My name is Emily Pillmore, and I am a Haskell Programmer by trade, recent alumnus of the ACT Adjoint 2019 school (focus: profunctor optics w/ Bartosz), and algebraic topology enthusiast.

I am currently working towards applying to Ph.D programs for the 2021 Fall Semester, and I've been very active in the NYC category theory community hosting/attending meetups both independently and through CUNY's Graduate Category Theory Seminar. We are reading Lambek's Introduction To Higher Categorical Logic, so hit me up if you're interested!

My categorical interests are mainly focused on topics in Homotopy Theory (HoTT, Homotopical Algebra, etc) and Topos theory, but I spend alot of my time in Computer Science land, so models of the lambda calculus, parametricity, and other FP-related concepts are constantly in view. I'm currently working on understanding fibrational models for System F and proof-relevant parametricity. You can approach me with any Haskell/optics questions you might have. I'd be happy to answer :)

As part of the ACT Adjoint 2019 school, we were able to produce the following:

- https://golem.ph.utexas.edu/category/2020/01/profunctor_optics_the_categori.html
- https://arxiv.org/pdf/2001.07488.pdf

Cheers, and I hope to learn lots with you all!

Emily

]]>I'm finishing a masters in computer science with a thesis in few-shot learning. I'm glad Bartosz, David and Brendan joined to make this possible, It's a great follow up on the first course by Bartosz (which I took).

My main interests are machine learning, philosophy of science and philosophy of mathematics. I found in category theory a sufficiently expressive language to make map common practices in different scientific fields, such as abstraction, inference, modeling, experimentation, imaging, etc. Hope to take this research interest forward in the future. Brendan Fong PhD thesis is fascinating!

Greetings from Mexico City.

]]>Though I am still at a very nascent stage in category theory but lots of Scala data type are now making sense.

Hopefully, by the end of the MIT course, I would be able to gather more knowledge with the help of discussion on this forum

]]>I'm also the administrator for the forum.

It's great that people from the Programming with Categories course will be chatting here. Let me know if there is anything I can do to help. I have created a new category for this course.

]]>Though I have used terms from financial accounting and economics, I do not think the costs and benefits of our actions can be reduced to monetary values. For example, I think Gross National Product is often a poor measure of the well-being of a nation. It can be just an indication of corpulence and waste.

As an engineer, I am inclined to convert most costs to energy input, regardless of whether this energy comes from renewable sources. Expending more energy generally means having bigger impacts on the environment, and with great energy expenditure comes great responsibility. Even if we were to find a way of making electricity without a significant environmental impact, the use of this magic resource would inevitably have an impact.

Some of this energy-equivalent accounting can be awkward. The manufacture of electric vehicles consumes mineral resources such as Cobalt for their batteries. We need some way to set this cost against the costs of burning fuel in internal combustion engines, before we say that EVs are so much better than ICEs. An interesting point here the true cost of mining Cobalt ore. Currently, a significant amount of this mining is ultimately done by poor people, including children, using their bare hands and only the crudest tools. This does not cost as much as mechanised mining in terms of direct energy inputs such as fuel, but must surely have a cost in other terms.

]]>My name is Glyn Adgie. I am an electronics design engineer by training and job title, but I frequently deal with other fields of engineering in the course of my work. I have investigated the causes and cures of corrosion, and contributed to the redesign of a product that suffered premature failures due to metal fatigue. I have some experience in software design, as many electronic products have a firmware component these days.

Outside of work, I have many interests, some of which have an ecological component. For example, I am concerned about the environmental costs of agriculture and the food supply chain as a whole. At a personal level, dietary choices and cooking methods come into this. One hopes that the right personal choices, if made by enough people, will change our agricultural and food supply methods for the better, just through economic forces.

I will now put some thought to some more specific discussion topics.

]]>I'm Ian White, and I'm trying to create a new political party based upon the principles of Jainism.

Jainism and Environmental Philosophy, Aidan Rankin's new book, seems to explain well the connection between Jainism and ecology.

I first became aware of John (Baez) by reading his 'Tale of n-Categories' about 18 years ago now, while learning Haskell.

I use the handle 'votejainism' and can be found at various locations on the internet such as twitter, github, vimeo, vk, bitchute, etc. ]]>

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 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

]]>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 :)

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