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A Categorical Look at Random Variables Posted by Tom Leinster guest post by Mark Meckes For the past several years I’ve been thinking on and off about whether there’s a fruitful category-theoretic perspective on probability theory, or at least a perspective with a category-theoretic flavor. (You can see this MathOverflow question by Pete Clark for some background, though I started thinking about t
In my final year at Harvard and again in my first year at Johns Hopkins, I had an opportunity to teach an advanced undergraduate/beginning graduate-level topics course entitled “Category Theory in Context.” Its aim was to provide a first introduction to the basic concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads,
Roux Cody recently posted an interesting article complaining about FOM — the foundations of mathematics mailing list: Roux Cody, on Foundations of Mathematics (mailing list), 29 March 2016. Cody argued that type theory and especially homotopy type theory don’t get a fair hearing on this list, which focuses on traditional set-theoretic foundations. This will come as no surprise to people who have p
Here are the notes from a basic course on category theory: Category Theory Course - Winter 2016. Unlike my Fall 2015 seminar, this quarter I tried to give a systematic introduction to the subject. However, many proofs (and additional theorems) were offloaded to another more informal seminar, for which notes are not available. So, many proofs here are left as ‘exercises for the reader’. If you disc
guest post by Jamie Vicary When you’re trying to prove something in a monoidal category, or a higher category, string diagrams are a really useful technique, especially when you’re trying to get an intuition for what you’re doing. But when it comes to writing up your results, the problems start to mount. For a complex proof, it’s hard to be sure your result is correct — a slip of the pen could lea
A 2-Categorical Approach to the Pi Calculus Posted by John Baez guest post by Mike Stay Greg Meredith and I have a short paper that’s been accepted for Higher-Dimensional Rewriting and Applications (HDRA) 2015 on modeling the asynchronous polyadic pi calculus with 2-categories. We avoid domain theory entirely and model the operational semantics directly; full abstraction is almost trivial. As a ni
Category Theory in Machine Learning Posted by David Corfield I was asked recently by someone for my opinion on the possibility that category theory might prove useful in machine learning. First of all, I wouldn’t want to give the impression that there are signs of any imminent breakthrough. But let me open a thread by suggesting one or two places where categories may come into play. For other area
My new book is out! Click the image for more information. It’s an introductory category theory text, and I can prove it exists: there’s a copy right in front of me. (You too can purchase a proof.) Is it unique? Maybe. Here are three of its properties: It doesn’t assume much. It sticks to the basics. It’s short. I want to thank the nn-Café patrons who gave me encouragement during my last week of wo
Guest post by Sam van Gool Monads provide a categorical setting for studying sets with additional structure. Similarly, 2-monads provide a 2-categorical setting for studying categories with additional structure. While there is really only one natural notion of algebra morphism in the context of monads, there are several choices of algebra morphism in the context of 2-monads. The interplay between
Guest post by Alexander Campbell We want to develop category theory in a general 2-category, in order to both generalise and clarify our understanding of category theory. The key to this endeavour is to express the basic notions of the theory of categories in a natural 2-categorical language. In this way we are continuing a theme present in previous posts from the Kan Extension Seminar, wherein mo
« The Electronic Frontier Foundation at the Joint Meetings | Main | Hilarious Takedown of Bonkers Maths in Top Psychology Journal » For the last couple years, people interested in quantum gravity have been arguing about the ‘firewall problem’. It’s a thought experiment involving black holes that claims to demonstrate an inconsistency in some widely held assumptions about how quantum mechanics and
Elaine Landry, in the philosophy department at U. C. Davis, is putting together a book called Categories for the Working Philosopher. Here are the contributors and their (perhaps tentative) topics: Samson Abramsky — Computer Science, etc. John Baez — Applied Mathematics John Bell — Logic/Model Theory Bob Coecke — Quantum Mechanics and Ontology Robin Cockett — Proof Theory/Linear Logic David Corfie
Summer saw the foundations of mathematics rocked by the publication of The HoTT Book. Here we are a few months later and the same has happened to physics with the appearance on the ArXiv of Urs’s Differential cohomology in a cohesive infinity-topos. Physics clearly needs more than the bare homotopy types of HoTT. Field configurations may be groupoids (1-types) under gauge equivalence, or indeed ∞−
The Continuation Passing Transform and the Yoneda Embedding Posted by John Baez Guest post by Mike Stay The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming. They’re the same thing! Why doesn’t anyone ever say so? Assume A and B are types; the continuation passing transform takes a function (here I’m using C++ notation) B f(A a
Category Theory in Homotopy Type Theory Posted by Mike Shulman Benedikt Ahrens, Chris Kapulkin, and I have just posted the following preprint: Univalent categories and the Rezk completion, arXiv:1303.0584 This is mainly a development of basic (1-)category theory using homotopy type theory (a.k.a. “univalent foundations”) as the foundational system. So for all of you readers who’ve been enjoying th
The discussion on Tom’s recent post about ETCS, and the subsequent followup blog post of Francois, have convinced me that it’s time to write a new introductory blog post about type theory. So if you’ve been listening to people talk about type theory all this time without ever really understanding it—or if you’re a newcomer and this is the first you’ve heard of type theory—this post is especially f
Over the last few years, I’ve been very slowly working up a short expository paper — requiring no knowledge of categories — on set theory done categorically. It’s now progressed to the stage where I’d like to get some feedback. Here’s the latest draft. (Edit: Now revised in the light of your helpful comments below — thanks! — and posted as arXiv:1212.6543.) Typos, clumsy wording, mathematical matt
Instiki 0.30.3 is a Security-Release. Please update. (See here for more information.) Introduction These pages are devoted to Instiki (current version Instiki 0.30.3), the nifty Ruby-on-Rails-based Wiki. There’s also a forum, where you can discuss issues with Instiki, ask questions, etc. At the beginning of 2007, I forked Instiki 0.11, and started to develop this MathML-enabled branch. Two years l
Recent Entries Magnitude 2023 Dec 4, 2023 Live-blogging a conference on magnitude in Osaka. Adjoint School 2024 Dec 1, 2023 The Adjoint School is a way to learn applied category theory. This year it will lead up to an in-person research week at the University of Oxford on June 10 - 14, 2024. Apply now! Seminar on This Week’s Finds Dec 1, 2023 You can see 18 videos of lectures based on my column Th
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