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: infinitesimal morphisms

Jules Hedges (May 28 2024 at 15:20):

Has anyone ever thought about a situation like this, and/or written it down?

In applied category theory it's common to have a category whose morphisms represent some kind of process (physical, computational, probabilistic etc). The nature of a category ties you to these being discrete time.

I'm imagining that for every $0 \leq t \leq 1$ you have an object $X_t \in \mathcal C$, plus an "infinitesimal outgoing morphism". I imagine those would be described by something like a (op?)fibration $p: \mathcal E \to \mathcal C$, where an object of $\mathcal E$ consists of an object of $\mathcal C$ together with some other stuff that you think of as an "infinitesimal outgoing morphism". The discrete version would be the domain (op?)fibration $\mathcal C^\to \to \mathcal C$, where you say that an infinitesimal outgoing morphism is the same thing as an ordinary outgoing morphism. So you have a functor (?) $f : [0,1] \to \mathcal E$, which maybe you'd want to be continuous for some topological structure on $\mathcal E$

What I'd like to be able to do with this is to turn this data into a "path integral", so for every $a < b \in [0,1]$ you get an ordinary morphism $\oint_a^b f$ in $\mathcal C$, from $p (X_a) \to p (X_b)$. I imagine they ought to be "sheafy" in the sense that overlapping integrals join together properly, and you get equations like $\oint_b^c f \circ \oint_a^b f = \oint_a^c f$

This vaguely feels like something that somebody might have done once...

Daniel Geisler (May 28 2024 at 15:40):

This sounds like a category theory version question about fractional iteration. I don't think a solid foundation exists for fractional iteration, much less a category version of one. A typical question here is if there is a fixed point, because most research begins with the existence of fixed points.

Paolo Perrone (May 28 2024 at 15:48):

The nLab has something along these lines in its article on differential forms.
I don't know if this is exactly what you are looking for, but in case it is, there seems to be a beautiful categorical story.

Dylan Braithwaite (May 28 2024 at 16:09):

At a guess, the notion of a [[Lie algebroid]] might be related to what you described, purely based on the intuition that a Lie algebra is like the space of infinitesimal elements of a Lie group

Daniel Geisler (May 28 2024 at 16:25):

@Paolo Perrone addresses the primary form these questions are typically considered in. I believe @Dylan Braithwaite comment addresses the historical Lie algebra/group approach where chaos is not an issue.

Jules Hedges (May 28 2024 at 16:27):

My question might be equivalent to "what is a differential 1-form valued in morphisms of a category rather than in real numbers?". But my differential geometry-fu is not strong so I'm not certain about that

Paolo Perrone (May 28 2024 at 16:29):

Jules Hedges said:

My question might be equivalent to "what is a differential 1-form valued in morphisms of a category rather than in real numbers?". But my differential geometry-fu is not strong so I'm not certain about that

The answer to that question (which might be what you really want to know) would be: it is a smooth functor from the path groupoid of X to a category C.

Daniel Geisler (May 28 2024 at 16:36):

I'm reviewing @Todd Trimble's postings for relevant material.

I found Terry Tao's posts on dynamics very helpful. Lecture 1, lecture 2.

Peva Blanchard (May 28 2024 at 18:22):

Jules Hedges said:

My question might be equivalent to "what is a differential 1-form valued in morphisms of a category rather than in real numbers?". But my differential geometry-fu is not strong so I'm not certain about that

Not a direct answer, but this question reminded my of non commutative differential geometry, where people define "differential calculus" using an abstract version of graded algebra $\Omega^* = \bigoplus_n \Omega^n$ (together with a graded derivation $d$ of $\Omega^*$ s.t. $d^2 = 0$, etc.). This yields examples of differential forms taking values, e.g., in matrices.

Some examples in this introduction (Section 3).
For a more thorough exposition (clearly out of my reach), there is Alain Connes's publication non-commutative differential geometry.

Jules Hedges (May 29 2024 at 13:23):

Thanks all! I'm getting the feeling that my exact question ("how do you integrate along a curve in a category?") probably hasn't been answered before, which is a bit unfortunate because I'm definitely not the right person to work out that kind of stuff

John Baez (May 29 2024 at 13:34):

If you want "curve" to literally mean a smooth map from the real line to... something... then there are various known answers that involve the theory of manifolds or more general smooth spaces (like diffeological spaces).

John Baez (May 29 2024 at 13:39):

For example we can look at a category internal to smooth manifolds. This will have a manifold of morphisms, say $\mathrm{Mor}$ and we can do things like look at a smooth curve

$\gamma: [0,\infty) \to \mathrm{Mor}$

such that $\gamma(0)$ is an identity morphism and the source of $\gamma(t)$ is the same for all $t \in [0,\infty),$ while the target is allowed to change. The tangent vector $\gamma'(0)$ will be an 'infinitesimal morphism' pointing in the direction of how the morphism starts moving away from the identity at $t = 0$. $\gamma'(t)$ at any time $t$ will be the 'velocity' of the morphism at that time.

We can recover $\gamma(t)$ from $\gamma(0)$ and $\gamma'(s)$ for $0 \le s \le t$ by doing an integral. It sounds like this is what you wanted to do... or at least, one interpretation of what you wanted.

John Baez (May 29 2024 at 13:43):

All this stuff is most widely studied when we have a groupoid internal to smooth manifolds. The nicer ones of these are called [[Lie groupoids]]. People use these in differential geometry and highbrow mathematical physics.