if it doesn't make enough sense!

There were a lot of interesting discussions after I left. Blake summarized them for me... but very very briefly, it seems that:

a) we need to think harder about how to model "levels of abstraction", because this phrase means several different things,

b) we need to think more about how our "operads of communication networks" are connected to Spivak's "operad of wiring diagrams",

c) we should continue thinking about Petri nets. The operad of wiring diagrams has an algebra whose elements are open Petri nets. These have a bit of computational power, at least if we equip the Petri nets with suitable bells and whistles. Thus, we can use them to model many real-world gadgets, which may be handy as we move forwards in this project.

Comment Source:24 April 2017: 1) This week Blake and I visited Metron Scientific Solutions. I only visited for a few hours, but Blake stayed for several days, including a hackathon where the Metron team tried to solve a search and rescue problem posed by DARPA! One thing I did during my short stay was try to sell people on the vision of using operads to design systems 'a bit at a time'. Here 'a bit at a time' is meant in two ways. The basic idea is that a 'system' is an element of some algebra $A$ of some operad $O$. Then: a) We can design small systems and stick them together using the operations of $O$ to get bigger systems, and b) We can design systems at a high level of abstraction (meaning: not much detail) and then move to a lower level of abstraction (meaning: add extra details) with the help of a map of operads $O' \to O$. We do this by starting with an element of some algebra $A$ of $O$, and then choosing a way to 'lift' it to an element of some algebra $A'$ of $O'$. Here the primed guys are the ones with more detail. The second point here is subtler than the first and I'm not really trying to explain it here, so don't feel bad if it doesn't make enough sense! There were a lot of interesting discussions after I left. Blake summarized them for me... but very very briefly, it seems that: a) we need to think harder about how to model "levels of abstraction", because this phrase means several different things, b) we need to think more about how our "operads of communication networks" are connected to Spivak's "operad of wiring diagrams", c) we should continue thinking about Petri nets. The operad of wiring diagrams has an algebra whose elements are open Petri nets. These have a bit of computational power, at least if we equip the Petri nets with suitable bells and whistles. Thus, we can use them to model many real-world gadgets, which may be handy as we move forwards in this project. 
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20.

28 April 2017:

1) Kenny Courser passed his oral exam! He gave a talk called A bicategory of decorated cospans, based on his paper with the same title. He survived my questions and also the fact that two members of committee didn't remember to come until he went and grabbed them. (One them was me.)

Congratulations, Kenny!

2) I gave a talk at the Stanford Complexity Group, called Biology as information dynamics. You can see the slides by clicking on the link, and you can also see a video here.

It went a lot better than my similar talk Arizona State University, probably because this audience was more interested in the subject, I covered more ground, and I was more confident.

Marc Harper attended - he came up with some of the math I discussed, and now he's working at Google. So did Vaughn Pratt - the computer scientist who helped come up with Pratt certificates and the Knuth-Morris-Pratt algorithm. He used to be very active on the category theory mailing list, but not much lately.

3) Daniel Cicala put a paper on the arXiv:

Abstract. This paper presents a symmetric monoidal and compact closed bicategory that categorifies the zx-calculus developed by Coecke and Duncan. The 1-cells in this bicategory are certain graph morphisms that correspond to the string diagrams of the zx-calculus, while the 2-cells are rewrite rules.

He is submitting this to CALCO, the Conference on Algebra and Coalgebra in Computer Science. After he put it on the arXiv, Duncan asked him to submit it to QPL, the conference on Quantum Physics and Logic run by Duncan, Coecke and others. (Even I am involved in it, slightly, and Jason Erbele spoke there in 2015.) Luckily he can submit an "extended abstract" to QPL and have the best of both worlds.

Comment Source:28 April 2017: 1) Kenny Courser passed his oral exam! He gave a talk called A bicategory of decorated cospans, based on his paper with the same title. He survived my questions and also the fact that two members of committee didn't remember to come until he went and grabbed them. (One them was me.) Congratulations, Kenny! 2) I gave a talk at the Stanford Complexity Group, called [Biology as information dynamics](http://math.ucr.edu/home/baez/bio_asu/). You can see the slides by clicking on the link, and you can also [see a video here](https://www.youtube.com/watch?v=IKetDJof8pk). It went a lot better than my similar talk Arizona State University, probably because this audience was more interested in the subject, I covered more ground, and I was more confident. Marc Harper attended - he came up with some of the math I discussed, and now he's working at Google. So did Vaughn Pratt - the computer scientist who helped come up with Pratt certificates and the Knuth-Morris-Pratt algorithm. He used to be very active on the category theory mailing list, but not much lately. 3) Daniel Cicala put a paper on the arXiv: * [Categorifying the zx-calculus](https://arxiv.org/abs/1704.07034). > **Abstract.** This paper presents a symmetric monoidal and compact closed bicategory that categorifies the zx-calculus developed by Coecke and Duncan. The 1-cells in this bicategory are certain graph morphisms that correspond to the string diagrams of the zx-calculus, while the 2-cells are rewrite rules. He is submitting this to [CALCO](http://coalg.org/mfps-calco2017/), the Conference on Algebra and Coalgebra in Computer Science. After he put it on the arXiv, Duncan asked him to submit it to [QPL](http://qpl.science.ru.nl/), the conference on Quantum Physics and Logic run by Duncan, Coecke and others. (Even I am involved in it, slightly, and Jason Erbele spoke there in 2015.) Luckily he can submit an "extended abstract" to QPL and have the best of both worlds.
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21.

3 May 2017:

This week's progress is all about the virtues of initiative: finding opportunities and seizing them.

1) Blake found an interesting ad on MathJobs for a research position at the Bay Area Complexity Institute. The institute is so new that it doesn't even exist yet, but the job description looks like it was written specially for Blake:

You must have a working knowledge of graduate-level category theory, measure theory, and probability theory. Bonus points if you are experienced with any of: statistical mechanics, dynamical systems, Markov chains, various theories and models of computation, organismal evolution, or complexity in any formal setting. You must also enjoy research and writing, and have the personality traits of a theory builder rather than a problem solver, but have the skills of both. Perhaps most importantly, you should be excited and passionate about working on a small team to pursue the connection between information-theoretic complexity, thermodynamics and entropy, and the evolution of complexity in cosmology and living systems.

He applied, and they sound interested! He's going to meet the people running this institute on May 11th, right after his talk to Crutchfield's group at U.C. Davis.

Moral: look around for jobs. Apply for jobs that sound cool.

2) Jason had been invited by Tobias Fritz to a conference at the Perimeter Institute, in Canada. That's where I used to hang out when I worked on quantum gravity - and indeed, my former student Derek Wise is giving a talk at this conference about his wonderful work on Hopf algebras and quantum gauge theories!

One problem: Tobias didn't offer him travel money. An invitation without money is a sad thing. But Jason hustled and overcame this problem:

I initially asked Tobias Fritz and his group if I could get any travel funds through the Perimeter Institute. The answer was a fairly definite "probably not." I tried various avenues at Victor Valley College, where I am currently teaching, all of which came back No. So I bought my plane tickets, reserved a hotel and rental car, etc. and informed Lucy Zhang, the person in Tobias' group with whom I have been primarily in contact. Two days later I had a sponsor: Daniel Gottesman. So I cancelled my hotel and rental car reservations, the cost of the plane tickets will be reimbursed, the entry fee is waived, and more. I think the only thing that isn't covered is parking at LAX. There is still some time, so I am looking into alternatives to that.

Daniel Gottesman is a bigshot in quantum computation at the Perimeter Institute.

Moral: it pays to ask for help - you just might get it!

3) As for me, I'm not doing anything useful - just hanging out in Hong Kong, checking out temples. But I got here because I wrote about Guowu Meng's amazing work connecting special relativity to Newton's inverse square force law... and he's here in Hong Kong, so he invited me here!

Moral: blog about cool stuff. If you explain it well, people will pay attention.

Comment Source:3 May 2017: This week's progress is all about the virtues of initiative: finding opportunities and seizing them. 1) Blake found an interesting ad on MathJobs for a [research position at the Bay Area Complexity Institute](https://www.mathjobs.org/jobs/jobs/10174). The institute is so new that it doesn't even exist yet, but the job description looks like it was written specially for Blake: > You must have a working knowledge of graduate-level category theory, measure theory, and probability theory. Bonus points if you are experienced with any of: statistical mechanics, dynamical systems, Markov chains, various theories and models of computation, organismal evolution, or complexity in any formal setting. You must also enjoy research and writing, and have the personality traits of a theory builder rather than a problem solver, but have the skills of both. Perhaps most importantly, you should be excited and passionate about working on a small team to pursue the connection between information-theoretic complexity, thermodynamics and entropy, and the evolution of complexity in cosmology and living systems. He applied, and they sound interested! He's going to meet the people running this institute on May 11th, right after his talk to Crutchfield's group at U.C. Davis. **Moral: look around for jobs. Apply for jobs that sound cool.** 2) Jason had been invited by Tobias Fritz to a conference at the Perimeter Institute, in Canada. That's where I used to hang out when I worked on quantum gravity - and indeed, my former student Derek Wise is giving a talk at this conference about his [wonderful work on Hopf algebras and quantum gauge theories](https://arxiv.org/abs/1512.03966)! One problem: Tobias didn't offer him travel money. An invitation without money is a sad thing. But Jason hustled and overcame this problem: > I initially asked Tobias Fritz and his group if I could get any travel funds through the Perimeter Institute. The answer was a fairly definite "probably not." I tried various avenues at Victor Valley College, where I am currently teaching, all of which came back No. So I bought my plane tickets, reserved a hotel and rental car, etc. and informed Lucy Zhang, the person in Tobias' group with whom I have been primarily in contact. Two days later I had a sponsor: Daniel Gottesman. So I cancelled my hotel and rental car reservations, the cost of the plane tickets will be reimbursed, the entry fee is waived, and more. I think the only thing that isn't covered is parking at LAX. There is still some time, so I am looking into alternatives to that. Daniel Gottesman is a bigshot in quantum computation at the Perimeter Institute. **Moral: it pays to ask for help - you just might get it!** 3) As for me, I'm not doing anything useful - just hanging out in Hong Kong, checking out temples. But I got here because I wrote about [Guowu Meng's amazing work connecting special relativity to Newton's inverse square force law](http://math.ucr.edu/home/baez/gravitational.html)... and he's here in Hong Kong, so he invited me here! **Moral: blog about cool stuff. If you explain it well, people will pay attention.** 
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22.

6 May 2017:

1) You may recall that Daniel Cicala got into the Kan Extension Seminar, which is an advanced online course on category theory run by Emily Riehl, Alexander Campbell and our very own Brendan Fong. Participants read famous papers on category theory, discuss them and blog about them. Daniel recently blogged about the paper he read:

2) Blake got offered a 3-month internship at the Princeton branch of Siemens, a big engineering firm! The advertisement was pretty interesting:

PhD Intern – Next-Generation Engineering with Category Theory and Sheaves

Responsibilities

• Develop concepts for unifying different engineering formalisms that describe different domains and aspects of complex systems using Category Theory.
• Implement the developed concepts in software prototypes to demonstrate novel languages and algorithms to represent and extract knowledge across engineering disciplines.
• Investigate the use of Sheaves for the creation of hybrid engineering models that combine data-driven and physics-based representations.
• Participate in the preparation of scientific publications.

Required Knowledge/Skills, Education, and Experience

• PhD student in Mathematics or Physics.
• Programming skills in Python and/or JavaScript and ability to quickly prototype in these languages.
• Good written and verbal communication skills in English are required, as well as excellent interpersonal skills in multi-cultural environments.
• Team player who can also be independent, prioritize work and thrive in a fast-paced dynamic environment.
• The successful candidate must be able to work with controlled technology in accordance with US Export Control Law.

The guy in charge is named Arquimedes Canedo, and he seems potentially interested in hiring Blake or other applied category theory people (hint hint!) for a longer-term project.

Comment Source:6 May 2017: 1) You may recall that Daniel Cicala got into the [Kan Extension Seminar](https://golem.ph.utexas.edu/category/2016/10/the_kan_extension_seminar_retu.html), which is an advanced online course on category theory run by Emily Riehl, Alexander Campbell and our very own Brendan Fong. Participants read famous papers on category theory, discuss them and blog about them. Daniel recently blogged about the paper he read: * [A discussion on notions of Lawvere theory](https://golem.ph.utexas.edu/category/2017/05/a_discussion_on_notions_of_law.html). 2) Blake got offered a 3-month internship at the Princeton branch of Siemens, a big engineering firm! The [advertisement](https://jobs.siemens-info.com/jobs/207833/PhD+Intern+%E2%80%93+Next-Generation+Engineering+with+Category+Theory+and+Sheaves?lang=en-US) was pretty interesting: > **PhD Intern – Next-Generation Engineering with Category Theory and Sheaves** > **Responsibilities** > * Develop concepts for unifying different engineering formalisms that describe different domains and aspects of complex systems using Category Theory. > * Implement the developed concepts in software prototypes to demonstrate novel languages and algorithms to represent and extract knowledge across engineering disciplines. > * Investigate the use of Sheaves for the creation of hybrid engineering models that combine data-driven and physics-based representations. > * Participate in the preparation of scientific publications. > **Required Knowledge/Skills, Education, and Experience** > * PhD student in Mathematics or Physics. > * Programming skills in Python and/or JavaScript and ability to quickly prototype in these languages. > * Good written and verbal communication skills in English are required, as well as excellent interpersonal skills in multi-cultural environments. > * Team player who can also be independent, prioritize work and thrive in a fast-paced dynamic environment. > * The successful candidate must be able to work with controlled technology in accordance with US Export Control Law. The guy in charge is named [Arquimedes Canedo](https://scholar.google.com/citations?user=r2zo9HQAAAAJ&hl=en), and he seems potentially interested in hiring Blake or other applied category theory people (hint hint!) for a longer-term project.
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23.

That first paper by Canedo is context-modeling, which is something we worked on via a DARPA project a few years ago

https://www.researchgate.net/publication/283579370_C2M2L_Final_Report

Basically covering the same territory and I recognize all the players

Comment Source:That first paper by Canedo is context-modeling, which is something we worked on via a DARPA project a few years ago https://www.researchgate.net/publication/283579370_C2M2L_Final_Report Basically covering the same territory and I recognize all the players 
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24.
edited June 2017

Interesting, WebHubTel! I will tell my student Blake Pollard about this, since he is going to do that 3-month internship with Canedo this summer.

Comment Source:Interesting, WebHubTel! I will tell my student Blake Pollard about this, since he is going to do that 3-month internship with Canedo this summer.
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25.

28 May 2017:

Lots of good news:

1) Brandon Coya got a "dissertation year fellowship" which offers him $7200 of funding for one quarter next year. All of you UCR grad students should apply for this at the appropriate time. 2) Blake also got one! He'll use it this summer. 3) I gave a plenary talk at the Hong Kong Mathematical Society, on the dodecahedron, the icosahedron and E8. Check it out - it's fun! I'm coming back to the US now. Comment Source:28 May 2017: Lots of good news: 1) Brandon Coya got a "dissertation year fellowship" which offers him$7200 of funding for one quarter next year. All of you UCR grad students should apply for this at the appropriate time. 2) Blake also got one! He'll use it this summer. 3) I gave a plenary talk at the Hong Kong Mathematical Society, on [the dodecahedron, the icosahedron and E8](https://johncarlosbaez.wordpress.com/2017/05/16/the-dodecahedron-the-icosahedron-and-e8/). Check it out - it's fun! I'm coming back to the US now.
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26.
edited June 2017

2 June 2017:

More good news:

1) Adam Yassine passed his oral exam, speaking on "Open systems in classical mechanics".

2) Blake Pollard passed his thesis defense, speaking on "Open Markov processes and reaction networks".

Comment Source:2 June 2017: More good news: 1) Adam Yassine passed his oral exam, speaking on "Open systems in classical mechanics". 2) Blake Pollard passed his thesis defense, speaking on "[Open Markov processes and reaction networks](http://math.ucr.edu/home/baez/thesis_defense_pollard.pdf)". 
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27.

9 June 2017:

More good news:

1) Kenny Courser's paper A bicategory of decorated cospans got accepted for publication in Theory and Applications of Categories!

He had to go through a couple rounds of revisions demanded by the referee. In the last round, the referee demanded that Kenny explain the difference between his use of double categories and Lerman and Spivak's use of double categories to study dynamical systems. Kenny wrote and explanation and sent a new version of his paper to the editor. He didn't hear back for many weeks. Finally, it turned out that the editor had never sent the new version to the referee! When Kenny discovered this, the matter was quickly resolved.

Moral: if you sent an email to a journal editor and don't hear back, it's possible they haven't read your email, or forgot to do anything about it. Don't be too shy to politely ask them what's up.

2) Daniel Cicala's paper on Categorifying the zx-calculus, rejected by CALCO, was accepted by QPL.

In computer science conference papers are considered more important than papers in journals - the opposite from math. It really helps your career to get papers accepted by important conferences. CALCO is the Category on Algebra and Coalgebra in Computer Science. QPL is Quantum Physics and Logic. Daniel's paper is a good fit for QPL since that conference is organized by people like Bob Coecke and Ross Duncan, who helped develop the zx-calculus (a diagrammatic method for dealing with certain categories that come up in quantum computation). So, if you write any sort of paper on diagrammatic methods for dealing with categories that show up in physics or engineering - signal flow diagrams, bond graphs, electrical circuits, etc. - you should consider submitting it to QPL. It comes around once every summer, but papers can be submitted earlier.

(I helped referee a bunch of papers for QPL, but I'm not allowed to referee papers by my own students.)

3) Daniel Cicala is now at Snowbird, Utah, learning about homotopy type theory.

It turns out two of the organizers of this workshop are friends of mine - Mike Shulman and Dan Christensen. Furthermore, they helped me answer a question that came up in a paper Daniel and Kenny are writing: is any functor from a groupoid to itself equivalent to an isofibration? It turns out the answer is yes and that this was not previously known (at least not by them).

Comment Source:9 June 2017: More good news: 1) Kenny Courser's paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100) got accepted for publication in _Theory and Applications of Categories_! He had to go through a couple rounds of revisions demanded by the referee. In the last round, the referee demanded that Kenny explain the difference between his use of double categories and Lerman and Spivak's use of double categories to study dynamical systems. Kenny wrote and explanation and sent a new version of his paper to the editor. He didn't hear back for many weeks. Finally, it turned out that the editor had never sent the new version to the referee! When Kenny discovered this, the matter was quickly resolved. Moral: if you sent an email to a journal editor and don't hear back, it's possible they haven't read your email, or forgot to do anything about it. Don't be too shy to politely ask them what's up. 2) Daniel Cicala's paper on [Categorifying the zx-calculus](https://arxiv.org/abs/1704.07034), rejected by CALCO, was accepted by QPL. In computer science conference papers are considered more important than papers in journals - the opposite from math. It really helps your career to get papers accepted by important conferences. [CALCO](http://coalg.org/mfps-calco2017/cfp-calco.html) is the Category on Algebra and Coalgebra in Computer Science. [QPL](http://qpl.science.ru.nl/) is Quantum Physics and Logic. Daniel's paper is a good fit for QPL since that conference is organized by people like Bob Coecke and Ross Duncan, who helped develop the zx-calculus (a diagrammatic method for dealing with certain categories that come up in quantum computation). So, if you write any sort of paper on diagrammatic methods for dealing with categories that show up in physics or engineering - signal flow diagrams, bond graphs, electrical circuits, etc. - you should consider submitting it to QPL. It comes around once every summer, but papers can be submitted earlier. (I helped referee a bunch of papers for QPL, but I'm not allowed to referee papers by my own students.) 3) Daniel Cicala is now at Snowbird, Utah, learning about [homotopy type theory](https://homotopytypetheory.org/2016/10/04/hott-mrc/). It turns out two of the organizers of this workshop are friends of mine - Mike Shulman and Dan Christensen. Furthermore, they helped me answer a question that came up in a paper Daniel and Kenny are writing: is any functor from a groupoid to itself equivalent to an isofibration? It turns out the answer is yes and that this was not previously known (at least not by them). 
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28.

14 June 2017:

1) It looks like we'll be having a new member on our team this fall! Christina Vasilakopoulou is being offered a visiting assistant professorship at UCR, and seems likely to accept. She was a student of Martin Hyland, the category theorist at Cambridge University who also advised Tom Leinster, Eugenia Cheng and Aaron Lauda (who worked with me when he was an undergrad at UCR). Since then she's worked with various people including David Spivak. She's done a mix of "pure" category theory (applied to algebra, actually) and "applied" category theory such as this:

I'm sure she'll bring a lot of energy and new ideas to our team.

2) Blake and I gave talks here:

You can see my talk slides here:

There are a lot of interesting people here, including Luca Peliti, who talked about an analogy I'm really interested in:

• Luca Peliti, On the value of information in gambling, evolution and thermodynamics.

Abstract. The connection between the information value of a message and capital gain was made by Kelly in 1953. In 1965 Kimura tried to evaluate the rate of information intake by a population undergoing Darwinian evolution by equating it with the substitutional load. Recently, the analogy between Kelly’s scheme and work extraction was pointed out in the context of stochastic thermodynamics. I shall try to connect these threads, highlighting analogies and differences between the meaning of information and its value in the different contexts.

and Hong Qian - Blake kept running into his work while working on his thesis:

• Hong Qian, The mathematical foundation of a landscape theory for living matter and life.

Abstract. The physicists’ notion of energy is derived from Newtonian mechanics. The theory of thermodynamics is developed based on that notion, and the realization of mechanical energy dissipation in terms of heat. Since the work of L. Boltzmann, who trusted that atoms were real as early as in 1884, the heat became intimately related to the stochastic motion of the invisible atoms and molecules. In this talk, starting from a stochastic description of a class of rather general dynamics that is not limited to mechanics, we show a notion of energy can be derived mathematically, in the limit of vanishing stochasticity, based on the Kullback-Leibler divergence, or relative entropy associated with the stochastic, Markov processes. With the emergent notion of an energy function, e.g., “landscape”, a mathematical structure inherent to the stochastic dynamics, which is akin to thermodynamics, is revealed. This analysis implies that an abstract “mathematicothermodynamics” structure exists, and can be formulated, for dynamics of complex systems independent of classical thermal physics, for example, in ecology.

I've gotten a bunch of ideas for new projects, which I'm listing in a notebook.

Comment Source:14 June 2017: 1) It looks like we'll be having a new member on our team this fall! [Christina Vasilakopoulou](https://arxiv.org/find/math/1/au:+Vasilakopoulou_C/0/1/0/all/0/1) is being offered a visiting assistant professorship at UCR, and seems likely to accept. She was a student of Martin Hyland, the category theorist at Cambridge University who also advised Tom Leinster, Eugenia Cheng and Aaron Lauda (who worked with me when he was an undergrad at UCR). Since then she's worked with various people including David Spivak. She's done a mix of "pure" category theory (applied to algebra, actually) and "applied" category theory such as this: * David I. Spivak, Christina Vasilakopoulou, Patrick Schultz, [Dynamical systems and sheaves](https://arxiv.org/abs/1609.08086). * Patrick Schultz, David I. Spivak, Christina Vasilakopoulou, Ryan Wisnesky, [Algebraic databases](https://arxiv.org/abs/1602.03501). I'm sure she'll bring a lot of energy and new ideas to our team. 2) Blake and I gave talks here: * [Dynamics, Thermodynamics and Information Processing in Chemical Networks](https://luxcnworkshop.wordpress.com/), 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini. You can see my talk slides here: * [The mathematics of open reaction networks](http://math.ucr.edu/home/baez/networks_luxembourg/open_reaction_networks_web.pdf). There are a lot of interesting people here, including Luca Peliti, who talked about an analogy I'm really interested in: * [Luca Peliti](http://www.peliti.org/), On the value of information in gambling, evolution and thermodynamics. > **Abstract.** The connection between the information value of a message and capital gain was made by Kelly in 1953. In 1965 Kimura tried to evaluate the rate of information intake by a population undergoing Darwinian evolution by equating it with the substitutional load. Recently, the analogy between Kelly’s scheme and work extraction was pointed out in the context of stochastic thermodynamics. I shall try to connect these threads, highlighting analogies and differences between the meaning of information and its value in the different contexts. and Hong Qian - Blake kept running into his work while working on his thesis: * [Hong Qian](https://arxiv.org/find/math-ph/1/au:+Qian_H/0/1/0/all/0/1), The mathematical foundation of a landscape theory for living matter and life. > **Abstract.** The physicists’ notion of energy is derived from Newtonian mechanics. The theory of thermodynamics is developed based on that notion, and the realization of mechanical energy dissipation in terms of heat. Since the work of L. Boltzmann, who trusted that atoms were real as early as in 1884, the heat became intimately related to the stochastic motion of the invisible atoms and molecules. In this talk, starting from a stochastic description of a class of rather general dynamics that is not limited to mechanics, we show a notion of energy can be derived mathematically, in the limit of vanishing stochasticity, based on the Kullback-Leibler divergence, or relative entropy associated with the stochastic, Markov processes. With the emergent notion of an energy function, e.g., “landscape”, a mathematical structure inherent to the stochastic dynamics, which is akin to thermodynamics, is revealed. This analysis implies that an abstract “mathematicothermodynamics” structure exists, and can be formulated, for dynamics of complex systems independent of classical thermal physics, for example, in ecology. I've gotten a bunch of ideas for new projects, which I'm listing in a notebook.
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29.

22 June 2017:

Here is this week's progress, as far as I know:

1) I'm visiting the University of Genoa. It's home to 3 well-known category theorists:

• Marco Grandis (who works on double and n-tuple categories),
• Giuseppe Rosolini (who does functorial semantics for programming languages) and
• Eugenio Moggi (the guy who introduced monads in computer science - a big deal in Haskell and some other languages).

Marco Grandis told me something very interesting. His advisor, Gabriele Darbo, introduced a "theory of devices" in 1970, based on the category of corelations! He applied it to linear electrical circuits using the "add currents, duplicate voltages" rule. All this is VERY similar to Brendan's thesis work and also some of my work with Blake. But it's also different!

Yesterday I summarized the ideas here:

I haven't had time yet to think hard about how his formalism connects to ours. Darbo's work was mostly ignored, perhaps because it's all in Italian. I think we can still learn something from it, even though we've gone further.

2) Today I gave a general talk for the math and science faculty here at Genoa:

3) I also gave a talk to the math department:

Comment Source:22 June 2017: Here is this week's progress, as far as I know: 1) I'm visiting the University of Genoa. It's home to 3 well-known category theorists: * Marco Grandis (who works on double and n-tuple categories), * Giuseppe Rosolini (who does functorial semantics for programming languages) and * Eugenio Moggi (the guy who introduced monads in computer science - a big deal in Haskell and some other languages). Marco Grandis told me something very interesting. His advisor, Gabriele Darbo, introduced a "theory of devices" in 1970, based on the category of corelations! He applied it to linear electrical circuits using the "add currents, duplicate voltages" rule. All this is VERY similar to Brendan's thesis work and also some of my work with Blake. But it's also different! Yesterday I summarized the ideas here: * [The theory of devices](https://johncarlosbaez.wordpress.com/2017/06/20/the-theory-of-devices/). I haven't had time yet to think hard about how his formalism connects to ours. Darbo's work was mostly ignored, perhaps because it's all in Italian. I think we can still learn something from it, even though we've gone further. 2) Today I gave a general talk for the math and science faculty here at Genoa: * [Tales of the dodecahedron: from Pythagoras to Plato to Poincar&eacute;](http://math.ucr.edu/home/baez/dodecahedron/genoa_talk/1.html). 3) I also gave a talk to the math department: * [Applied category theory](http://math.ucr.edu/home/baez/control/applied_category_theory.pdf). 
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30.
edited July 2017

24 June 2017:

1) Daniel Cicala submitted an abstract for my AMS special session on Applied Category Theory:

• A bicategorical syntax for pure state qubit quantum mechanics

Abstract. We begin by constructing a framework used to study open networks modeled by graphs and their rewritings. This consists of a symmetric monoidal compact closed bicategory built by combining spans and cospans inside a topos. Into this bicategorical framework, we fit Coecke and Duncan’s zx-calculus, a graphical language used to reason about pure state qubit quantum mechanics. After viewing the zx-calculus through this lens, we highlight several benefits over the 1-categorical approach: the presence of a symmetric monoidal compact closed structure and a better representation of rewriting information. (Received June 22, 2017)

2) So did Brendan Fong:

• Black boxes and decorated corelations

Abstract. Consider an electric circuit. Suppose this circuit has chosen terminals, which we may connect with the terminals of another circuit. That is to say, consider that we may compose two circuits to obtain another circuit. This suggests we might model circuits as morphisms in a category. Next, suppose I want to compose a circuit with a resistor of resistance 2 ohms. If I have no such resistors, I could substitute with a pair of 1 ohm resistors in series. This suggests a coarser representation of circuits, one that keeps track of only how the circuit behaves, and not their [its] constituent components. In this talk I shall introduce decorated corelations as a tool for constructing categories that model circuits, and constructing ‘black box’ functors that shift between these models. This framework is applicable not only to circuits, but to open systems in general. (Received June 23, 2017)

Comment Source:24 June 2017: 1) Daniel Cicala submitted an abstract for my AMS special session on Applied Category Theory: * A bicategorical syntax for pure state qubit quantum mechanics > **Abstract.** We begin by constructing a framework used to study open networks modeled by graphs and their rewritings. This consists of a symmetric monoidal compact closed bicategory built by combining spans and cospans inside a topos. Into this bicategorical framework, we fit Coecke and Duncan’s zx-calculus, a graphical language used to reason about pure state qubit quantum mechanics. After viewing the zx-calculus through this lens, we highlight several benefits over the 1-categorical approach: the presence of a symmetric monoidal compact closed structure and a better representation of rewriting information. (Received June 22, 2017) 2) So did Brendan Fong: * Black boxes and decorated corelations > **Abstract.** Consider an electric circuit. Suppose this circuit has chosen terminals, which we may connect with the terminals of another circuit. That is to say, consider that we may compose two circuits to obtain another circuit. This suggests we might model circuits as morphisms in a category. Next, suppose I want to compose a circuit with a resistor of resistance 2 ohms. If I have no such resistors, I could substitute with a pair of 1 ohm resistors in series. This suggests a coarser representation of circuits, one that keeps track of only how the circuit behaves, and not their [its] constituent components. In this talk I shall introduce decorated corelations as a tool for constructing categories that model circuits, and constructing ‘black box’ functors that shift between these models. This framework is applicable not only to circuits, but to open systems in general. (Received June 23, 2017) 
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31.
edited July 2017

27 June 2017:

More abstracts for our AMS special session on Applied Category Theory on November 4-5.

1) Kenny Courser:

• A bicategory of coarse-grained Markov processes.

Abstract. If C is a category with finite colimits, D is a symmetric monoidal category and F is a lax symmetric monoidal functor from C to D, Fong has developed a theory of F-decorated cospans which are suitable for representing open dynamical systems. Indeed, Fong has shown the existence of a symmetric monoidal category consisting of objects of C and isomorphism classes of F-decorated cospans in C as morphisms. One application of this result is given by Baez, Fong and Pollard in which they construct a symmetric monoidal category whose morphisms are given by isomorphism classes of open Markov processes. Using a result of Shulman, we present a symmetric monoidal bicategory consisting of finite sets as objects,open Markov processes as morphisms and coarse-grainings of open Markov processes as 2-morphisms. (Received June 25, 2017)

• Open systems in classical mechanics

Abstract. Using the framework of category theory, we formalize the heuristic principles that physicists employ in constructing the Hamiltonians for open classical systems as sums of Hamiltonians of subsystems. First we construct a category where the objects are symplectic manifolds and the morphisms are spans whose legs are surjective Poisson maps. Using a slight variant of Fong’s theory of ”decorated” cospans, we then decorate the apices of our spans with Hamiltonians. This gives a category where morphisms are open classical systems, and composition allows us to build these systems from smaller pieces. (Received June 26, 2017)

Keep 'em coming! I'm still hoping that Brandon, Joseph and Christina will submit abstracts. John Foley may submit one too: it would be great to have him give a talk about how our Metron project is using operads for "compositional tasking". This would be a nice sequel to a talk by Joseph on the underlying math - namely, "network models" and the operads they give rise to.

Comment Source:27 June 2017: More abstracts for our AMS special session on Applied Category Theory on November 4-5. 1) Kenny Courser: * A bicategory of coarse-grained Markov processes. > **Abstract.** If C is a category with finite colimits, D is a symmetric monoidal category and F is a lax symmetric monoidal functor from C to D, Fong has developed a theory of F-decorated cospans which are suitable for representing open dynamical systems. Indeed, Fong has shown the existence of a symmetric monoidal category consisting of objects of C and isomorphism classes of F-decorated cospans in C as morphisms. One application of this result is given by Baez, Fong and Pollard in which they construct a symmetric monoidal category whose morphisms are given by isomorphism classes of open Markov processes. Using a result of Shulman, we present a symmetric monoidal bicategory consisting of finite sets as objects,open Markov processes as morphisms and coarse-grainings of open Markov processes as 2-morphisms. (Received June 25, 2017) 2) Adam Yassine: * Open systems in classical mechanics > **Abstract.** Using the framework of category theory, we formalize the heuristic principles that physicists employ in constructing the Hamiltonians for open classical systems as sums of Hamiltonians of subsystems. First we construct a category where the objects are symplectic manifolds and the morphisms are spans whose legs are surjective Poisson maps. Using a slight variant of Fong’s theory of ”decorated” cospans, we then decorate the apices of our spans with Hamiltonians. This gives a category where morphisms are open classical systems, and composition allows us to build these systems from smaller pieces. (Received June 26, 2017) Keep 'em coming! I'm still hoping that Brandon, Joseph and Christina will submit abstracts. John Foley may submit one too: it would be great to have him give a talk about how our Metron project is using operads for "compositional tasking". This would be a nice sequel to a talk by Joseph on the underlying math - namely, "network models" and the operads they give rise to.
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32.

I'm way behind on these, but I'll continue where I left off:

30 June 2017

We're getting more abstracts for our AMS special session on Applied Category Theory on November 4-5! I'm happy to announce that David Spivak has submitted one, because he generally doesn't like to travel.

1) Joseph Moeller submitted one on his work with the Metron project:

Abstract. A network is a complex of interacting systems which can often be represented as a graph equipped with extra structure. Networks can be combined in many ways, including by overlaying one on top of the other or sitting one next to another. We introduce network models - which are formally a simple kind of lax symmetric monoidal functor - to encode these ways of combining networks. By applying a general construction to network models, we obtain operads for the design of complex networked systems. (Received June 29, 2017)

2) Christina Vasilakopoulou submitted on on her work with David Spivak and Patrick Schultz:

• Abstract dynamical systems

Abstract. We describe a categorical framework of modeling and analyzing systems in a broad sense. The latter can be thought of as ‘machines’ with inputs and outputs, carrying some sort of signal that occurs through some notion of time; special cases include discrete and continuous dynamical systems. Modeling them as algebras for the wiring diagram operad, a central goal is to understand the behavior of composite systems, formed as arbitrary interconnections of component subsystems. This shall be accomplished using lax monoidal functors, which provide a coherent formalization of systems, as well as sheaf theory, which captures the crucial notion of time. (Received June 29, 2017)

3) David Spivak submitted one:

• A higher-order temporal logic for dynamical systems

Abstract. We consider a very general class of dynamical systems—including discrete, continuous, hybrid, deterministic, non- deterministic, etc.—based on sheaves. We call these sheaves behavior types: they tell us the set of possible behaviors over any interval of time. A machine can be construed as a wide span of such sheaves, and these machines can be composed as morphisms in a hypergraph category. The topos of sheaves has an internal language, which we use as a new sort of higher-order internal logic for talking about behaviors. We can use this logic to prove properties about a composite system of systems from properties of the parts and how they are wired together. (Received June 28, 2017)

4) A third-year statistics grad student at Stanford named Evan Patterson submitted an interesting one:

• Knowledge representation in bicategories of relations

Abstract. We introduce the relational ontology log, or relational olog, a categorical framework for knowledge representation based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a knowledge representation system based on the category of sets and functions. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. On a practical level, we demonstrate that relational ologs have an intuitive yet fully precise graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. (Received June 29, 2017)

On other news, I got back to Riverside yesterday! I have some big news to report from my visit to Turin, but I'll report that separately. If anyone at UCR wants to talk to me in person, I'll be here until around June 14th - set up an appointment! We've got a lot of projects going that are worth talking about. I'm especially eager to accelerate Kenny's work on coarse-graining Markov processes, and help Brandon finish his increasingly deep and interesting paper on bond graphs, and help Joseph finish a paper on network models, and get him started on a paper on compositional tasking.

Comment Source:I'm way behind on these, but I'll continue where I left off: 30 June 2017 We're getting more abstracts for our [AMS special session on Applied Category Theory](http://math.ucr.edu/home/baez/ACT2017) on November 4-5! I'm happy to announce that David Spivak has submitted one, because he generally doesn't like to travel. 1) Joseph Moeller submitted one on his work with the Metron project: * Operads for modeling networks > **Abstract.** A network is a complex of interacting systems which can often be represented as a graph equipped with extra structure. Networks can be combined in many ways, including by overlaying one on top of the other or sitting one next to another. We introduce network models - which are formally a simple kind of lax symmetric monoidal functor - to encode these ways of combining networks. By applying a general construction to network models, we obtain operads for the design of complex networked systems. (Received June 29, 2017) > 2) Christina Vasilakopoulou submitted on on her work with David Spivak and Patrick Schultz: * Abstract dynamical systems > **Abstract.** We describe a categorical framework of modeling and analyzing systems in a broad sense. The latter can be thought of as ‘machines’ with inputs and outputs, carrying some sort of signal that occurs through some notion of time; special cases include discrete and continuous dynamical systems. Modeling them as algebras for the wiring diagram operad, a central goal is to understand the behavior of composite systems, formed as arbitrary interconnections of component subsystems. This shall be accomplished using lax monoidal functors, which provide a coherent formalization of systems, as well as sheaf theory, which captures the crucial notion of time. (Received June 29, 2017) 3) David Spivak submitted one: * A higher-order temporal logic for dynamical systems > **Abstract.** We consider a very general class of dynamical systems—including discrete, continuous, hybrid, deterministic, non- deterministic, etc.—based on sheaves. We call these sheaves behavior types: they tell us the set of possible behaviors over any interval of time. A machine can be construed as a wide span of such sheaves, and these machines can be composed as morphisms in a hypergraph category. The topos of sheaves has an internal language, which we use as a new sort of higher-order internal logic for talking about behaviors. We can use this logic to prove properties about a composite system of systems from properties of the parts and how they are wired together. (Received June 28, 2017) 4) A third-year statistics grad student at Stanford named Evan Patterson submitted an interesting one: * Knowledge representation in bicategories of relations > **Abstract.** We introduce the relational ontology log, or relational olog, a categorical framework for knowledge representation based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a knowledge representation system based on the category of sets and functions. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. On a practical level, we demonstrate that relational ologs have an intuitive yet fully precise graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. (Received June 29, 2017) On other news, I got back to Riverside yesterday! I have some big news to report from my visit to Turin, but I'll report that separately. If anyone at UCR wants to talk to me in person, I'll be here until around June 14th - set up an appointment! We've got a lot of projects going that are worth talking about. I'm especially eager to accelerate Kenny's work on coarse-graining Markov processes, and help Brandon finish his increasingly deep and interesting paper on bond graphs, and help Joseph finish a paper on network models, and get him started on a paper on compositional tasking. 
• Options
33.

7 July 2017:

Three pieces of news. The third one is a big deal for me.

1) Brandon has submitted an abstract for our AMS special session on the weekend of November 4-5. ) * Frobenius monoids, weak bimonoids, and corelations

Abstract. In this talk we consider object 2 in the category FinCorel, whose objects are finite sets and whose morphisms are “corelations.” The object 2 can be equipped with two different Frobenius monoid structures. We show that the two Frobenius monoids interact to form a “weak bimonoid” as defined by Pastro and Street. Baez and Fong have shown that FinCorel is useful for modeling circuits made of wire as morphisms in a category. In this analogy the object 1 is viewed as a single wire. We show how the two Frobenius monoids associated to the object 2 relate to placing pairs of wires into series and parallel connections. (Received July 06, 2017)

2) Kenny and Daniel have finished writing a paper and put it on the arXiv.

Abstract. If C is a category with chosen pullbacks and a terminal object then, using a result of Shulman, we obtain a fully dualizable and symmetric monoidal bicategory Sp(Sp(C))} whose objects are those of C whose morphisms are spans in C, and whose 2-morphisms are isomorphism classes of spans of spans in C. If C is a topos, the first author has previously constructed a bicategory MonicSp(Csp(C)) whose objects are those of C, whose morphisms are cospans in C, and whose 2-morphisms are isomorphism classes of spans of cospans in C with monic legs. We prove this bicategory is also symmetric monoidal and even compact closed. We discuss applications of such bicategories to graph rewriting as well as to Morton and Vicary's combinatorial approach to Khovanov's categorified Heisenberg algebra.

3) At the end of my trip to Europe I spent a week in Turin at the Institute for Scientific Interchange. This is a research institute with about 56 scientists run by Mario Rasetti, a physicist who used to do statistical mechanics at the Institute for Advanced Studies. 30 years ago he went back to Italy to set up a roughly similar institute in his home town. They study complex networks and data science.

He's been reading our work and liking it. He asked me to set up a mathematics group there!

If I do this, I'll be hiring a number of postdocs to work on networks, applied category theory and the like. That could be some of you - though of course I don't want to hire just my own students; I want to hire the best people I can find. There should also be opportunities for shorter visits, like conferences and workshops, which some of you might attend.

If I do this I won't be giving up my position at UCR, at least not soon. Instead, I'll visit the place repeatedly, for example during summers. Turin is a wonderful city, so I like this idea. But mainly I'm excited at the idea of being able to put together a team of people working on the kinds of math I like.

It's not 100% certain this will happen: I need to decide I really want to do it, and write a proposal, which the board of the ISI will then read, etc.

Comment Source:7 July 2017: Three pieces of news. The third one is a big deal for me. 1) Brandon has submitted an abstract for our [AMS special session](http://math.ucr.edu/home/baez/ACT2017) on the weekend of November 4-5. ) * Frobenius monoids, weak bimonoids, and corelations > **Abstract.** In this talk we consider object 2 in the category FinCorel, whose objects are finite sets and whose morphisms are “corelations.” The object 2 can be equipped with two different Frobenius monoid structures. We show that the two Frobenius monoids interact to form a “weak bimonoid” as defined by Pastro and Street. Baez and Fong have shown that FinCorel is useful for modeling circuits made of wire as morphisms in a category. In this analogy the object 1 is viewed as a single wire. We show how the two Frobenius monoids associated to the object 2 relate to placing pairs of wires into series and parallel connections. (Received July 06, 2017) 2) Kenny and Daniel have finished writing a paper and put it on the arXiv. * [Bicategories of spans and cospans](https://arxiv.org/abs/1707.02098v1) > **Abstract.** If C is a category with chosen pullbacks and a terminal object then, using a result of Shulman, we obtain a fully dualizable and symmetric monoidal bicategory Sp(Sp(C))} whose objects are those of C whose morphisms are spans in C, and whose 2-morphisms are isomorphism classes of spans of spans in C. If C is a topos, the first author has previously constructed a bicategory MonicSp(Csp(C)) whose objects are those of C, whose morphisms are cospans in C, and whose 2-morphisms are isomorphism classes of spans of cospans in C with monic legs. We prove this bicategory is also symmetric monoidal and even compact closed. We discuss applications of such bicategories to graph rewriting as well as to Morton and Vicary's combinatorial approach to Khovanov's categorified Heisenberg algebra. 3) At the end of my trip to Europe I spent a week in Turin at the Institute for Scientific Interchange. This is a research institute with about 56 scientists run by Mario Rasetti, a physicist who used to do statistical mechanics at the Institute for Advanced Studies. 30 years ago he went back to Italy to set up a roughly similar institute in his home town. They study complex networks and data science. He's been reading our work and liking it. He asked me to set up a mathematics group there! If I do this, I'll be hiring a number of postdocs to work on networks, applied category theory and the like. That could be some of you - though of course I don't want to hire just my own students; I want to hire the best people I can find. There should also be opportunities for shorter visits, like conferences and workshops, which some of you might attend. If I do this I won't be giving up my position at UCR, at least not soon. Instead, I'll visit the place repeatedly, for example during summers. Turin is a wonderful city, so I like this idea. But mainly I'm excited at the idea of being able to put together a team of people working on the kinds of math I like. It's not 100% certain this will happen: I need to decide I really want to do it, and write a proposal, which the board of the ISI will then read, etc.
• Options
34.

13 July 2017:

This week's progress:

1) I just finished a cool paper with Brandon Coya and Franciscus Rebro:

Abstract. Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the 'behavior' of these circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved by Fong and the first author; unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also give use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Mathematically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.

I started this project with Franciscus over a year ago, so it's great to be done.

Except we're not done! There are probably lots of mistakes and suboptimalities, so I'd be really grateful if all of you could look over it and send me comments. Also, at one point I promised Jason that the appendix would contain some lemmas on props that he needed. It doesn't have those yet - just the monadic adjunction between props and signatures, which is the key to all those further lemmas.

2) Daniel Cicala is doing lots of good stuff:

​Also, just to keep you updated, I got back last week from the QPL conference, which was lots of fun. My talk seemed to go well. Next week, I'm traveling to Vancouver for CT 2017 where I'm giving two talks. The first is a ten minute expository talk attached to the Kan seminar. It's on Lack & Rosicky's Notions of Lawvere theories paper. The second is an accepted talk about my span of cospans stuff. ​

3) Brendan is doing lots of stuff - but I can't keep track of it all, so I invite him to tell us.

4) I'm going to Singapore tonight, and staying at the Centre of Quantum Technologies until September 15th.

5) I got a referee's report on a paper I'd almost given up on. It's so old most of you guys probably don't know it:

Abstract. Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum field theory can be adapted to apply to such systems. Here we use these techniques to study how the 'master equation' describing stochastic time evolution for a reaction network is related to the 'rate equation' describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a 'coherent state', meaning that the numbers of entities of each kind are described by independent Poisson distributions.

I finished writing this in 2013, as a kind of 'prequel' to a more interesting paper that Brendan and I wrote. Since then it's been in journal hell - or perhaps purgatory or limbo. It's a long and boring story, but in 2014 it was solicited for a special issue in Natural Computing, and I've been trying to get a referee's report from them ever since. I finally got it - and luckily, they say they'll accept it after I make some small changes, which are really small.

Moral: when you're trying to publish something, never give up.

Comment Source:13 July 2017: This week's progress: 1) I just finished a cool paper with Brandon Coya and Franciscus Rebro: * [Props in network theory](https://arxiv.org/abs/1707.08321) > **Abstract.** Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the 'behavior' of these circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved by Fong and the first author; unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also give use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Mathematically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props. I started this project with Franciscus over a year ago, so it's great to be done. Except we're not done! There are probably lots of mistakes and suboptimalities, so I'd be really grateful if all of you could look over it and send me comments. Also, at one point I promised Jason that the appendix would contain some lemmas on props that he needed. It doesn't have those yet - just the monadic adjunction between props and signatures, which is the key to all those further lemmas. 2) Daniel Cicala is doing lots of good stuff: > ​Also, just to keep you updated, I got back last week from the QPL conference, which was lots of fun. My talk seemed to go well. Next week, I'm traveling to Vancouver for CT 2017 where I'm giving two talks. The first is a ten minute expository talk attached to the Kan seminar. It's on Lack & Rosicky's Notions of Lawvere theories paper. The second is an accepted talk about my span of cospans stuff. ​ 3) Brendan is doing lots of stuff - but I can't keep track of it all, so I invite him to tell us. 4) I'm going to Singapore tonight, and staying at the Centre of Quantum Technologies until September 15th. 5) I got a referee's report on a paper I'd almost given up on. It's so old most of you guys probably don't know it: * [Quantum techniques for reaction networks](http://arxiv.org/abs/1306.3451) > **Abstract.** Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum field theory can be adapted to apply to such systems. Here we use these techniques to study how the 'master equation' describing stochastic time evolution for a reaction network is related to the 'rate equation' describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a 'coherent state', meaning that the numbers of entities of each kind are described by independent Poisson distributions. I finished writing this in 2013, as a kind of 'prequel' to a more interesting paper that Brendan and I wrote. Since then it's been in journal hell - or perhaps purgatory or limbo. It's a long and boring story, but in 2014 it was solicited for a special issue in Natural Computing, and I've been trying to get a referee's report from them ever since. I finally got it - and luckily, they say they'll accept it after I make some small changes, which are really small. Moral: when you're trying to publish something, never give up.
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35.

20 July 2017:

1) The paper Blake and I wrote on A compositional framework for reaction networks has been accepted by Reviews in Mathematical Physics. The best part: it seems no corrections were demanded!

2) More news from Daniel, who is at the big annual category theory conference:

I ran into Rick Blute who was my masters adviser and the editor of TAC that I sent my Spans of Cospans paper to. He told me he recently sent an email to the referee reminding them to get me a report. We'll see how that goes.

I gave my talk on Lack and Rosicky's Notions of Lawvere Theories paper today. I got great feedback from Emily Riehl, so that was nice. I presented believing the entire time that this white haired and white bearded man sitting front and center was Bill Lawvere (he was supposed to be here) but it turned out to be Michael Barr. Oddly, I was not the only one of the Kan seminar group to make this mistake.

Us Kan folk, along with some alumni from the first instance of the online seminar went out to dinner and it turns out Christina Vasilakopoulou was one of these alumni, so we got to chat a bit.

3) James Haydon is applying decorated cospans to computer science:

What I've done is set up a framework for composing coroutines (= asynchronous cooperating processes) using a category of decorated cospans. Furthermore, I've implemented the whole thing in code: actual composition of concurrent processes via pushouts!

As an underlying category I take typed channel contexts; this represents a support for a pi-calculus process: channel names and types which it may read and write from. The morphisms map names while respecting the typing structure.

For such a context $$X$$, I define a restricted pi-calculus $$\Pi(X)$$, which is the set of well typed pi-calculi processes that may only read and write to the channel names specified in $$X$$.

This defines a monoidal functor

$$\Pi : (\mathrm{TyCh}, +) \to (\mathrm{Set}, \times)$$ with the required properties to form a category of decorated cospans. I have implemented all this in the Idris programming language, the source code is here:

https://github.com/jameshaydon/cospanProc

I've experimented with several examples and I think this provides a nice framework for organising code, and composing processes in a safe way. While you compose the processes, you simultaneously compute, via the pushout, the communication interface the resulting process will expose.

Comment Source:20 July 2017: 1) The paper Blake and I wrote on [A compositional framework for reaction networks](https://arxiv.org/abs/1704.02051) has been accepted by _Reviews in Mathematical Physics_. The best part: it seems no corrections were demanded! 2) More news from Daniel, who is at the big annual category theory conference: > Some updates from Vancouver: > I ran into Rick Blute who was my masters adviser and the editor of TAC that I sent my Spans of Cospans paper to. He told me he recently sent an email to the referee reminding them to get me a report. We'll see how that goes. > I gave my talk on Lack and Rosicky's Notions of Lawvere Theories paper today. I got great feedback from Emily Riehl, so that was nice. I presented believing the entire time that this white haired and white bearded man sitting front and center was Bill Lawvere (he was supposed to be here) but it turned out to be Michael Barr. Oddly, I was not the only one of the Kan seminar group to make this mistake. > Us Kan folk, along with some alumni from the first instance of the online seminar went out to dinner and it turns out Christina Vasilakopoulou was one of these alumni, so we got to chat a bit. 3) James Haydon is applying decorated cospans to computer science: > What I've done is set up a framework for composing coroutines (= asynchronous cooperating processes) using a category of decorated cospans. Furthermore, I've implemented the whole thing in code: actual composition of concurrent processes via pushouts! > As an underlying category I take typed channel contexts; this represents a support for a pi-calculus process: channel names and types which it may read and write from. The morphisms map names while respecting the typing structure. > For such a context \$$X\$$, I define a restricted pi-calculus \$$\Pi(X)\$$, which is the set of well typed pi-calculi processes that may only read and write to the channel names specified in \$$X\$$. > This defines a monoidal functor > $\Pi : (\mathrm{TyCh}, +) \to (\mathrm{Set}, \times)$ > with the required properties to form a category of decorated cospans. I have implemented all this in the Idris programming language, the source code is here: > https://github.com/jameshaydon/cospanProc > I've experimented with several examples and I think this provides a nice framework for organising code, and composing processes in a safe way. While you compose the processes, you simultaneously compute, via the pushout, the communication interface the resulting process will expose.
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36.
edited June 2018

28 July 2017:

This week's progress!

1) First, some good news from Jason Erbele:

A. Just a quick reminder that I will be attending the Hopf Algebras etc. conference at the Perimeter Institute this coming week. Based on the list on the conference website, just under 50 people are participating. The relatively small crowd should make it a bit easier for me to meet people and network, despite not being one of the speakers.

B. I haven't gotten word one way or the other yet regarding the Air Force Research Laboratory postdoc in Dayton, OH, but I have been invited to spend a couple of days at the beginning of September at U. Penn. to get to know some of the people that I would be working with, assuming I do get the job. It'll be Tuesday and Wednesday, September 5-6. They also want me to give a talk, and Dan Koditschek had this to say about it:

"[M]y suggestion would be that you aim for a 45 min. talk with your major effort being to explain the nature and impact (as distinct from the details of the technical accomplishments) of your ideas and results to an audience interested in but very unfamiliar with your area. My group is typically quite active and you will likely be interrupted throughout with questions."

You will likely recall that Dan Koditschek is the head of the group that Brendan was with before he took David Spivak's offer at MIT.

The second item is especially promising. He's telling you to focus on the big picture and not sink into too many technical details, and he's telling you to discuss the "impact" of your work, which means you have to make it sound like a big deal. I urge you to get some advice from Brendan about how to do this, because he knows Koditschek. He also knows Sobocinski and company, who seem to be doing a reasonable job of convincing people of the importance of their work.

In trying to sell your ideas and accomplishments, it will help to interpret "your" rather broadly, to include the work of Bonchi, Sobocinski and Zanasi on similar themes. I don't mean to pretend you did their work - you should of course credit them. But you should learn their work, and if they've done interesting things, especially "practical" things, you should feel free to explain those things in your talk.

In short, think of yourself not as a mere individual, but as a representative of a movement. If the movement is doing interesting things, and you represent it ably, Koditschek may want to hire you - to have "someone who knows that stuff" around, especially now that Brendan is no longer there.

2) Second, Nina Otter is at a workshop on Macaulay2, which is a software package for algebraic geometry. I get the feeling that this workshop is focused on using Macaulay2 in statistics, phylogenetics and other areas - "applied algebraic geometry".

3) Third, Blake and I put our paper on A compositional framework for Markov processes into the suitable format for Reviews in Mathematical Physics and gave it to that journal. They should give us the proofs back in a week or two.

4) Fourth, Brandon and Franciscus and I put our paper on Props in network theory onto the arXiv. Shortly afterward, I got a comment from the category theorist Thomas Holder. He said:

This proves that the first thing people do is look at your bibliography. I fixed this.

The other thing is, since you pay considerable attention to the history of the subject, I'd like to point out that the use of monoidal categories in network theory including the use of string diagrams was pioneered by the German computer scientist Günter Hotz in 1965/66 and that by the end of the 1960s there was considerable activity by his group in Saarbrücken with some nice results e.g. Hans Langmaarck discovered the first ordering of the braid group in the process of giving normal forms for the morphisms in 1969. At the same time this was taken up by a research group in East Germany around the algebraicist Lothar Budach, a former student of Krull's, in collaboration with Hans-Joachim Hoehnke leading to a monumental monograph on 'Automaten und Funktoren' in 1975. David Benson at that time also worked in this framework.

Most of this work is unfortunately in German (e.g. a monograph of Hotz on 'Schaltkreistheorie' or Boolean circuit theory in 1974) - you find some of the relevant references at the nLab entry on Hotz.

Permit me to unload here an observation which for a lack of proper understanding of QFT I've never known much what to make of: One of the results in the Budach-Hoehnke book is a generalization of the Chomsky normal form for context-free grammars which says that the morphisms (=syntactical derivations) in the monoidal category corresponding to your (not necessarily context-free) grammar have a normal form provided the rewriting rules that generate your category don't include at the same time a (creation) rule that rewrites the empty string to a non empty string e->X1...Xn and the reverse annihilation rule X1...Xn->e. This suggests that (at least to a non cognoscente like me) that renormalization of a QFT is tied to the normalization of the corresponding theory of Feynman diagrams.

thanks for your patience&best wishes Thomas Holder

Personally I think the relation between "normalization of strings" and "renormalization in QFT" is mainly just a pun, but the remarks on Hotz are interesting and I may decide to add something about his work to our paper.

Comment Source:28 July 2017: This week's progress! 1) First, some good news from Jason Erbele: > A. Just a quick reminder that I will be attending the [Hopf Algebras etc.](https://www.perimeterinstitute.ca/conferences/hopf-algebras-kitaevs-quantum-double-models-mathematical-connections-gauge-theory) conference at the Perimeter Institute this coming week. Based on the list on the conference website, just under 50 people are participating. The relatively small crowd should make it a bit easier for me to meet people and network, despite not being one of the speakers. > B. I haven't gotten word one way or the other yet regarding the Air Force Research Laboratory postdoc in Dayton, OH, but I have been invited to spend a couple of days at the beginning of September at U. Penn. to get to know some of the people that I would be working with, assuming I do get the job. It'll be Tuesday and Wednesday, September 5-6. They also want me to give a talk, and Dan Koditschek had this to say about it: > > "[M]y suggestion would be that you aim for a 45 min. talk with your major effort being to explain the nature and impact (as distinct from the details of the technical accomplishments) of your ideas and results to an audience interested in but very unfamiliar with your area. My group is typically quite active and you will likely be interrupted throughout with questions." > You will likely recall that Dan Koditschek is the head of the group that Brendan was with before he took David Spivak's offer at MIT. The second item is especially promising. He's telling you to focus on the big picture and not sink into too many technical details, and he's telling you to discuss the "impact" of your work, which means you have to make it sound like a big deal. I urge you to get some advice from Brendan about how to do this, because he knows Koditschek. He also knows Sobocinski and company, who seem to be doing a reasonable job of convincing people of the importance of their work. In trying to sell your ideas and accomplishments, it will help to interpret "your" rather broadly, to include the work of Bonchi, Sobocinski and Zanasi on similar themes. I don't mean to pretend you did their work - you should of course credit them. But you should learn their work, and if they've done interesting things, especially "practical" things, you should feel free to explain those things in your talk. In short, **think of yourself not as a mere individual, but as a representative of a movement**. If the movement is doing interesting things, and you represent it ably, Koditschek may want to hire you - to have "someone who knows that stuff" around, especially now that Brendan is no longer there. 2) Second, Nina Otter is at a workshop on [Macaulay2](https://sites.google.com/view/macaulay2-gatech-2017/home), which is a software package for algebraic geometry. I get the feeling that this workshop is focused on using Macaulay2 in statistics, phylogenetics and other areas - "applied algebraic geometry". 3) Third, Blake and I put our paper on [A compositional framework for Markov processes](https://arxiv.org/abs/1704.02051) into the suitable format for _Reviews in Mathematical Physics_ and gave it to that journal. They should give us the proofs back in a week or two. 4) Fourth, Brandon and Franciscus and I put our paper on [Props in network theory](https://arxiv.org/abs/1707.08321) onto the arXiv. Shortly afterward, I got a comment from the category theorist Thomas Holder. He said: > Please excuse me for making a couple of comments on your arXiv.1707.08321. For one your first reference in the bibliography is empty. This proves that the first thing people do is look at your bibliography. I fixed this. > The other thing is, since you pay considerable attention to the history of the subject, I'd like to point out that the use of monoidal categories in network theory including the use of string diagrams was pioneered by the German computer scientist Günter Hotz in 1965/66 and that by the end of the 1960s there was considerable activity by his group in Saarbrücken with some nice results e.g. Hans Langmaarck discovered the first ordering of the braid group in the process of giving normal forms for the morphisms in 1969. At the same time this was taken up by a research group in East Germany around the algebraicist Lothar Budach, a former student of Krull's, in collaboration with Hans-Joachim Hoehnke leading to a monumental monograph on 'Automaten und Funktoren' in 1975. David Benson at that time also worked in this framework. > Most of this work is unfortunately in German (e.g. a monograph of Hotz on 'Schaltkreistheorie' or Boolean circuit theory in 1974) - you find some of the relevant references at the nLab entry on Hotz. > Permit me to unload here an observation which for a lack of proper understanding of QFT I've never known much what to make of: One of the results in the Budach-Hoehnke book is a generalization of the Chomsky normal form for context-free grammars which says that the morphisms (=syntactical derivations) in the monoidal category corresponding to your (not necessarily context-free) grammar have a normal form provided the rewriting rules that generate your category don't include at the same time a (creation) rule that rewrites the empty string to a non empty string e->X1...Xn and the reverse annihilation rule X1...Xn->e. This suggests that (at least to a non cognoscente like me) that renormalization of a QFT is tied to the normalization of the corresponding theory of Feynman diagrams. > thanks for your patience&best wishes Thomas Holder Personally I think the relation between "normalization of strings" and "renormalization in QFT" is mainly just a pun, but the remarks on Hotz are interesting and I may decide to add something about his work to our paper.
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37.

2 August 2017:

This week's progress:

1) Nina Otter is at Banff International Research Station, the place in Canada where they have tons of math workshops, attending one on Topological Data Analysis: Developing Abstract Foundations. Topological data analysis uses things like persistent homology to calculate the topology of clouds of data points. It's very fashionable these days.

2) Blake and I gave Reviews in Mathematical Physics a final version of our paper "A compositional framework for reaction networks". I celebrated by blogging about it all over:

• Category theory in chemistry, on G+, a "pop" explanation that got lots of likes but also lots of interesting comments.

• A compositional framework for reaction networks, on Azimuth, a more technical explanation that also got some good comments.

• A compositional framework for reaction networks, on the n-Category Cafe, an almost identical technical explanation that got completely different comments, including a really fascinating one from Mike Shulman: he's been looking at open Petri nets in logic, without knowing that's what they're called! Instead of chemicals reacting to give other chemicals, he's looking at assumptions "reacting" to give conclusions! In other words, he's looking at proofs. /This is the kind of analogy Mike Stay and I really like, and I bet it will produce interesting new ways of looking at both chemistry and logic.

3) Nina and I just sent a revised version of our paper "Operads and phylogenetic trees" to Steve Lack, who is an editor at TAC. The referee had wanted tons of big changes, so this was a lot of work. Our reply letter listing what we've done is 11 pages long! I won't be surprised if the referee points out mistakes in this.

But the paper is better now - if you were too scared to read it before, read it now! It's about Markov processes and operads.

The referee had earlier said "Overall, I believe that an improved version of this paper would make an excellent, unusual and interesting contribution to TAC", so I hope they accept it now. We've been working on this paper since December 2013.

Moral: it sometimes takes insane amounts of persistence to publish things.

Comment Source:2 August 2017: This week's progress: 1) Nina Otter is at Banff International Research Station, the place in Canada where they have tons of math workshops, attending one on [Topological Data Analysis: Developing Abstract Foundations](https://www.birs.ca/events/2017/5-day-workshops/17w5108). Topological data analysis uses things like [persistent homology](https://en.wikipedia.org/wiki/Persistent_homology) to calculate the topology of clouds of data points. It's very fashionable these days. 2) Blake and I gave _Reviews in Mathematical Physics_ a final version of our paper "A compositional framework for reaction networks". I celebrated by blogging about it all over: * [Category theory in chemistry](https://plus.google.com/u/0/+johncbaez999/posts/hhyzhu1mB7F), on G+, a "pop" explanation that got lots of likes but also lots of interesting comments. * [A compositional framework for reaction networks](https://johncarlosbaez.wordpress.com/2017/07/30/a-compositional-framework-for-reaction-networks/), on Azimuth, a more technical explanation that also got some good comments. * [A compositional framework for reaction networks](https://golem.ph.utexas.edu/category/2017/07/a_compositional_framework_for_2.html), on the n-Category Cafe, an almost identical technical explanation that got completely different comments, including a really fascinating one from Mike Shulman: he's been looking at open Petri nets in logic, without knowing that's what they're called! Instead of chemicals reacting to give other chemicals, he's looking at assumptions "reacting" to give conclusions! In other words, he's looking at proofs. /This is the kind of analogy Mike Stay and I really like, and I bet it will produce interesting new ways of looking at both chemistry and logic. 3) Nina and I just sent a revised version of our paper "[Operads and phylogenetic trees](http://math.ucr.edu/home/baez/phylo.pdf)" to Steve Lack, who is an editor at TAC. The referee had wanted tons of big changes, so this was a lot of work. Our reply letter listing what we've done is 11 pages long! I won't be surprised if the referee points out mistakes in this. But the paper is better now - if you were too scared to read it before, read it now! It's about Markov processes and operads. The referee had earlier said "Overall, I believe that an improved version of this paper would make an excellent, unusual and interesting contribution to TAC", so I hope they accept it now. We've been working on this paper since December 2013. **Moral: it sometimes takes insane amounts of persistence to publish things.** 
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38.

10 August 2017:

This week's progress:

1) Jason Erbele wrote an entertaining blog article about the conference he attended at the Perimeter Institute:

2) Nina Otter gave a talk at a week-long conference in Sapporo, Japan, called Applied Algebraic Topology 2017:

• New invariants for multi-parameter persistent homology.

Abstract. Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance.

PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA.

In this talk I will briefly introduce persistent homology, give an overview of the complexity of the theory in the multi-parameter case, and then discuss how tools from commutative algebra give invariants able to capture homology classes with large persistence.

No prior knowledge on the subject is assumed. This talk is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann.

3) I also gave a talk at this conference:

Abstract. As algebraic topology becomes more important in applied mathematics, it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as "the same" if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the "homotopification" of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract "spaces" (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of "robustness" in applications will influence algebraic topology.

Quite a few other talks used category theory! But as this slide from Tom Leinster's talk, they did so with a bit of hesitancy:

Comment Source:10 August 2017: This week's progress: 1) Jason Erbele wrote an entertaining blog article about the conference he attended at the Perimeter Institute: * [Hopf Algebras in Kitaev’s Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics](https://jasonmaths.wordpress.com/2017/08/09/haikqdmmcfgtttqcacqm-postmortem/). 2) Nina Otter gave a talk at a week-long conference in Sapporo, Japan, called Applied Algebraic Topology 2017: * New invariants for multi-parameter persistent homology. > **Abstract.** Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance. > PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA. > In this talk I will briefly introduce persistent homology, give an overview of the complexity of the theory in the multi-parameter case, and then discuss how tools from commutative algebra give invariants able to capture homology classes with large persistence. > No prior knowledge on the subject is assumed. This talk is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann. 3) I also gave a talk at this conference: * [The rise and spread of algebraic topology](http://math.ucr.edu/home/baez/alg_top/) > **Abstract.** As algebraic topology becomes more important in applied mathematics, it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as "the same" if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the "homotopification" of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract "spaces" (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of "robustness" in applications will influence algebraic topology. Quite a few other talks used category theory! But as this slide from Tom Leinster's talk, they did so with a bit of hesitancy: <center><img width = "600" src = "http://math.ucr.edu/home/baez/alg_top/leinster_talk.jpg"></center>
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39.
edited June 2018

17 August 2017:

1) Kenny's first paper, A bicategory of decorated cospans, has been published by Theory and Applications of Categories. He put a lot of work into this! It looks great in its final form.

2) Nina Otter is visiting me at the Centre for Quantum Technologies - we got some money from the center for her to do this. We're trying to write a quick, short paper on the relation between persistent homology and magnitude homology.

3) I wrote a blog article that starts explaining how our project with Metron is using operads to design networks:

I'll need to write more to really explain the idea! DARPA is having a seminar on operads on September 14th and they may use some of these blog articles as reading material.

4) And now for the really big news: there will be a workshop on Applied Category Theory at the Lorentz Center in Leiden, the Netherlands, April 23 to May 4, 2018. This is being organized by Bob Coecke, Aleks Kissinger, Martha Lewis and Josh Tan... and our very own Brendan Fong!

They've put together an ambitious program where grad students will join online seminars in January and start doing projects. Nina Otter is helping organize this. This phase will culminate in a few days of sessions at the Lorentz Center, which come right before the actual conference.

At the conference there will be speakers roughly like this:

• Samson Abramsky
• John Baez
• Mikhail Gromov
• Kathryn Hess
• Jean Krivine
• Tom Leinster

This list is likely to change a bit, so please don't spread it around. (Gromov is such a bigshot that I'll be amazed if he comes, but it would be cool.)

I hope all of you apply when the application opens up, though I can't promise all of you will be accepted. I'll be running one of the seminars, but for my grad students this will be a great chance to learn from someone else and - if you're well-organized - even write a paper. It should be a wonderful experience: I don't think there's ever been a conference that covered such wide-ranging applications of category theory.

Now I'm supposed to apply for an NSF grant to help pay for US grad students to attend this conference. Brendan should keep pestering me to actually do this.

There's a bit more information on the conference below. I imagine some of this is tentative.

Comment Source:17 August 2017: 1) Kenny's first paper, <a href = "http://www.tac.mta.ca/tac/volumes/32/29/32-29.pdf">A bicategory of decorated cospans</a>, has been published by <i>Theory and Applications of Categories</i>. He put a lot of work into this! It looks great in its final form. 2) Nina Otter is visiting me at the Centre for Quantum Technologies - we got some money from the center for her to do this. We're trying to write a quick, short paper on the relation between persistent homology and magnitude homology. 3) I wrote a blog article that starts explaining how our project with Metron is using operads to design networks: * <a href = "https://johncarlosbaez.wordpress.com/2017/08/17/complex-adaptive-system-design-part-3/">Complex adaptive system design (part 3)</a>. I'll need to write more to really explain the idea! DARPA is having a seminar on operads on September 14th and they may use some of these blog articles as reading material. 4) And now for the really big news: there will be a workshop on Applied Category Theory at the <a href = "http://www.lorentzcenter.nl/">Lorentz Center</a> in Leiden, the Netherlands, April 23 to May 4, 2018. This is being organized by Bob Coecke, Aleks Kissinger, Martha Lewis and Josh Tan... and our very own Brendan Fong! They've put together an ambitious program where grad students will join online seminars in January and start doing projects. Nina Otter is helping organize this. This phase will culminate in a few days of sessions at the Lorentz Center, which come right before the actual conference. At the conference there will be speakers roughly like this: * Samson Abramsky * John Baez * Mikhail Gromov * Kathryn Hess * Jean Krivine * Tom Leinster * Nicoletta Sabadini * Mehrnoosh Sadrzadeh This list is likely to change a bit, so please don't spread it around. (Gromov is such a bigshot that I'll be amazed if he comes, but it would be cool.) I hope all of you apply when the application opens up, though I can't promise all of you will be accepted. I'll be running one of the seminars, but for my grad students this will be a great chance to learn from someone else and - if you're well-organized - even write a paper. It should be a wonderful experience: I don't think there's ever been a conference that covered such wide-ranging applications of category theory. Now I'm supposed to apply for an NSF grant to help pay for US grad students to attend this conference. Brendan should keep pestering me to actually do this. There's a bit more information on the conference below. I imagine some of this is tentative.
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40.

26 August 2017:

Here is this week's progress:

1) Jason Erbele submitted an abstract for the Nov. 4-5 special session on applied category theory at UCR:

• Controllability and observability: diagrams and duality

Abstract. Diagrams of systems appear in many different fields of study, and for good reason: they can dramatically simplify communication of and calculations with those systems. In many cases, large diagrams can be viewed as coming from piecing together smaller diagrams in ways that preserve important data, and complicated diagrams can be rewritten to produce simpler diagrams that represent the same behavior. Category theory provides a framework to reason with diagrams as mathematical objects that can be composed and transformed by rewrite rules. In particular, for linear, time independent control systems, the dual notions of controllability and observability can be expressed in terms of a the dual notions of epimorphism and monomorphism, as applied to certain composite diagrams. (Received August 25, 2017)

2) Nina Otter put a new paper on the arXiv with Hal Schenk and her two advisors, the mathematical biologist Heather Harrington and the homotopy theorist Ulrike Tillmann:

Abstract. A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

3) Joseph Moeller has been visiting Metron Scientific Solutions in Reston since last Friday. He's talking to John Foley about our project, for example the use of "graphic monoids" for describing networks of commitments among a collection of agents.

Graphic monoids were introduced by Lawvere. These are monoids obeying the identity xyx = xy, which means that if you try to commit to x, then to y, then to x it's the same as trying to commit to x and then trying to commit to y. You may or may not be able to make these commitments, which is why I say "try". But if you succeed in committing to x the first time, committing to it a second time doesn't change anything, even if you've made some other commitments in the meantime!

I'm hoping Joseph will get pulled into the more applied aspects of our project when he's there, since the cutting edge right now is using our math to get things done. But the applied aspects, done right, will probably involve a lot of brand new pure math, like what Joseph and John are doing right now with graphic monoids.

4) I wrote another blog article about our Metron project:

This is about the simplest example of a "network operad" and its simplest algebra.

Comment Source:26 August 2017: Here is this week's progress: 1) Jason Erbele submitted an abstract for the Nov. 4-5 special session on applied category theory at UCR: * Controllability and observability: diagrams and duality > <b>Abstract.</b> Diagrams of systems appear in many different fields of study, and for good reason: they can dramatically simplify communication of and calculations with those systems. In many cases, large diagrams can be viewed as coming from piecing together smaller diagrams in ways that preserve important data, and complicated diagrams can be rewritten to produce simpler diagrams that represent the same behavior. Category theory provides a framework to reason with diagrams as mathematical objects that can be composed and transformed by rewrite rules. In particular, for linear, time independent control systems, the dual notions of controllability and observability can be expressed in terms of a the dual notions of epimorphism and monomorphism, as applied to certain composite diagrams. (Received August 25, 2017) 2) Nina Otter put a new paper on the arXiv with Hal Schenk and her two advisors, the mathematical biologist Heather Harrington and the homotopy theorist Ulrike Tillmann: * <a href = "https://arxiv.org/abs/1708.07390">Stratifying multiparameter persistent homology</a> > **Abstract.** A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case. 3) Joseph Moeller has been visiting Metron Scientific Solutions in Reston since last Friday. He's talking to John Foley about our project, for example the use of "graphic monoids" for describing networks of commitments among a collection of agents. <a href = "https://ncatlab.org/nlab/show/graphic+category">Graphic monoids</a> were introduced by Lawvere. These are monoids obeying the identity xyx = xy, which means that if you try to commit to x, then to y, then to x it's the same as trying to commit to x and then trying to commit to y. You may or may not be able to make these commitments, which is why I say "try". But if you succeed in committing to x the first time, committing to it a second time doesn't change anything, even if you've made some other commitments in the meantime! I'm hoping Joseph will get pulled into the more applied aspects of our project when he's there, since the cutting edge right now is using our math to get things done. But the applied aspects, done right, will probably involve a lot of brand new pure math, like what Joseph and John are doing right now with graphic monoids. 4) I wrote another blog article about our Metron project: * <a href = "https://johncarlosbaez.wordpress.com/2017/08/22/complex-adaptive-system-design-part-4/">Complex adaptive system design (part 4)</a> This is about the simplest example of a "network operad" and its simplest algebra.
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41.
edited June 2018

1 September 2017:

Here is this week's progress, as far as I know:

1) I made some final changes to a paper with Brandon Coya and Franciscus Rebro, Props in network theory, and we submitted it to Theory and Application of Categories.

The hardest part was proving a result Jason Erbele wanted for his thesis: if we have a prop P presented by some generators and relations, and we add some extra generators, we get a new prop P' having P as a sub-prop.

There's clearly a morphism P → P'. The challenge is to show it's monic. This seems obvious, but I needed Todd Trimble to help me prove it, and eventually Charles Rezk explained why the proof is subtle: while this result is true for props, it's not true for all algebraic gadgets described by multi-sorted Lawvere theories.

Yes, sometimes adding an extra generator can make an algebraic gadget "collapse" and become smaller in some ways! This doesn't happen for the familiar gadgets that have just one sort of element, like groups or rings, so it seems bizarre, but Rezk came up with a cool example of what can go wrong when we have more sorts.

The story is tersely told in Appendix A.2 of our paper, but you can see the whole saga in this discussion on the n-Café:

Moral: it's good to solve problems in public, on blogs. More brains make things go faster, it's more fun, and people get to know you, which is essential for getting jobs. If you have questions you want help with, write a blog article and I'll post it.

2) Brendan Fong visited Blake Pollard at Princeton, where Blake is interning with Arquimedes Canedo at the engineering firm Siemens. Brendan gave a talk. Blake wrote a nice description of what happened, but the most important part for you grad students is:

Canedo wants to hire more interns in the future: ideally people who know some applied category theory and can do some scientific programming! This could be YOU.

The word "Princeton" looks really good on your CV - believe me. Blake writes:

Brendan's visit and accompanying talk went really well. Afterwards (over beers), Arqui proposed that the Siemens Princeton office become the first corporate 'Applied Category Theory Hub.' He is also interested in coming to the Netherlands workshop next summer, or at least to contribute some new problems from the Siemens domain.

It was great to get Brendan, Dimitris, and myself together in a room with Arqui to discuss the potential role that applied category theory could play in helping Siemens realize this vision of a kind of universal knowledge base' for their engineering and manufacturing processes and systems.

It is clear that their vision is still a ways out, but in addition to trying to formalize such a knowledge base, riding the current wave of interest in applied category theory might help the Siemens folk continue to secure both internal and government funding to realize such a vision. Apparently there is a long-time Siemens guy in Germany who is also interested in exploring potential applications of category theory.

I think we've done a good job helping shift the perceived benefits of category theory from 'provides magic wand' to 'provides new, potentially unifying perspective.'

Some good analogies were thrown around with respect to the question: What is category theory buying us?

This is like asking: What has set theory done for us?

Or if you headed back and showed calculus to the folks designing and building the Roman aqueducts, they'd most likely respond, 'So what? Check out those aqueducts.'

Brendan's talk went really well. I convinced him to include his result translating the condition of controllability for linear time invariant systems to a diagrammatic condition and I think people liked that a lot. Feeling that you understand something more intuitively by looking at pictures is a big initial draw for category theory.

Afterwards we all got to chat a bit about categories and where exactly they might be useful.

I like the perspective that we're trying to find the analogue of the functional programming style in engineering.

Arqui hasn't had any 'ah ha' moments yet, but he's a dreamer, he sees the potential, and now he understands better the work required to get to a more realistic vision of how categories can help corporate technology. I really hope we can get beyond 'category theory as a way to secure DARPA funds.'

Arqui wants more interns in the future exploring this stuff. So tell the gang! Ideally they should know basic scientific programming for prototyping ideas. I'm trying to provide evidence that folks with my kind of background are the ones you want around, not just to dream about categories, but also to push forward more practical research projects which are only tangentially related (hence my current interlude with machine learning). I'm giving a talk tomorrow about my work so far on my internship as kind of an interview for a full-time position.

I invited Jason to give a talk next week since he'll be in Philly. The Siemens gang is up for a weekly category theory seminar!

Also Brendan and I revived the idea of a Banff Oaxaca workshop 'Sticking things together in Mexico.' I think the applications are due in late September.

I forgot about that workshop. I'd be happy to help a bit with applying, though I'm getting insanely busy.

Finally, some non-progress. Last week Lisa and I took a vacation in Ubud, the "cultural capital" of Bali. We listened to music:

and much more. We'd like to rent a house and spend a month there sometime.

Comment Source:1 September 2017: Here is this week's progress, as far as I know: 1) I made some final changes to a paper with Brandon Coya and Franciscus Rebro, <a href = "https://arxiv.org/abs/1707.08321">Props in network theory</a>, and we submitted it to <i>Theory and Application of Categories</i>. The hardest part was proving a result Jason Erbele wanted for his thesis: if we have a prop P presented by some generators and relations, and we add some extra generators, we get a new prop P' having P as a sub-prop. There's clearly a morphism P &rarr; P'. The challenge is to show it's monic. This seems obvious, but I needed Todd Trimble to help me prove it, and eventually Charles Rezk explained why the proof is subtle: while this result is true for props, it's not true for all algebraic gadgets described by multi-sorted Lawvere theories. Yes, sometimes adding an extra generator can make an algebraic gadget "collapse" and become smaller in some ways! This doesn't happen for the familiar gadgets that have just one sort of element, like groups or rings, so it seems bizarre, but Rezk came up with a cool example of what can go wrong when we have more sorts. The story is tersely told in Appendix A.2 of our paper, but you can see the whole saga in this discussion on the n-Caf&eacute;: * <a href = "https://golem.ph.utexas.edu/category/2017/08/a_puzzle_on_multisorted_lawver.html">A puzzle on multi-sorted Lawvere theories</a> <b>Moral: it's good to solve problems in public, on blogs.</b> More brains make things go faster, it's more fun, and people get to know you, which is essential for getting jobs. If you have questions you want help with, write a blog article and I'll post it. 2) Brendan Fong visited Blake Pollard at Princeton, where Blake is interning with Arquimedes Canedo at the engineering firm Siemens. Brendan gave a talk. Blake wrote a nice description of what happened, but the most important part for you grad students is: <b>Canedo wants to hire more interns in the future: ideally people who know some applied category theory and can do some scientific programming! This could be YOU.</b> The word "Princeton" looks really good on your CV - believe me. Blake writes: > Brendan's visit and accompanying talk went really well. Afterwards (over beers), Arqui proposed that the Siemens Princeton office become the first corporate 'Applied Category Theory Hub.' He is also interested in coming to the Netherlands workshop next summer, or at least to contribute some new problems from the Siemens domain. > It was great to get Brendan, Dimitris, and myself together in a room with Arqui to discuss the potential role that applied category theory could play in helping Siemens realize this vision of a kind of universal knowledge base' for their engineering and manufacturing processes and systems. > It is clear that their vision is still a ways out, but in addition to trying to formalize such a knowledge base, riding the current wave of interest in applied category theory might help the Siemens folk continue to secure both internal and government funding to realize such a vision. Apparently there is a long-time Siemens guy in Germany who is also interested in exploring potential applications of category theory. > I think we've done a good job helping shift the perceived benefits of category theory from 'provides magic wand' to 'provides new, potentially unifying perspective.' > Some good analogies were thrown around with respect to the question: What is category theory buying us? > This is like asking: What has set theory done for us? > Or if you headed back and showed calculus to the folks designing and building the Roman aqueducts, they'd most likely respond, 'So what? Check out those aqueducts.' > Brendan's talk went really well. I convinced him to include his result translating the condition of controllability for linear time invariant systems to a diagrammatic condition and I think people liked that a lot. Feeling that you understand something more intuitively by looking at pictures is a big initial draw for category theory. > Afterwards we all got to chat a bit about categories and where exactly they might be useful. > I like the perspective that we're trying to find the analogue of the functional programming style in engineering. > Arqui hasn't had any 'ah ha' moments yet, but he's a dreamer, he sees the potential, and now he understands better the work required to get to a more realistic vision of how categories can help corporate technology. I really hope we can get beyond 'category theory as a way to secure DARPA funds.' > Arqui wants more interns in the future exploring this stuff. So tell the gang! Ideally they should know basic scientific programming for prototyping ideas. I'm trying to provide evidence that folks with my kind of background are the ones you want around, not just to dream about categories, but also to push forward more practical research projects which are only tangentially related (hence my current interlude with machine learning). I'm giving a talk tomorrow about my work so far on my internship as kind of an interview for a full-time position. > I invited Jason to give a talk next week since he'll be in Philly. The Siemens gang is up for a weekly category theory seminar! > Also Brendan and I revived the idea of a Banff Oaxaca workshop 'Sticking things together in Mexico.' I think the applications are due in late September. I forgot about that workshop. I'd be happy to help a bit with applying, though I'm getting insanely busy. Finally, some non-progress. Last week Lisa and I took a vacation in Ubud, the "cultural capital" of Bali. We listened to music: <center><img width = "600" src = "http://math.ucr.edu/home/baez/diary/bali/drummer_barong_dance_in_batu_balan_small.jpg"></center> hiked through rice paddies: <center><img width = "600" src = "http://math.ucr.edu/home/baez/diary/bali/ubud_rice_fields_small.jpg"></center> and <a href = "http://math.ucr.edu/home/baez/diary/september_2017.html">much more</a>. We'd like to rent a house and spend a month there sometime.
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42.

Reading that Siemens has an outpost at Princeton and might be into ACT I thought I'd mention Schneider Electric which is a European leader in energy and environmental control. People might like to look at some of their circuits for practical ideas.

Comment Source:Reading that Siemens has an outpost at Princeton and might be into ACT I thought I'd mention [Schneider Electric](https://www.schneider-electric.co.uk/en/) which is a European leader in energy and environmental control. People might like to look at some of their circuits for practical ideas.
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It would be great if someone at Schneider Electric were willing to talk. Usually it's just one or two people at a firm who are interested in applied category theory... but that's all it takes for something to happen! Arquimedes Canedo not only gave my student Blake a summer internship at Siemens, he's also looking for more math students to work on applied category theory as interns there. I haven't managed to get any of my students to do it this summer, mainly because I found this out so late.

Comment Source:It would be great if someone at Schneider Electric were willing to talk. Usually it's just one or two people at a firm who are interested in applied category theory... but that's all it takes for something to happen! Arquimedes Canedo not only gave my student Blake a summer internship at Siemens, he's also looking for more math students to work on applied category theory as interns there. I haven't managed to get any of my students to do it this summer, mainly because I found this out so late.
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"It would be great if someone at Schneider Electric were willing to talk." That's what I thought so I'll try and find if there's somebody there who might be interested.

Comment Source:"It would be great if someone at Schneider Electric were willing to talk." That's what I thought so I'll try and find if there's somebody there who might be interested.
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45.

Jim what due you mean with ACT? The https://en.wikipedia.org/wiki/ACT_(NASDAQ) ?

yes schneider electric seems to have automated reporting tools that is on:

they write:

The Schneider Electric share price and the CAC 40 index are 15 minutes delayed and are automatically updated during the trading day from 9:00 am to 5:30 pm, (Paris time). Note that we are not responsible for any typing or transmission errors. Source : SYMEX ECONOMICS

or do you mean with ACT something completely different? Maybe: https://en.wikipedia.org/wiki/Advanced_Concepts_Team ?

Schneider Electric has a lot of products, as it seems so to say from A (-ct(Nasdaq)) to Z-ealandian dairy that is on the front page they write:

To go from local milk co-op to global industry giant is a bold idea. Fonterra redefined New Zealand's milk industry and became one of the most efficient dairy plants in the world, optimizing EFFICIENCY with Schneider EcoStruxure™ Plant.

in Dahlem Dorf they have an Agroculture Museum they may be interested in new dairy tech, but John?

Comment Source:Jim what due you mean with ACT? The https://en.wikipedia.org/wiki/ACT_(NASDAQ) ? yes schneider electric seems to have automated reporting tools that is on: https://www.schneider-electric.com/en/about-us/investor-relations/share-information/share-price.jsp they write: >The Schneider Electric share price and the CAC 40 index are 15 minutes delayed and are automatically updated during the trading day from 9:00 am to 5:30 pm, (Paris time). Note that we are not responsible for any typing or transmission errors. Source : SYMEX ECONOMICS or do you mean with ACT something completely different? Maybe: https://en.wikipedia.org/wiki/Advanced_Concepts_Team ? Schneider Electric has a lot of products, as it seems so to say from A (-ct(Nasdaq)) to Z-ealandian dairy that is on the front page they write: >To go from local milk co-op to global industry giant is a bold idea. Fonterra redefined New Zealand's milk industry and became one of the most efficient dairy plants in the world, optimizing EFFICIENCY with Schneider EcoStruxure™ Plant. in <a href="https://www.google.com/maps/@52.4519219,13.2860502,17z">Dahlem Dorf </a> they have an <a href="https://en.wikipedia.org/wiki/Dahlem_Manor">Agroculture Museum</a> they may be interested in new dairy tech, but John?
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46.

@nad Sorry to send you up the garden of the forking paths*

ACT = Applied Category theory.

Schneider's work is interesting, isn't it?

• (Borgesian p-value fishing)
Comment Source:@nad Sorry to send you up the garden of the forking paths* ACT = Applied Category theory. Schneider's work is interesting, isn't it? * (Borgesian p-value fishing)
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47.

7 September 2017:

Two bits of progress:

1) Brendan Fong has arrived at MIT and is starting to enjoy the 3:30 math department teas, and Jacob Lurie's weekly seminar on higher categories and homotopy theory.

2) There's a postdoc for someone working in applied category theory! I strongly urge Jason to apply for this, because it looks exactly like the stuff he does. So, Jason, even if you're hoping to get the U. Penn job, you should apply for this too. I told the guy who emailed me, Spencer Breiner, that you would be good for this position.

Dr. Baez,

Hello. My name is Spencer Breiner, and I am a researcher at the US National Institute for Standards and Technology.

I am writing because my colleague, Eswaran Subrahmanian, has recently obtained funding for a one-year postdoc in applied category theory. We are currently searching for candidates, and I am hoping that you would be willing to post an advertisement on the n-Category Café and/or your Azimuth blog. The project itself will draw from your own work with Brendan Fong on passive linear networks.

An announcement for the position is listed below. If you have any question or requests please let me know and I will be happy to provide additional information.

Spencer Breiner

NIST

One Year Postdoc Position at Carnegie Mellon/NIST

We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant (EAGER) from the NSF's Systems Science program. The position will be managed through Carnegie Mellon University [PI: Eswaran Subrahmanian], but the position itself will be located at the US National Institute for Standards and Technology (NIST), located in Gaithersburg, MD outside of Washington, DC.

The project aims to develop a compositional semantics for electrical networks which is suitable for system prediction, analysis and control. This work will extend existing methods for linear circuits (featured on this blog!) to include (i) probabilistic estimates of future consumption and (ii) top-down incentives for load management. We will model a multi-layered system of such distributed energy resources'' including loads and generators (e.g., solar array vs. power plant), different types of resource aggregation (e.g., apartment to apartment building), and across several time scales. We hope to demonstrate that such a system can balance local load and generation in order to minimize expected instability at higher levels of the electrical grid.

This post is available full-time (40 hours/5 days per week) for 12 months, and can begin as early as October 1st.

Comment Source:7 September 2017: Two bits of progress: 1) Brendan Fong has arrived at MIT and is starting to enjoy the 3:30 math department teas, and Jacob Lurie's weekly seminar on higher categories and homotopy theory. 2) There's a postdoc for someone working in applied category theory! I strongly urge Jason to apply for this, because it looks exactly like the stuff he does. So, Jason, even if you're hoping to get the U. Penn job, you should apply for this too. I told the guy who emailed me, Spencer Breiner, that you would be good for this position. Here's the email I got about this: > Dr. Baez, > Hello. My name is Spencer Breiner, and I am a researcher at the US National Institute for Standards and Technology. > I am writing because my colleague, Eswaran Subrahmanian, has recently obtained funding for a one-year postdoc in applied category theory. We are currently searching for candidates, and I am hoping that you would be willing to post an advertisement on the n-Category Café and/or your Azimuth blog. The project itself will draw from your own work with Brendan Fong on passive linear networks. > An announcement for the position is listed below. If you have any question or requests please let me know and I will be happy to provide additional information. > Thanks for your help, > Spencer Breiner > NIST &nbsp; > One Year Postdoc Position at Carnegie Mellon/NIST > We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant (EAGER) from the NSF's Systems Science program. The position will be managed through Carnegie Mellon University [PI: Eswaran Subrahmanian], but the position itself will be located at the US National Institute for Standards and Technology (NIST), located in Gaithersburg, MD outside of Washington, DC. > The project aims to develop a compositional semantics for electrical networks which is suitable for system prediction, analysis and control. This work will extend existing methods for linear circuits (featured on this blog!) to include (i) probabilistic estimates of future consumption and (ii) top-down incentives for load management. We will model a multi-layered system of such distributed energy resources'' including loads and generators (e.g., solar array vs. power plant), different types of resource aggregation (e.g., apartment to apartment building), and across several time scales. We hope to demonstrate that such a system can balance local load and generation in order to minimize expected instability at higher levels of the electrical grid. > This post is available full-time (40 hours/5 days per week) for 12 months, and can begin as early as October 1st. > For more information on this position, please contact Dr. Eswaran Subrahmanian (sub@cmu.edu) or Dr. Spencer Breiner (spencer.breiner@nist.gov).
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48.
edited August 2018

11 September 2017:

1) You can now see the webpage for the workshop and "summer school" that Brendan and others are running on Applied Category Theory next April in the Netherlands.

Click the link to read more! You grad students should apply for the summer school now. If the math department were serving pizza from 12 to 1 you would not show up at 12:50. Apply the same level of intelligence to this situation.

Accompanying the workshop will be a 4-day summer school for a limited number of early-career researchers, also at the Lorentz Center from April 23 to April 27, 2018, as well as a 16-week series of online seminars for up to 16 PhD students and postdocs called the Kan Extension Lab. Applications are due November 1, and admissions will be notified on November 15.

While attendance at the summer school is not required to attend the online seminar, or vice versa, our intention is for participants to attend both. Participants will have the opportunity to work with established mentors in the field, and will have the opportunity to present their research at the full workshop. Applications are due on November 1.

As I mentioned before, I'll be one of those "established mentors" - but if you get in you should work with someone else, to expand your scope.

2) It hasn't happened yet, but I'll be flying back from Singapore when it does, so:

On Friday, John Foley and Tom Mifflin will give a presentation to some DARPA bigshots about our work on operads. With luck, this will get the more interested in using modern mathematics to design complex networked systems. Here is John's outline, just so you get the idea:

A) Tom Mifflin: Why operads? (5 min)

B) John Foley: An introduction to operads and their algebras (10 min)

C) Examples: point-to-point communications (15 min)

a. An operad based on simple graphs

b. An algebra to compose simple graphs

c. An algebra to compose entities with location

d. An algebra for range-limited communications

D) Working at multiple levels with operads (10 min)

a. Layers of abstraction: granularity and detail

b. Formalizing a tool chain to construct operads and compose networks

c. Example of the potential of operads: the recognition principle (time permitting)

E) Conclusion: An R&D path provided by operads (3 min)

a. Match a composition formalism to your domain

b. Leverage the formalism to automate 'recipe authorship’

c. Exploit composition and attempt to integrate with optimization

Comment Source:11 September 2017: 1) You can now see the webpage for the workshop and "summer school" that Brendan and others are running on <a href = "http://www.appliedcategorytheory.org/">Applied Category Theory</a> next April in the Netherlands. Click the link to read more! You grad students should apply for the summer school now. If the math department were serving pizza from 12 to 1 you would not show up at 12:50. Apply the same level of intelligence to this situation. > Accompanying the workshop will be a 4-day summer school for a limited number of early-career researchers, also at the Lorentz Center from April 23 to April 27, 2018, as well as a 16-week series of online seminars for up to 16 PhD students and postdocs called the Kan Extension Lab. Applications are due November 1, and admissions will be notified on November 15. > While attendance at the summer school is not required to attend the online seminar, or vice versa, our intention is for participants to attend both. Participants will have the opportunity to work with established mentors in the field, and will have the opportunity to present their research at the full workshop. Applications are due on November 1. As I mentioned before, I'll be one of those "established mentors" - but if you get in you should work with someone else, to expand your scope. 2) It hasn't happened yet, but I'll be flying back from Singapore when it does, so: On Friday, John Foley and Tom Mifflin will give a presentation to some DARPA bigshots about our work on operads. With luck, this will get the more interested in using modern mathematics to design complex networked systems. Here is John's outline, just so you get the idea: A) Tom Mifflin: Why operads? (5 min) B) John Foley: An introduction to operads and their algebras (10 min) a. Operads b. Algebras of an operad C) Examples: point-to-point communications (15 min) a. An operad based on simple graphs b. An algebra to compose simple graphs c. An algebra to compose entities with location d. An algebra for range-limited communications D) Working at multiple levels with operads (10 min) a. Layers of abstraction: granularity and detail b. Formalizing a tool chain to construct operads and compose networks c. Example of the potential of operads: the recognition principle (time permitting) E) Conclusion: An R&D path provided by operads (3 min) a. Match a composition formalism to your domain b. Leverage the formalism to automate 'recipe authorship’ c. Exploit composition and attempt to integrate with optimization
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49.
edited June 2018

13 September 2017:

I've finalized the schedule for our Applied Category Theory special session on the weekend of November 4th and 5th. See below for when your talk will take place! Unless your name is Brendan Fong or David Spivak, your talk is 20 minutes long.

We got a couple more talks, including one from Peter Gates on databases built using algebraic theories. Peter works for a company with the delightful name Categorical Informatics, which David Spivak is also involved with.

Brendan, David, Dmitry and perhaps others will be staying on until Monday, Tuesday or Wednesday and talking with us. So, be prepared for lots of interesting conversations!

Special Session SS 4A, Applied Category Theory

Saturday November 4, AM:

9:00-9:40am: David Spivak, 1134-18-32

10:00-10:20am: Dmitry Vagner, 1134-18-21

10:30-10:50am: Christina Vasilakopoulou, 1134-18-34

Saturday, PM:

3:00-3:40pm: Brendan Fong, 1134-18-27

4:00-4:20pm: Blake Pollard, 1134-18-37

4:30-4:50pm: Kenny Courser, 1134-18-30

5:00-5:20pm: Daniel Michael Cicala, 1134-18-26

Sunday November 5, AM:

9:00-9:20am: Jason Erbele, 1134-93-103

10:30-10:50 am: Brandon Coya, 1134-18-38

10:00-10:20 am: John Foley, 1134-18-182

10:30-10:50 am: Joseph Moeller, 1134-94-36

Sunday, PM:

2:00-2:40 pm: Vin de Silva, 1134-18-122

3:00-3:20 pm: Evan Patterson, 1134-18-35

3:30-3:50 pm: David P. Ellerman, 1134-94-9

4:00-4:20pm: Ralph L. Wojtowicz, 1134-60-11

4:30-4:40pm: Peter Y. Gates, 1134-68-336

Comment Source:13 September 2017: I've finalized the schedule for our Applied Category Theory special session on the weekend of November 4th and 5th. See below for when your talk will take place! Unless your name is Brendan Fong or David Spivak, your talk is 20 minutes long. We got a couple more talks, including one from Peter Gates on databases built using algebraic theories. Peter works for a company with the delightful name Categorical Informatics, which David Spivak is also involved with. Brendan, David, Dmitry and perhaps others will be staying on until Monday, Tuesday or Wednesday and talking with us. So, be prepared for lots of interesting conversations! Special Session SS 4A, Applied Category Theory Saturday November 4, AM: 9:00-9:40am: David Spivak, 1134-18-32 10:00-10:20am: Dmitry Vagner, 1134-18-21 10:30-10:50am: Christina Vasilakopoulou, 1134-18-34 Saturday, PM: 3:00-3:40pm: Brendan Fong, 1134-18-27 4:00-4:20pm: Blake Pollard, 1134-18-37 4:30-4:50pm: Kenny Courser, 1134-18-30 5:00-5:20pm: Daniel Michael Cicala, 1134-18-26 5:30-5:50pm: Adam Yassine, 1134-18-31 Sunday November 5, AM: 9:00-9:20am: Jason Erbele, 1134-93-103 10:30-10:50 am: Brandon Coya, 1134-18-38 10:00-10:20 am: John Foley, 1134-18-182 10:30-10:50 am: Joseph Moeller, 1134-94-36 Sunday, PM: 2:00-2:40 pm: Vin de Silva, 1134-18-122 3:00-3:20 pm: Evan Patterson, 1134-18-35 3:30-3:50 pm: David P. Ellerman, 1134-94-9 4:00-4:20pm: Ralph L. Wojtowicz, 1134-60-11 4:30-4:40pm: Peter Y. Gates, 1134-68-336 
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21 September 2017:

1) Blake finished his thesis and turned it in last Friday!

2) Blake got two offers of postdoc positions! One is from Spencer Breiner at the National Institute of Standards in Gaithersburg Maryland, and one at Siemens - I guess at Princeton? But he may instead go surfing in Peru.

3) John Foley's talk to the DARPA bigshots went well! DARPA may start up more projects connected to operads and categories.

4) Daniel and Kenny finished a paper, put it on the arXiv, and submitted it to Susan Niefield for publication in Theory and Applications of Categories!

Abstract. For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph).

5) Here's what I wrote on Azimuth about our special session on category theory - you can see people's abstracts by clicking on their talk titles. I hope you're all getting ready for this big pow-wow!

Comment Source:21 September 2017: 1) Blake finished his thesis and turned it in last Friday! 2) Blake got two offers of postdoc positions! One is from Spencer Breiner at the National Institute of Standards in Gaithersburg Maryland, and one at Siemens - I guess at Princeton? But he may instead go surfing in Peru. 3) John Foley's talk to the DARPA bigshots went well! DARPA may start up more projects connected to operads and categories. 4) Daniel and Kenny finished a paper, put it on the arXiv, and submitted it to Susan Niefield for publication in _Theory and Applications of Categories_! * [Spans of cospans in a topos](https://arxiv.org/abs/1707.02098) > **Abstract.** For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph). 5) Here's what I wrote on Azimuth about our special session on category theory - you can see people's abstracts by clicking on their talk titles. I hope you're all getting ready for this big pow-wow! * [Applied Category Theory at UCR (Part 2)](https://johncarlosbaez.wordpress.com/2017/09/21/applied-category-theory-at-ucr-part-2/)