Mastodawn

Niles Johnson Dec 19, 2023

New paper 🎉🎉
About coherence 🥱💤⁉️

joint work with Nick Gurski
https://arxiv.org/abs/2312.11261

The title is:
Universal pseudomorphisms, [*deep breath*]
with applications to diagrammatic coherence for braided and symmetric monoidal functors 🙃😸

I've always thought coherence theorems sound boring, but actually they're good! In this paper we take a problem that is hard (coherence for structured functors), do a *bunch* of really abstract stuff (2-monad theory), and come out with a solution that makes your life* significantly better.

[*Here, "your life" means the part of your life you spend checking diagrams of braided monoidal functors. Or, more generally, pseudomorphisms for algebras over a 2-monad.]

Almost 1/5 of this paper is dedicated to real, genuine examples, and that's what I want to focus on below. I'll say just a bit about the more abstract machinery on which the examples are based. If you've been following along, this is the culmination of my series "weird facts about monoidal functors and coherence" [1,2,3,4].

[1] https://mathstodon.xyz/@nilesjohnson/110741323263984146
[2] https://mathstodon.xyz/@nilesjohnson/110876487813747736
[3] https://mathstodon.xyz/@nilesjohnson/110979458364785667
[4] https://mathstodon.xyz/@nilesjohnson/111070640771166081

#CategoryTheory #MonoidalFunctor #Coherence #Braided #Symmetric #PseudomorphismClassifier

(1/14)

Universal pseudomorphisms, with applications to diagrammatic coherence for braided and symmetric monoidal functors

This work introduces a general theory of universal pseudomorphisms and develops their connection to diagrammatic coherence. The main results give hypotheses under which pseudomorphism coherence is equivalent to the coherence theory of strict algebras. Applications include diagrammatic coherence for plain, symmetric, and braided monoidal functors. The final sections include a variety of examples.

arXiv.org

Niles Johnson Sep 25, 2023

If you've been following my weird monoidal functor / coherence posts [1,2,3,4]... we are really getting close to finishing this project! It could be a matter of weeks. I'm excited because this project contains a cool blend of some mind-wringing 2-monad theory, followed by some (imo) genuinely useful applications to symmetric/braided monoidal functors, and then some real, detailed, actual examples. The doubling functor I mentioned a while back makes an appearance, along with (if we finally have it figured out, and we don't have to cut it) their even weirder friend, quadrupling!

We're working on getting the introduction and examples to be as clear as possible for readers who want to skip all of the not-entirely-easy middle part. I'm looking forward to saying more about it :)

#CategoryTheory
#Monad #MonoidalFunctor #PseudomorphismClassifier

Niles Johnson (@[email protected])

I'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) #CategoryTheory [1] https://unsworks.unsw.edu.au/server/api/core/bitstreams/6473f1c8-8890-4c05-9dfd-1ee957002b1e/content [2] https://arxiv.org/abs/2109.01249

Mathstodon

Niles Johnson Aug 30, 2023

Here's a weird fact about lists that I think is actually meaningful. I'd be interested if any of the programming language folks could tell me more!

📜 First, some background. Let M be the list monad. For a set X, MX is the set of lists whose entries are elements of X. This is an associative and unital monoid under concatenation.

Now, here are two different functions (natural transformations)
MX → M²X

The first, I'll call F, sends a list w in MX to the length-one list of lists whose single entry is the list w.

The second, I'll call G, sends a list w to the list of length-one lists, whose entries are the entries of w.

So, if w = [a,b,c], then these are

Fw = [[a,b,c]]
Gw = [[a],[b],[c]].

🐉 Ok, that's background. Now the weird fact:

G is a monoid homomorphism, with respect to concatenation of lists, but F is not! They both seem like really great constructions, and F is the unit for the monad M, but G is compatible with concatenation.

Why do I think this is actually meaningful? I don't think I can fully explain here, but I can rant a bit! :) Start by replacing the outer M in M²X with a slightly different construction Q that has (iso)morphisms between lists of lists that are obtained by concatenating the inner lists. If you do this, then F defines a *strong* monoidal functor MX → QMX, and G is a strict monoidal functor. These two things are related to something called (pseudo)morphism classifiers and "flexibility" of free algebras in Blackwell-Kelly-Power.

⁉️ If you don't know about that stuff, but you *do* know about list monad stuff, I'd be interested to hear whether these F and G show up in different guises, or with different vocabulary!

#CategoryTheory #Monad #PseudomorphismClassifier