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.

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.