Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: id my structure

Spencer Breiner (Mar 13 2024 at 14:52):

I am interested in thinking about a physics construction in terms of enriched categories. I have a poset of local configurations (states) $s\in S$ with all meets and joins, and a functor $H:S\to([0,\infty],\leq)$.

$H$ satisifies an inclusion-exclusion principle, so we have, e.g.,
$H(\bigcup s_i)=\sum_i H(s_i) - \sum_{i,i'} H(s_i\cap s_{i'}) + \ldots$
We can construct a modified family $s_j'$ with $\bigcup s_i=\bigcup s_j'$ for which the sequence terminates after a finite number of alternations, and we can calculate how many based on features of the state space.

What kind of thing is $H$? If I squint this looks a bit like a (decategorified?) sheaf/stack condition enriched in $([0,\infty],+)$ with (I guess?) the subtraction coming in from the closed monoidal structure.

If it helps, I am happy to think of $U_s\in S$ as the opens of a (nice) topological space with a given Borel measure $\mu$, and $H$ as the integral of a known function $H:U_s\mapsto \int_{U_s} h\ d\mu$.

Eric M Downes (Mar 13 2024 at 16:11):

Short answer:
You need to equip your domain with more structure in order to have anything categorical (right now $H$ is just a morphism in $\mathsf{Set}$), thankfully refactoring the domain into a more category-adjacent structure is easy. See below

Long answer:
Looking at categorifying entropy, are we? :)

You already have a lattice on $S$ implicitly, because you're taking unions and intersections of states, and this can be loosened somewhat to encode inclusion-exclusion in the domain. So let's move to the powerset on $S$, $2^S$, which could make $H$ into a monotone map; preserves meets or joins, you get to choose, but since you have identified a monoid on the codomain, lets see if we can preserve that.

Let's stick to the algebra for a moment, before categorifying. Intersections/meets are idempotent $x\wedge x=x$, but the reals are not, so $H$ wouldn't be a monoid homomorphism to the abelian monoid. (prove this) So lets give ourselves a sufficiently unrestricted monoid that we can map from, that still respects your equation. Free abelian group, I choose you! Essentially its a formal sum over elements with formal inverses.

Consider if you have a meet semi-lattice (idempotent monoid; $\forall x\in 2^S;~x\land x=x=x\land\top$) $(2^S,\land,\top)$. Convince yourself this is a good description of an intersection operation (what is $\top$). Now you can always assert that $\land$ distributes over the free abelian group $(2^S,+,-,\bot)$ to form a ring $(2^S,\pm,\land,\bot,\top)$ where $\land$ is a kind of multiplication.

Consider the ring automorphism $n:2^S\to 2^S; ~n(x)=\top-x$. This has the nice property that it yields the terms of inclusion-exclusion immediately; observe
$\forall x,y\in 2^S;~~x\vee y :=(\top-x)\wedge(\top-y)\implies x\vee y = x+y-x\wedge y$,
and so on to all higher orders.

So, this is how I would start by viewing the structure on the domain; you're actually mapping from a residuated lattice $(\vee,\wedge,+)$ to the reals. But you are only preserving addition right now, and only preserving it as a monoid.

If you assert that $H$ preserves $\vee,\wedge$ what do you get? I believe you can restrict things to get what is known as "probability", where $H(\bot)=0, H(\top)=1$. I cannot remember what structure you want $H$ to preserve, though, to get probability vs. entropy, vs. merely a lattice homomorphism into operations on the codomain; the last wont have the entropy/probability-like behavior in the codomain you said you wanted, but is still interesting! So that's worth figuring out.

You can actually go a bit weaker -- you really only need a modular lattice, not a fully distributive lattice in the domain to get useful IE images in the codomain; $H$ is then like a generalized rank function for a modular lattice then, except it takes real not integer values. All these terms you can google.

Right now the lattice over $S$ is a thin category (poset); the arrows are pretty simplistic. This makes $H$ close to being a functor that preserves limits or colimits, if you did add more interesting arrows.

Anyway, at this point you have some choices to make, depending on what you're trying to do. I would read the first two chapters at least of Seven Sketches (Fong & Spivak) before going further, so you get the necessary language and tools under your belt; it talks about lattices and monotone maps. It's very accessible and available for free online. Also try to prove / disprove some of the above statements if you can, it will help.

Also if I guessed correctly about what you're trying to do, I would try to approach what Tai Danae Bradley and Tom Leinster have written on entropy. Also John Baez and Tom (?) had a nice paper on kullback leibler divergence (relative entropy) being categorical ~ a decade or so ago. Those are all really important insights, and I'd b happy to talk further or help as needed; I consider these matters important. But read Ch 1-2 of Seven Sketches first!

John Baez (Mar 13 2024 at 17:17):

Also John Baez and Tom (?) had a nice paper on Kullback Leibler divergence (relative entropy) being categorical ~ a decade or so ago.

That was @Tobias Fritz and me: A Bayesian characterization of relative entropy. We thought "Bayesian" would sell better than "categorical".

Spencer Breiner (Mar 13 2024 at 18:39):

Thanks @Eric M Downes!

I'm familiar with most of that material, but I'm having trouble fitting the pieces together.

I agree that closure (as in residuated lattices) is important, and $H$ should send relative complements to differences in $[0,\infty]$. This seems to point toward enrichment in $([0,\infty],\leq,+,-)$, viewed as a closed monoidal category.

Then there is the inclusion-exclusion principle, which is shaped like a (truncated) simplicial diagram, which makes me think of sheaves/stacks. I would also like to relate different spaces of things-like-$H$ using sheafy transformations like direct/inverse image.

Hence my guess that this (very simple) thing $H$ should be an $(n,1)$-sheaf enriched in $[0,\infty]$. However, as they say, "I'm not sure that means what you think it means". If it does, I would expect this relationship with inclusion/exclusion to be familiar (to someone), maybe even folklore, since it comes up integrating positive functions on an open cover.

Here are some other ways I might hope to get at this question

• Is inclusion/exclusion a decategorification of higher sheaves/stacks?
• Positive measures are ______ enriched in $[0,\infty]$?
• What is a good resources for enriched sheaves/stacks, with bonus points for attention to the closure and/or a focus on Lawvere-style numeric enrichments.

Eric M Downes (Mar 13 2024 at 20:11):

These are great questions!

Here is a very short and unsatisfying answer:

Sheaf-topoi can be formed on (semi)simplicial sets, so yes all of your intuitions are in a good direction, and they do join together.

The good news: AFAICT this subject is exactly what Jacob Lurie is going on about in Higher Topos Theory!

The Bad News: ... this subject is exactly what Jacob Lurie is going on about in Higher Topos Theory :/

That is to say, I will probably spend the rest of my life trying to understand what he and Emily Reihl have done. Your princess might be in a different castle...

Mike Shulman (Mar 13 2024 at 20:13):

I don't know if this is helpful, but there's also a different thing that inclusion-exclusion is a decategorification of, namely the additivity of traces / Euler characteristics.

Eric M Downes (Mar 13 2024 at 20:17):

Yes 100% (and thank you) afaik Its due to the map Spencer hinted at, namely

WLOG map $\overbrace{x,y,z,\ldots}^n$ to $n$-faces; then $x\wedge y$ maps to the $n-1$-face (edge) between faces $x,y$, $x\wedge y\wedge z$ to the $n-2$-face (vertex). This picture is dual with another where you start with $0$-faces, corresponding to the sets, and make the $1$-faces their intersection, instead.

This simplicial set is exactly what Euler is tracing over. So if your map enriches them, you have a generalized Euler sum, which I believe people have studied.

Another approachable subject to look at is a category algebra.

• consider a topology / sigma algebra that is locally finite as a lattice
• Möbius inversion for locally finite posets (IE : Euler totient :: $2^S$ : $\mathbb{Z}^\times$)
• generalize this "incidence algebra over a poset" to a "category algebra" over your ring $R$ of choice, where the product of two homsets that do not match on (co)domains is $0_R$; otherwise the product is composition.

Eric M Downes (Mar 13 2024 at 20:25):

As far as sheaf resources, I am working on Ieke Moerdijk's book and this one
https://stacks.math.columbia.edu/browse

Eric M Downes (Mar 13 2024 at 20:40):

This looks relevant as per Mike Shulman's recommendation, though I haven't read it yet
https://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf

If anyone wants to work through any of this material slowly, provings things as we go, reach out, I'd be psyched.

James Deikun (Mar 15 2024 at 19:21):

The inclusion-exclusion principle looks a lot like triviality for some sort of decategorified [[Cech cohomology]] ...