Coends and Integrals

Posted on by Owen Lynch


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!

This website supports webmentions, a standard for collating reactions across many platforms. If you have a blog that supports sending webmentions and you put a link to this post, your blog post will show up as a response here. You can also respond via twitter or respond via mastodon (on your preferred mastodon server); through the magic of all tweets or toots with links to this post will show up below (subject to moderation).

Site proudly generated by Hakyll with stylistic inspiration from Tufte CSS