Coends and Integrals

Posted on November 21, 2020

Introduction

Here I was, browsing through twitter, and I came across an innocuous tweet by Alex Kontorovitch…. Or was it?

First of all, a warning. This is more of a “new result” blog post than it is a “explaining something that already exists”, so I hate to say it but if you are not familiar with coends or differential forms, the rest of this will be pretty gobbledigookish.

As I assume the reader is familiar with, there’s a well-known construction in category theory called an coend, which produces an element of \mathsf{D} from a profunctor P \colon \mathsf{C}^{\mathrm{op}} \times \mathsf{C} \to \mathsf{D}. The notation for it looks like this

\int^{c \in \mathsf{C}} P(c,c) \in \mathsf{D}

It has this notation because it has a lot of properties that are very reminiscent of integrals. Coends (and their dual, ends) are very useful for a lot of constructions in category theory, except for, funnily enough, calculus. Or at least, so I’ve been told; perhaps the correspondence that I am about to make has already been done.

Anyways, plowing ahead, I am going to do a construction that relates coends and actual integrals, and it may or may not be original, so hold onto your seats folks.

The Construction

Let M be a manifold. Then for any k, define C_{k}(M) to be the \mathbb{R}-algebra freely generated by all differentiable maps \Delta^{k} \to M, where \Delta^{k} is the k -simplex. As we are familiar with from simplicial homology, this turns into a chain complex, with a “boundary map” \partial_{k} \colon C_{k}(M) \to C_{k-1}(M).

We also have another chain complex, this one coming from de Rham cohomology, and given in degree k by \Omega^{k}(M), the \mathbb{R}-algebra of k-dimensional differential forms on M.

If \mathbb{N} is the poset of natural numbers, seen as a category by adding a unique arrow m \to n when m \leq n, then C_{-}(M) is a contravariant functor from \mathbb{N}, and \Omega^{-}(M) is a covariant functor from \mathbb{N}.

Now, if \sigma \colon \Delta^{k} \to M is a differentiable map, then we can integrate a k-form \omega \in \Omega^{k}(M) along it to get

\int_{\sigma(\Delta^{k})} \omega \in \mathbb{R}

For notational reasons, I define \mathrm{int}_{k}(\sigma,\omega) to be this integral; from henceforth the integral sign will be reserved for coends. We extend \mathrm{int}_{k}(-,\omega) to be a map C_{k}(M) \to \mathbb{R} using the fact that C_{k}(M) is the free \mathbb{R}-algebra on maps \Delta^{k} \to M. With a bit of work, one can show that this is a map \mathrm{int}_{k} \colon C_{k}(M) \otimes \Omega^{k}(M) \to \mathbb{R}, because \mathrm{int}_{k} \colon C_{k}(M) \times \Omega^{k}(M) \to \mathbb{R} is bilinear.

We can glue together these maps for all k to get a map

\mathrm{int} \colon \coprod_{k \in \mathbb{N}} C_{k}(M) \otimes \Omega^{k}(M) \to \mathbb{R}

Now comes the fun part. Stoke’s theorem says that \mathrm{int}_{k}(\partial \sigma, \omega) = \mathrm{int}_{k+1}(\sigma, \delta \omega). This precisely satisfies the universal property of the coend! That is, recall that the coend of the profunctor C_{-}(M) \otimes \Omega^{-}(M) is the colimit of

\coprod_{n \leq m} C_{m}(M) \otimes \Omega^{n}(M) \rightrightarrows \coprod_{k} C_{k}(M) \otimes \Omega^{k}(M)

where the two maps apply the morphism n \leq m contravariantly to C_{m}(M) (i.e., \partial^{m - n}), or covariantly to \Omega^{n}(M) (i.e. \delta^{m-n}). As \delta^{2} = 0 and \partial^{2} = 0, the only interesting case is m = n+1, and Stoke’s theorem precisely states that in that case, the images of those two maps integrate to the same thing! Therefore, we can extend \mathrm{int} to a \mathbb{R}-algebra morphism

\mathrm{int} \colon \int^{k \in \mathbb{N}} C_{k}(M) \otimes \Omega^{k}(M) \to \mathbb{R}

What is this strange \mathbb{R}-algebra called \int^{k \in \mathbb{N}} C_{k}(M) \otimes \Omega^{k}(M)? I have no idea. I don’t know if this algebra goes by another name, or is interesting for any reason. But I do believe that this construction is functorial in M, so this gives a functor from manifolds to \mathbb{R}-algebras, which one might hope preserves things that are important.

Mainly, I’m just happy that coends have any relation to integrals. Also, I wonder if the different properties that coends have (i.e., Fubini’s theorem, Hom-functor as Dirac delta) can somehow be brought to bear on this construction.

Send me thoughts about this!


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