I started researching quite a bit then, and I found category theory which I just loved. I found out that that's the stuff I'd like to work on in the future. I'm now learning mostly from the videos of David Spivak at LambdaConf and from 'Category Theory' by Steve Awodey.

Since I started learning category theory (pretty recently), I have begun finding categories in really non-mathematical things, and that only made me love Category Theory even more.

I have always been fascinated by mathematics and theoretical computer science, and I'm planning to continue my career as a researcher in these fields. I have founded clubs, and discussion groups and other things like this forum to try to find like-minded people to work with them on Category Theory. And that's what made me join this forum. In fact, I'm planning to change to a double major of mathematics and computer science next year to help push myself into research and academia. In this regard, Category Theory and Homotopy Type Theory are prime candidates for future research fields I could work on because I simply loved them.

Incidentally, it turned out a main focus of this project is environmental protection, which is a thing I'm really really interested in. So that's 2 in 1 :P

I hope that I can get to know people and learn more about category theory from this forum as well as contribute to the community.

]]>I've been trying to apply category theory to a couple topics:

Building a library / example elegantly embedding anyon vector spaces in Haskell - http://www.philipzucker.com/a-touch-of-topological-computation-3-categorical-interlude/

Automatic differentiation - http://www.philipzucker.com/reverse-mode-differentiation-is-kind-of-like-a-lens-ii/

Conal Elliot's compiling to categories - http://www.philipzucker.com/compiling-to-categories-3-a-bit-cuter/

Also I've recently been tinkering with some ideas about how to do something "category"eque with convex programming. Definitely be interested if you've got any tips/ references. https://github.com/philzook58/ConvexCat

]]>My main interests are Physics, Math and CS, so I'm curious of the applications of CT in these fields. However I'm also curious about how could it be used in other less theoretical fields.

]]>I am applying Category Theory to business, specifically in the way we design software. We are using OLOGs and String Diagrams to build graphical representations directly connected to the functional requirements.

Graphics are used by Subject Matter Experts with no understanding of the underlying math or science for the system. Instead they are given a ubiquitous language (developed interactively at the domain level) with which to communicate and collaborate across specific subject areas and specialties. I am applying Conceptual Spaces, Topology, and the ideas from Gardenfors (along with Brouwer, Heyting, and Goldblatt) to bridge the Language semantics of graphics, business processes, and software development. Most of our software is used for internal process communication and Category Theory has proven to be the proper tool for the job.

Adding guidance for how to apply that to business processing and software design is my focus. I am not referring to "functional programming" though that does play a role. I am referring more to the business process and understanding the information science behind the process outside of any applied technology.

]]>I tried reading Conceptual Mathematics: A First Introduction to Categories by Lawvere and Schaunel about a year ago and, while I found the "flight of a bird as a map from time to space" diagram to be very charming, I was ultimately unable to really understand the text. I'm hopeful that this course and forum will be able to guide me (and others!) through Seven Sketches.

I live and work in Brooklyn, where I spend as much of my time as possible seeing live music and drinking with friends. I studied architecture (as in buildings, not software) in undergrad, and it is probably the subject I enjoy talking about the most. If you have an interest, let's talk bricks!

My only other active side project is a visual demo of sorts for the Raft consensus algorithm using a few raspberry pis and some LED matrices to represent state. I just finished up the LED control code in Python and am moving on to Raft itself now. I'm on the fence for language choice, but may give Elixir a go.

Thank you for hosting the course John! Excited to give category theory another crack.

]]>The quantitative version of Boole's logic of subsets is just finite logical probability theory as developed by Boole in his *Laws of Thought*. Rota always held that probability (i.e., the normalized size of a subset) is to subsets as information is to partitions.

$$\frac{\text{Probability}}{\text{Subsets}}=\frac{\text{Information}}{\text{Partitions}}$$

The key to working that out was to see the analogy between an "element of a subset" and a "distinction of a partition," where a *distinction* or *dit* of a partition is an ordered pair of elements of the underlying set \(X\) that are in different blocks or parts of the partition. The *indistinctions* or *indits* of a partition are the ordered pairs of elements in the same block or part of the partition. Hence the set of all indits of a partition is just the binary equivalence relation associated with the partition, and the set of all dits of a partition, the *ditset* \(dit(P)\), is the complementary binary relation called an *apartness relation* or just a *partition relation*. The *logical entropy* \(h(P)\) of a partition is just the normalized size of the ditset:

$$h(P)=\frac{|dit(P)|}{|X^2|}.$$

Logical entropy is related in a definite way to Shannon entropy. All the Shannon definitions of simple, joint, conditional, and mutual entropy can be obtained by a uniform requantifying transformation from the corresponding definitions in logical information theory. Moreover, logical entropy (unlike Shannon entropy) is a measure in the sense of measure theory, and thus the logical definitions of simple, joint, conditional, and mutual logical entropy of partitions is just the measure applied respectively to the ditsets, the union of ditsets, the difference of ditsets, and the intersection of ditsets. Indeed, logical entropy is a probability measure with the interpretation that \(h(P)\) is the probability of getting a distinction in two independent draws from the set \(X\). The basic paper developing logical information theory (and spelling out the comparisons with the closely related Shannon theory) was also published in the Logic Journal of the IGPL.

I went into a little detail about logical information theory as the quantitative version of partition logic since it bears on the question of which of the opposite partial orderings, the coarseness ordering or the refinement ordering, should be used on partitions. Since refinement increases the number of distinctions, it increases logical (and Shannon) entropy so the refinement ordering seems best if one is emphasizing the ordering of partitions according to their information content measured by logical or Shannon entropy.

Another topic of my research has been "heteromorphisms" which are morphisms between the objects of different categories, e.g., the canonical injection of the set of generators into the free group on that set. Homomorphisms may be treated formally by hom-functors, and heteromorphisms are treated formally by the "het-functors" or *profunctors* that appear later in the course. For one reason or another, Mac Lane and Eilenberg did not include heteromorphisms in their basic "ontology" for category theory even though they are just as much a tool for the proverbial "working mathematician" (referenced in the title of Mac Lane's text) as are homomorphisms. The French school of category theory, e.g., the Grothendieck school, is much less prudish and let hets 'out of the closet' as just "morphisms" routinely. The differing attitude about hets comes out in the treatment of adjoint functors. The simpler and I think more natural treatment of adjoints was developed by Bodo Pareigis in the late 60's and published in his text translated into English as: Pareigis, Bodo. 1970. *Categories and Functors*. New York: Academic Press. It also comes out in what one takes as more important or more basic: adjoint functors or representable functors. Grothendieck gives representable functors the pride of place, whereas the American school puts the emphasis on adjoint functors. Hets, representable functors, and adjoints are all tied together in the Pareigis treatment of adjoints. Take the het-functor whose value \(\text{het}(X,G)\) is the set of set-to-group maps from a set \(X\) to a group \(G\). The underlying set functor is just the right-representation of those hets by homs in the category of sets, and the free-group functor is just the left-representation of those *same hets* by homs in the category of groups. Thus one has the natural isomorphisms:

$$ \text{hom}(F(X),G)\cong \text{het}(X,G)\cong \text{hom}(X,U(G)).$$

The heterophobic American school always leaves out the middle term of hets and just defines an adjunction using the natural isomorphism between the hom-sets. All left- and right-representations of a het-functor define a pair of adjoint functors, and given a pair of adjoint functors, there is a het-functor so that the given adjoints are isomorphic to the left- and right-representations of the het-functor. All this holds as well for adjunctions on preorders or Galois connections. An introductory paper is here.

]]>My plan is to work through the 7 sketches book first without focussing on the exercises, because I first want to know the goals and the important waypoints. Then I'd like to do a second pass of going through the paper where I delve into the depths.

]]>Here's some of my background for anyone who is interested. I studied actuarial science in my undergraduate. It's a degree program for people preparing to be actuaries. I took a minor in mathematics and found myself more inclined to math than insurance. Then I pursued MPhil (people in Hong Kong commonly do an MPhil and then PhD) in math, with my shaky background making the process kind of tough. Luckily I met my current supervisor. I do a little bit dessin d'enfant.

]]>I am a graduate student at Kyoto University. My background contains learnings in mathematics and physics, also some working experience in the finance industry.

Category theory has been continually intriguing me, so I am very excited to see it being applied to fields outside of mathematics and wish to find more and deeper involvement in the future.

My interests include category theory itself and my research field, which is applied algebraic topology.

Cheers!

]]>I am a PhD student at Lund University cognitive science in Sweden. From before, I have a master in cognitive science also from Lund, and a master in information systems engineering from Surrey in England.

My PhD project concerns modeling working memory, and I am particularly interested in aspects related to processes of abstraction and concretization.

I have been interested in category theory for a few years, but not really had the time to read up on it until now. I have been making a self-study course out of the 7 sketches book for the last three months, and will make a seminar of it here in a few weeks.

My hope is that I will be able to make use of category theory to think more productively about the computer models I am building, since these models are essentially operads, as far as I can tell. But my understanding is still very coarse, and hence I am hoping to get a better grasp of these concepts by coming here!

]]>I am a postdoc fellow studying the molecular basis of learning and memory in Drosophila at the University of Oxford.

I did my PhD in neurobiology, also studying learning and memory in Drosophila, but then was doing neural circuit stuff. And my undergraduate major is Computer Science.

I did Computer Science because I was pursuing after AI at that time, and later transferred to neuroscience because I started to believe perhaps we should understand the biological intelligence first, in order to create real intelligent machines. But ever since I got involved in real biological things, I realized more and more strongly that uncovering the secret of the genome is my true love (destiny).

Recently I got fascinated in Applied Category Theory because I felt it might be the hope for us to understand the genome as a whole. My math background is weak, and I am trying hard to catch up. I am so much grateful for Brendon and David’s 7S, which actually opens the door of the holy Category Theory to me. Also Prof. Baez’s lectures on 7S, the Catsters’ CT videos, and the new “What is ACT” book by Tai-Danae, etc…So much for me to learn and digest……

Last but most importantly, I found Azimuth an amazing place to take part in, and I will be extremely happy to discuss with you anything related to both category theory and the genes.

Cheer, Yanying

]]>I am a Master student in Mathematics and in Computational Science at the University of Amsterdam. I have a Bachelor in Mathematics, Physics and in Neuroscience.

I started fantasizing about applications of Category Theory in linguistics during my first course on the subject. I was happy to stumble upon the paper "Towards a Categorical Theory of Creativity for Music, Discourse, and Cognition" through which I came in contact with Ehresmann. She gave me some nice research directions how to apply Memory Evolutive Systems to neuroscience. Still working on this. Since, I had been considering different possibilities of applying Category Theory, but so far I am just trying to read up on the existing literature. More recently, I learned about John and the Azimuth project through a poster of the Applied Category Theory workshop in Leiden.

Currently, I am looking for a project for my Master's thesis on Applied Category Theory in Neuroscience.

]]>Several months ago I found Brendan Fong's Master's Thesis and thought to myself: wow, some category theory not just about homologies and other magic things but about something related to my work! Because without mathematical foundations my work is just writing code with quite indecent portion of primitive occultism:) I would like to change that and understand how ML is related to other seemingly abstract parts of math. Because I really believe that math is about relatedness. That's why I love it after all.

I don't think that I'll be especially successful in going through "7 Sketches" and/or these lectures because of spare time shortage. But now I'm reading Chapter 2 already and quite happy solving exercises and understanding not only proofs but also some connections to engineering-friendly applications.

I want to thank John Baez for this course effort, Brendan Fong and David Spivak for their book and all active participants of this forum for useful hints I've already discovered. I also wish good luck to other students:)

]]>Mr. Baez has made the analogy between left-right adjoints and liberal-conservative. Could certain issues in politics be seen as giving an approximation to an inverse, and the political divide comes from the two created opposing adjoints?

For instance, suggesting that a social issue be solved at an individual basis versus being solved at a community level feels very adjointy.

I'm curious what other people have to think.

]]>There are also lots of nice comments that I need to reply to. But first, let me write a lecture.

]]>I am PhD student in plant biology at the University of Georgia, Savannah River Ecology Lab. The mathematics I do is part hobby and part work. I don't have much formal training (in terms of courses), but I have self-studied quite a bit (undoubtedly there are plenty of gaps in my knowledge).

I primarily got interested in category theory through John Baez' papers (and some attempts at programming in Haskell). I saw John's paper Relative entropy in biological systems linked on the /r/math subreddit of reddit. I really liked that paper because it used some interesting math to change the way I conceive of evolution and ecology. From that paper, I was led to papers on using category theory to analyze/formalize diagrams in science (e.g., A compositional framework of passive linear networks). So I started to think I might be able to find applications of category theory to the systems I already study for work.

I think compositionality will be a useful idea in studying complex biological systems, especially ones where it is difficult to measure or control phenomena of interest. By describing how systems (e.g., plant tissues) composed to form bigger systems (e.g., whole plants), we can then deduce global system properties from local ones, or vice-versa. That way data, observations, and hypotheses at different scales (e.g., observations of forests vs. observations of tissues in a lab) can be related in a systematic way.

Anyways, that's how I ended up here.

]]>I'm currently a graduate student at UIC, preparing to apply to PhD programs in mathematics this fall. I plan to study category theory for my PhD, but I am not yet sure what sort of problems to work on.

At the moment I happen to be organizing an informal course in category theory for programmers while studying for the GRE math test over the summer.

]]>I really just miss doing math and so I hope i can use this to do that :)

]]>I look forward to getting to know others here and seeing what we can do as I learn!

I'm a software consultant, mostly working in the realm of scientific software and some other software encountered at the university setting. In graduate school, I studied computational biology, and my dissertation work focused on using convex optimization techniques to simulate a cell, mostly at the metabolic level (I was merely adding on work to this field). As an undergraduate I studied math and computer science - I wish I had studied more diligently at that time ;-).

My reason for joining Azimuth is primarily to take the course on Applied Category Theory; I had gotten most of the way through the first chapter of the version up on arXiv.org (that I incidentally work on as part of my job) when I e-mailed one of the authors and found out about the course on Azimuth. I'm primarily interested in category theory for the abstraction it can help bring to programming. I had heard of it in graduate school, but, I never studied it at the time. I'm also interested in some of the biological and network-oriented aspects of category theory, since I still do some biology (would like to do a bit more if I had the time and the right project(s)). Due to many demands in life currently I will probably be slow in catching up to people here, but I realize it is important to keep working at something or it is all too easy to let more immediate demands always take the place of long-term goals.

As for programming, I've been doing a lot of Scala over the last 3 years and in general have been very happy with it, but I am also a huge fan of ATS (http://www.ats-lang.org/ ) which I used briefly (with much help from the author) in graduate school. I think it may be of interest to some of the software oriented folks here. I've also slowly been learning Haskell and Idris; I'd tried out Haskell over a decade ago, and was very impressed (story for another time), and wish I had taken more opportunity to learn it in the past.

One more thing for now: I realize this site was primarily created for issues that closely relate to weather modeling. I have been working with researchers that use WRF (https://www.mmm.ucar.edu/weather-research-and-forecasting-model) in case that is of interest to anyone. I can't claim to be at all competent with it, as my primary role has been to containerize it. I do not know Fortran, and don't particularly want to learn it ;-). But if this is of interest to anyone here for there modeling, I can perhaps find out something.

]]>I'm a new postdoctoral researcher at Emory University in Atlanta, Georgia. I work in developing computer models of neurons, and I'm especially interested in metasimulation, or the science of integrating simulations and experimentation with emphasis on computational neuroscience. I'm investigating applying category theory to this process and look forward to the discussions here as well as learning more about category theory. I've been reading Mac Lane and watching the Milewski videos on YouTube. Please don't hesitate to drop me a line.

Nice to meet you all!

Dave

]]>I'm James K. Alcala, about to begin my graduate studies in mathematics this fall at UCR. I recently completed a B.S. in pure math from UCR, so I'm happy to be returning to familiar territory. Right now, the topics I enjoy learning are analysis (real and complex) and some applied math, which I think is why I'm here: the more I hear about category theory, why it exists, and how it's being applied today, the more I want to learn about it. A little over a year ago, I worked on a project supervised by Dr. Fernando López-García, now of Cal Poly Pomona, studying the optimal constant of the Poincaré inequality on convex subsets of Euclidean space with respect to the \(L^{2}\) metric. I was able to give a talk in once of Dr. Michel Lapidus' seminars focusing on the proof in the 1D and 2D cases. While my background is primarily mathematical, I also spent a year as an English major who intended to minor in math, so I enjoy critical reading and humanities-type work, as well.

]]>My name is Gien. My background is in systems design of electronic systems. I've been involved in R+D for most of my adult life, as it pertains to electronics, industrial electronics, and wireless , industrial and SCADA communications. However, my interest is no longer in that domain. I'm transferring my systems approach from engineering to working on the wicked problems faced by our species and especially in citizen-based approaches which can be applied locally, but scale globally. The organization I co-founded, Stop Reset Go, is focused on mobilizing citizens for sense-making, developing and piloting rapid whole system change solutions for the planet. Our small collective is working with a growing solidarity network of open source scientists, engineers, academics, ecologists, entrepreneurs, makers, economists, social activists, agroforesters, industrial designers as well as ordinary citizens to develop local circular economic models as well as programs for psychological shift towards a society based upon holistic wellbeing. We have developed a very specific multi-discplinary strategy that we would like to share with this community to get feedback and see how viable it is and if there are synergies. This is a work in progress, but members of our solidarity network are at various stages of activating their specific solutions.

]]>I am developing a language for expressing categories and category families. The specific sub goals are:

Data model [category] migration and integration. A data model is an instance of a schema [which are categories].

Identify/Develop a categorical language using computational data structures. In particular I am doing this in Clojure.

Publish categories This builds on David Spivak's idea of published ologs. https://johncarlosbaez.wordpress.com/2015/03/27/spivak-part-1/ Specifically I have a github repository. https://github.com/babeloff/categories

Investigate visualizers, editors and other interpreters. Too many to call out, but I am currently reevaluating the https://webgme.org/ tooling concepts in categorical terms. The main GME concepts are: containment, inheritance, sets, pointers, class-diagrams, and rules. We have two versions of GME, a C++ .Net desktop version and a javascript browser version. Presently these two versions have similar but distinct data-models.

At the same time, I've been paying attention to the world around me: social, political, ecological and economic systems, and would be looking to try my hand at applying math to these problems (possibly via dynamical systems theory and related ideas) in the hopes of finding new sustainable solutions. This has also lead to studying the history of economic thought. As a software developer, I'm also investigating how dependently-typed programming can also help. As of now, I'm independently studying all this, so I'm very open to collaboration.

]]>I'm an American physicist/mathematician now working in finance. In my academic life I worked mainly on packing problems, such as trying to find the best way to pack regular tetrahedra, trying to figure out if any convex solid packs less efficiently than the ball, and developing methods to search for new and interesting packing arrangements of spheres in higher dimensions. I also found time to work on how to move a sofa around a corner, modeling trade of nutrients between microbes, and looking at the thermodynamics of clusters formed by sticky spherical colloids. I'm now working on quantitative aspects of trading equities, options, futures, and ETFs at a proprietary trading firm.

]]>]]>Those in your category theory course may be interested to know that Samuel Eilenberg was far more than a great mathematician. He was also an expert in antique South Asian and SE Asian bronzes and stone sculpture. This sort of endeavor requires considerable knowledge of iconographies, patenation of metals and alloys, and regional history. He was well known in the Asian art collecting community and had a major collection and considerable following. He was affectionately known amongst South Asian art collectors as Sammy. He is sorely missed by the art world as well as the mathematics community. A link to a free download of a museum catalog representing a portion of his collection follows below.

Best regards,

Robert Schlesinger

Hypergraph categories, decorated cospan categories, decorated corelation categories - lots of nice category theory, all in the service of electrical engineering.

I'm dying to tell you about how to compose enriched profunctors, and I'll do it tomorrow - Thursday! In the meantime you could read about quantales in the book.

]]>This week I've been doing some recreational mathematics with some Greg Egan, Dan Piponi, John D. Cook and other people on Twitter. We've shown things like this:

- the average distance between \(xy\) and \(yx\) for unit quaternions \(x,y\) is

$$ \displaystyle{\frac{32}{9 \pi}} \approx 1.13176848421 \dots $$

- the average distance between \((xy)z\) and \(x(yz)\) for unit octonions \(x,y\) is

$$ \displaystyle{ \frac{2^{14}3^2}{5^3 7^3 \pi} \approx 1.0947335878 \dots } $$

- You can take the unit sphere in \(\mathbb{R}^n\), randomly choose two points on it, and compute their distance. This gives a random variable, whose moments you can calculate. When \(n = 1, 2\) or \(4\), and seemingly in no other cases, all the even moments are
*integers*. However, for every even \(n\), if we compute the first \(k\) even moments, figure out the fraction of them that are integers, and take the limit as \(n \to \infty\), this fraction approaches 1.

You can see the whole story here:

]]>You don't need to know anything about quaternions to do this. Another way to phrase the question is: what's the average distance between two randomly chosen points of the unit sphere in \(\mathbb{R}^4\)?

Yet another way to put it is this: if you sit at the north pole of this sphere, and randomly pick another point on the sphere, what is the expected value of its distance from you?"

(Here and in all that follows, distances are measured using straight lines in \(\mathbb{R}^n\), not using geodesics on the sphere. There is a unique rotation-invariant probability measure on a sphere of any dimension, and that's what we use to randomly choose a point on a sphere.)

Greg computed the answer, which is \(64/15\pi \approx 1.35812218105\). But he went further and computed the whole probability distribution:

I tried to figure out why the answer looked like this. I noticed, that if if you take a unit sphere in any dimension, and you sit at the north pole, the points on the equator are at distance \(\sqrt{2} \approx 1.41421356\). This is pretty close to \(1.35812218105\).

I wrote:

Neat! I'm kind of surprised the mode doesn't occur at $\sqrt(2)$, because if you're at the north pole it'll seem like "most" points are at the equator. Maybe I'm making an elementary mistake, confusing parametrization by angle and by distance.

That was indeed the problem.

But as you go to higher dimensions, points near the equator should become more and more probable (you can see this using a little calculation). So, I predicted that if we repeat this calculation for spheres of different dimensions, the probability distribution should spike more and more strongly near \(\sqrt{2}\).

]]>