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: community: our work

Topic: Kevin Arlin

Kevin Arlin (Jan 25 2024 at 19:47):

Evan Patterson, Tim Hosgood, James Fairbanks and I posted our new paper on systems of equations in categories last week: https://arxiv.org/abs/2401.09751

Kevin Arlin (Jan 25 2024 at 19:48):

The basic idea is that you can describe an equation, say a system of PDE's, by drawing a diagram in a relevant category, say, a category of vector bundles and differential operators. The indexing shape of such a diagram describes the spaces that participate in the equation.

Kevin Arlin (Jan 25 2024 at 19:48):

Roughly speaking, these are the variables that would show up in a traditional presentation of the equation.

Kevin Arlin (Jan 25 2024 at 19:49):

(Eg $df\wedge dg=\omega$ might correspond to a diagram indexed by a cospan.)

Kevin Arlin (Jan 25 2024 at 19:50):

We shouldn't really imagine a diagram in a category, here; to allow for multilinear operators, as we want, the diagram is probably really in a multicategory of some kind. Most of the paper is written for a category but happily it turns out that the theory is the same for diagrams in an object of any locally presentable 2-category!

Kevin Arlin (Jan 25 2024 at 19:51):

So what's the theory? Well, once you've decided that an equation involving objects of a category (or whatever) $X$ is a diagram $D:J\to X,$ you suddenly have a category of equations given by the category of diagrams in $X,$ which is a well-known lax slice kind of construction.

Kevin Arlin (Jan 25 2024 at 19:51):

In particular this category lets you change the shape $J$, so you can start to wonder about relationships between equations that have totally different sets of variables, something that's hard to do in traditional syntax.

Kevin Arlin (Jan 25 2024 at 19:52):

But are maps in diagram categories meaningful for solving equations? Well, solving a system of differential equations means something like consistently choosing a section of every vector bundle showing up in the system.

Kevin Arlin (Jan 25 2024 at 19:53):

That is, it's choosing a cone tipped by the trivial line bundle $\mathbb R.$

Kevin Arlin (Jan 25 2024 at 19:53):

A fancier but equivalent way to say this is solving an equation is lifting the diagram $D$ against the projection from the coslice $\mathbb R/X\to X.$

Kevin Arlin (Jan 25 2024 at 19:54):

Indeed it's not hard to interpret such liftings against any coslice projection in terms of solving equations; better yet, you can generalize from coslice projections, which are the representable discrete opfibrations, to lifts against arbitrary discrete opfibrations, and still interpret the result quite naturally in terms of solving equations [see the paper intro.]

Kevin Arlin (Jan 25 2024 at 19:55):

So getting back to maps of diagrams, you would want to know whether they transport lifts against discrete opfibrations, and indeed, for the correct variance of diagram category, they always do! This was shown in the preceding paper by Evan, Tim, James, and Andrew Baas: https://arxiv.org/abs/2204.01843

Kevin Arlin (Jan 25 2024 at 19:56):

Thus the diagram category does meaningfully talk about equations, but it isn't really a category of equations because different diagrams can present "the same" equation in that there are maps of diagrams that induce bijections on sets of lifts against all discrete opfibrations--so the presented equations have all the same solutions of every possible shape.

Kevin Arlin (Jan 25 2024 at 19:57):

Such maps are what we study in the paper, and we find that if you invert those maps, what you discover is that every diagram presents a canonical equation given by the discrete opfibration it generates via the comprehensive factorization system (something we have to propose a definition of in a locally presentable 2-category.)

Kevin Arlin (Jan 25 2024 at 19:57):

What this implies is that two diagrams present "the same" equation if and only if there's a cospan of initial functors over $X$ between them!

Kevin Arlin (Jan 25 2024 at 19:58):

We're hoping to use this to find simpler presentations of systems of PDE's in our Decapodes software project for compositional multiphysics: https://github.com/AlgebraicJulia/Decapodes.jl

Kevin Arlin (Jan 25 2024 at 20:00):

Other than the applied connections, I'm curious to think more about how the idea of solving an equation relative to an arbitrary discrete opfibration over the base connects to the algebraic geometry story of solving an equation relative to an arbitrary scheme over a base.

Kevin Arlin (Jan 25 2024 at 20:00):

I'm excited to have this paper up as it's my first publication since moving from higher category theory to ACT!