Shakedown Social

Counting Is Hard<a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a>

Counting Is HardThanks to the work of <a href="https://mastodon.acm.org/@mspstrath" class="u-url mention" rel="nofollow noopener" target="_blank">@mspstrath</a> all the TYPES 2025 talks are available (including mine) Update: as fred points out the playlist was not supposed to be public yet, so, er watch this space<a href="https://www.youtube.com/watch?v=W-lYwG3E_x4&" rel="nofollow noopener" translate="no" target="_blank">https://www.youtube.com/watch?v=W-lYwG3E_x4&</a>Like comment and subscribe, ring the bell, all that stuff<a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://mathstodon.xyz/tags/computability" class="mention hashtag" rel="nofollow noopener" target="_blank">#computability</a>

Counting Is HardAs promised. Here is the sequel to my Weihrauch reductions are Containers post, this time relating strong reductions to dependent adaptors. Enjoy!<a href="https://www.countingishard.org/blog/strong-reducibility-as-an-adaptor" rel="nofollow noopener" translate="no" target="_blank">https://www.countingishard.org/blog/strong-reducibility-as-an-adaptor</a><a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://mathstodon.xyz/tags/computability" class="mention hashtag" rel="nofollow noopener" target="_blank">#computability</a>

Ramin Honary<a class="u-url mention" href="https://chaos.social/@das_g" rel="nofollow noopener" target="_blank">@das_g</a> True. It is certainly magical that there is a programming language which defines a state monad called “IO” (or sometimes “Effect”) which carries around with it a symbol of the entire Real World in order to model the idea that any evaluation of a function of that type of monad may (or may not) create a change somewhere out in the real world, as opposed to “pure” functions which can only ever manipulate the stack.<a class="hashtag" href="https://fe.disroot.org/tag/tech" rel="nofollow noopener" target="_blank">#tech</a> <a class="hashtag" href="https://fe.disroot.org/tag/software" rel="nofollow noopener" target="_blank">#software</a> <a class="hashtag" href="https://fe.disroot.org/tag/haskell" rel="nofollow noopener" target="_blank">#Haskell</a> <a class="hashtag" href="https://fe.disroot.org/tag/programminglanguage" rel="nofollow noopener" target="_blank">#ProgrammingLanguage</a> <a class="hashtag" href="https://fe.disroot.org/tag/typetheory" rel="nofollow noopener" target="_blank">#TypeTheory</a> <a class="hashtag" href="https://fe.disroot.org/tag/categorytheory" rel="nofollow noopener" target="_blank">#CategoryTheory</a>

Chris Grossack (she/they)Here's a cute <a href="https://sunny.garden/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://sunny.garden/tags/puzzle" class="mention hashtag" rel="nofollow noopener" target="_blank">#puzzle</a>:(1) Prove the forgetful functor from (ℤ-)graded vector spaces (with all linear maps) to vector spaces admits infinitely many distinct left adjoints.(2) How do you square this with the fact that left adjoints are unique when they exist?

amen zwa, esq.There are many <a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> textbooks with <a href="https://mathstodon.xyz/tags/programmer" class="mention hashtag" rel="nofollow noopener" target="_blank">#programmer</a> or <a href="https://mathstodon.xyz/tags/programming" class="mention hashtag" rel="nofollow noopener" target="_blank">#programming</a> in their titles. Invariably, they all start out with the precise definitions of categories, limits, colimits, functors, natural transformations, adjunctions, presheaves, .... Then, in the last page of the last section of the last chapter, they make vague references to programming language semantics, rather purfunctorily (CAUTION: that English word is not related to CT functor, profunctor, and the like).No, not <a href="https://mathstodon.xyz/tags/Bartosz" class="mention hashtag" rel="nofollow noopener" target="_blank">#Bartosz</a>'s guidebook, "Category Theory for Programmers". This one is, true to its title, for programmers.<a href="https://github.com/hmemcpy/milewski-ctfp-pdf" rel="nofollow noopener" translate="no" target="_blank">https://github.com/hmemcpy/milewski-ctfp-pdf</a>

Giulio<a href="https://mathstodon.xyz/@BartoszMilewski" class="u-url mention" rel="nofollow noopener" target="_blank">@BartoszMilewski</a> <a href="https://mathstodon.xyz/@madnight" class="u-url mention" rel="nofollow noopener" target="_blank">@madnight</a> <a href="https://mathstodon.xyz/@johncarlosbaez" class="u-url mention" rel="nofollow noopener" target="_blank">@johncarlosbaez</a> Thanks for sharing!P.S.A few people eventually moved from <a href="https://mastodon.world/tags/twitter" class="mention hashtag" rel="nofollow noopener" target="_blank">#twitter</a> to <a href="https://mastodon.world/tags/mastodon" class="mention hashtag" rel="nofollow noopener" target="_blank">#mastodon</a>.Suddenly, the old <a href="https://mastodon.world/tags/buzzwords" class="mention hashtag" rel="nofollow noopener" target="_blank">#buzzwords</a> (*) of <a href="https://mastodon.world/tags/stringtheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#stringtheory</a> and <a href="https://mastodon.world/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> pop up in the new, <a href="https://mastodon.world/tags/distributed" class="mention hashtag" rel="nofollow noopener" target="_blank">#distributed</a> <a href="https://mastodon.world/tags/fediverse" class="mention hashtag" rel="nofollow noopener" target="_blank">#fediverse</a>.(*) The above link is an example of a "history" of <a href="https://mastodon.world/tags/physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#physics</a> that mentions <a href="https://mastodon.world/tags/CarloRovelli" class="mention hashtag" rel="nofollow noopener" target="_blank">#CarloRovelli</a> but not <a href="https://mastodon.world/tags/EnricoFermi" class="mention hashtag" rel="nofollow noopener" target="_blank">#EnricoFermi</a>.

Counting Is HardIn his famous paper, "on proof and progress in mathematics", Thurston lists 8 (and implies 29 other) ways to think of the derivative.I was bored waiting for a bus, so I tried listing the different ways I could think of what a category is (see below). Please feel free to help me add more!A Category is... 1. The usual definition (omitted for space) 2. an abstract theory of functions / arrows (or as Awodey would say "archery") 3. a monoidoid 4. a poset with evidence (wording stolen from Alex Kavvos) 5. a set-enriched category 6. an object in CAT 7. a syntax for a programming language 8. a maze of twisted arrows all alike 9. a "path-complete" digraph (if there is a path x -> y there is an edge x -> y) 10. a multicategory where every arrow has arity 1 11. a polynomial comonad (spivak et al)<a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a>

Oscar CunninghamIs there a construction like the spectrum of a ring, but that gives you an ∞-groupoid rather than a topological space?<a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> <a href="https://mathstodon.xyz/tags/Topology" class="mention hashtag" rel="nofollow noopener" target="_blank">#Topology</a> <a href="https://mathstodon.xyz/tags/AlgebraicTopology" class="mention hashtag" rel="nofollow noopener" target="_blank">#AlgebraicTopology</a>

Oscar CunninghamMathematicians sometime talk about algebra and geometry being dual to each other. One way to formalise this is by talking about opposite categories. If the objects of a category act like algebras, then in the opposite category they act like spaces.But the category of finite dimensional vector spaces is its own opposite! This suggests that linear algebra is in some sense the place where algebra and geometry meet. Perhaps that explains why it's so tractable and efficacious.<a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#Math</a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#Maths</a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> <a href="https://mathstodon.xyz/tags/LinearAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#LinearAlgebra</a>

Hobson LaneMind blown by Paul Lessard's explanation of how <a href="https://mstdn.social/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> applies to <a href="https://mstdn.social/tags/DeepLearning" class="mention hashtag" rel="nofollow noopener" target="_blank">#DeepLearning</a>, on <a href="https://mstdn.social/tags/MLST" class="mention hashtag" rel="nofollow noopener" target="_blank">#MLST</a> (<a href="https://mstdn.social/tags/MachineLearning" class="mention hashtag" rel="nofollow noopener" target="_blank">#MachineLearning</a> Street Talk podcast). The extra level of abstraction of CT gives you a "Lego set for the universe" (and <a href="https://mstdn.social/tags/AGI" class="mention hashtag" rel="nofollow noopener" target="_blank">#AGI</a>).Glad the <a href="https://mstdn.social/tags/EA" class="mention hashtag" rel="nofollow noopener" target="_blank">#EA</a> folks are still thinking about this stuff and starting to be a bit more sceptical about <a href="https://mstdn.social/tags/LLMs" class="mention hashtag" rel="nofollow noopener" target="_blank">#LLMs</a>, recognizing that LLMs need to be augmented with <a href="https://mstdn.social/tags/GOFAI" class="mention hashtag" rel="nofollow noopener" target="_blank">#GOFAI</a>, like <a href="https://oval.cs.stanford.edu" rel="nofollow noopener" translate="no" target="_blank">https://oval.cs.stanford.edu</a> & <a href="https://symbolica.ai" rel="nofollow noopener" translate="no" target="_blank">https://symbolica.ai</a>

Chris Grossack (she/they)Hm... Recently I've been having a lot of trouble trying to tell the difference between1. Things that are "well known" and published in a paper I haven't read2. Things that are "well known" and lots of people have proven it privately, but for some reason or other nobody has actually published it3. Things that are "well known" and lots of people kind of see a sketch of a proof, but there's plenty of details to be checked4. Things that are "well known" mainly in the sense that some expert or other basically conjectured it at some point and everyone believed them. Lots of people can see the moral truth, but even finding a framework in which to check the details is semi-openI'm curious if other people in <a href="https://sunny.garden/tags/math" class="mention hashtag" rel="nofollow noopener" target="_blank">#math</a>, especially <a href="https://sunny.garden/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a>, experience this (I think people do). I'm also curious how people handle it when deciding what to think about (and later, what to publish).Obviously feel free to boost this and reply with your Thoughts™. I'm interested in getting as many opinions as possible. Including thoughts from people outside CT, and even outside Math/CS if you've experienced similar feelings!

Cass AlexandruPainted a mug with my preferred tech stack: <a href="https://types.pl/tags/emacs" class="mention hashtag" rel="nofollow noopener" target="_blank">#emacs</a> <a href="https://types.pl/tags/agda" class="mention hashtag" rel="nofollow noopener" target="_blank">#agda</a> <a href="https://types.pl/tags/haskell" class="mention hashtag" rel="nofollow noopener" target="_blank">#haskell</a> <a href="https://types.pl/tags/nix" class="mention hashtag" rel="nofollow noopener" target="_blank">#nix</a> <a href="https://types.pl/tags/nixos" class="mention hashtag" rel="nofollow noopener" target="_blank">#nixos</a> (Also featured: the Σ ⊣ Δ ⊣ Π adjunction sequence from the categorical semantics of (dependent) type theory) <a href="https://types.pl/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://types.pl/tags/typetheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#typetheory</a>

Oscar CunninghamCan you define a 'simplicial set of small simplicial sets' by defining Δⁿ → Simp to be the set of small simplicial sets over Δⁿ, i.e. A → Δⁿ?Would we then have that the maps B → Simp were in correspondence with the simplicial sets over B, for all B?<a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> <a href="https://mathstodon.xyz/tags/SimplicialSets" class="mention hashtag" rel="nofollow noopener" target="_blank">#SimplicialSets</a> <a href="https://mathstodon.xyz/tags/Homotopy" class="mention hashtag" rel="nofollow noopener" target="_blank">#Homotopy</a>

RanaldCloustonI've been on Mastodon for a year, so it's time for a new pinned <a href="https://fediscience.org/tags/introduction" class="mention hashtag" rel="nofollow noopener" target="_blank">#introduction</a> post with an updated dog pic! I'm a lecturer in <a href="https://fediscience.org/tags/computerScience" class="mention hashtag" rel="nofollow noopener" target="_blank">#computerScience</a> at Australian National University <a href="https://fediscience.org/tags/ANU" class="mention hashtag" rel="nofollow noopener" target="_blank">#ANU</a> in <a href="https://fediscience.org/tags/Canberra" class="mention hashtag" rel="nofollow noopener" target="_blank">#Canberra</a> , <a href="https://fediscience.org/tags/Ngunnawal" class="mention hashtag" rel="nofollow noopener" target="_blank">#Ngunnawal</a> / <a href="https://fediscience.org/tags/Ngambri" class="mention hashtag" rel="nofollow noopener" target="_blank">#Ngambri</a> country. I research <a href="https://fediscience.org/tags/typeTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#typeTheory</a> , <a href="https://fediscience.org/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> , <a href="https://fediscience.org/tags/proofTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#proofTheory</a> and a little <a href="https://fediscience.org/tags/categoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categoryTheory</a> , and teach an intro to programming class in <a href="https://fediscience.org/tags/Haskell" class="mention hashtag" rel="nofollow noopener" target="_blank">#Haskell</a> . Sometimes I post about work; when I'm busy at work I'm more likely to post about <a href="https://fediscience.org/tags/books" class="mention hashtag" rel="nofollow noopener" target="_blank">#books</a> , my <a href="https://fediscience.org/tags/labradoodle" class="mention hashtag" rel="nofollow noopener" target="_blank">#labradoodle</a> , and other pleasant distractions

CorbinThe syntax-semantics adjunction yields weird machines when we apply it to computers. Each computer has a specification, which describes legal syntax and logical rules; however, the physical implementation is either a weird machine (e.g. undocumented behavior, emergent behavior) or an abstract interpreter which emulates the specification without any side effects.This is already known, but not usually expressed with <a href="https://defcon.social/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a>. And why might I do that? Well, either a CPU is in the center of the adjunction or not; implementations in the center must be equivalent to the original specification. This is an instance of the initiality conjecture: <a href="https://ncatlab.org/nlab/show/initiality+conjecture" rel="nofollow noopener" target="_blank">https://ncatlab.org/nlab/show/initiality+conjecture</a>

chris martenshas anyone written up an explanation of Yoneda in terms of logic? in particular it seems like a “metatheorem” about category theory with a similar kind of structure (and implications) to cut and identity admissibility as metatheorems about logics. is there anything there?<a href="https://hci.social/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> <a href="https://hci.social/tags/logic" class="mention hashtag" rel="nofollow noopener" target="_blank">#logic</a> <a href="https://hci.social/tags/typetheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#typetheory</a>

Niles JohnsonI've just learned another weird fact about units in monoidal categories. 🌠 Or, more specifically, it's a weird fact about monoidal functors, F, and their unit constraints, F⁰.The coherence theorems for *strong* or *normal* monoidal functors assert that every formal diagram commutes. But for general monoidal functors, there is a standard example of a formal diagram that doesn't commute. It's not even complicated!For notation, suppose F : A → A' is a monoidal functor with unit constraint F⁰ and monoidal constraint F². I'll write the monoidal products of A and A' as a dot, like x·y, and I'll write I and I' for the monoidal units. I'll use λ/ρ for the left/right unit isomorphisms.Now consider a square diagram, where the top-right composite is F(I) —{λ⁻¹}⟶ I'·F(I) —{F⁰·1}⟶ F(I)·F(I) and the left-bottom composite is F(I) —{ρ⁻¹}⟶ F(I)·I' —{1·F⁰}⟶ F(I)·F(I).This square doesn't commute in general!This diagram, and the general coherence for monoidal functors, is given in the 1974 Ph.D. thesis of Geoffrey Lewis [1]. There's also a more recent general treatment of coherence, with lots of applications, in "Coherence for bicategories, lax functors, and shadows" by Malkiewich-Ponto [2].(1/3)<a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> [1] <a href="https://unsworks.unsw.edu.au/server/api/core/bitstreams/6473f1c8-8890-4c05-9dfd-1ee957002b1e/content" rel="nofollow noopener" target="_blank">https://unsworks.unsw.edu.au/server/api/core/bitstreams/6473f1c8-8890-4c05-9dfd-1ee957002b1e/content</a>[2] <a href="https://arxiv.org/abs/2109.01249" rel="nofollow noopener" target="_blank">https://arxiv.org/abs/2109.01249</a>

tuttleturtleAnyone have a recommended introduction to <a href="https://mathstodon.xyz/tags/CategoryTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#CategoryTheory</a> for someone who knows super random bits and pieces of intro from friends taking classes as part of their grad program when I was in undergrad and partner being a functional programmer? I've been meaning to get to teaching myself an intro class equivalent for over a decade now and I keep getting stuck on what I have is a paper book to teach myself from and that's not accessible to me anymore.

Alyssa Renata<a href="https://mathstodon.xyz/tags/introduction" class="mention hashtag" rel="nofollow noopener" target="_blank">#introduction</a>Hi, I'm Alyssa and I'm a masters student at the Institute for Logic, Language & Computation.I'm mostly here to look at <a href="https://mathstodon.xyz/tags/categorytheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#categorytheory</a> and <a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener" target="_blank">#math</a> stuff, also I've been experimenting with making Japanese curry so I might post about that.Btw, I'm <a href="https://mathstodon.xyz/tags/trans" class="mention hashtag" rel="nofollow noopener" target="_blank">#trans</a>.

Recent searches

Search options

Administered by:

Server stats:

#categorytheory