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(In what follows, $\Omega = \{0,1\}.$)

In set theory, we can define that a relation $X \rightarrow Y$ is a function $X \rightarrow \mathcal{P}(Y)$. This is the same as a subset of $X \times Y$, by the following argument. $$\mathcal{P}(X \times Y) \cong \Omega^{X \times Y} \cong (\Omega^Y)^X \cong \mathcal{P}(Y)^X$$

It turns out relations can be composed, and they're pretty useful. I was thinking of setting up something similar in the world of topological spaces. Define that a relation $X \rightarrow Y$ is a sheaf on $X$ valued in the topos $\mathrm{Sh}(Y)$. I'm not quite sure why I'm defining this, but I'm hopeful that there might be a connection to multivalued-functions.

I'm not sure if this is the same as a sheaf of sets on $X \times Y$.

Anyhoo:

Question. Can "relations" between topological spaces be composed?

goblin GONE
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  • This is a very odd notion of relation... Does your sheaf case give rise to some kind of monad on $\mathbf{Top}$? – Malice Vidrine Sep 09 '18 at 09:03
  • Your "relation" is usually called a multivalued function (Or multifunction) . A relation is usually defined on a single set $X$ and is a subset of $X \times X$. Such relations on a fixed set can also be composed. – Henno Brandsma Sep 09 '18 at 10:12
  • E.g. see wikiepdia – Henno Brandsma Sep 09 '18 at 10:13
  • @HennoBrandsma the category $\mathbf{Rel}$ is equivalent to the category of the OP (simply by noticing that $\hom(X\times Y, 2) \simeq \hom(X, 2^Y)$ ), so I would argue that it is usually called a relation (see e.g. at the nLab https://ncatlab.org/nlab/show/Rel ) – Maxime Ramzi Sep 09 '18 at 10:19
  • @Max I wouldn't call a subset of $X \times Y$ a relation either... But I get the point. – Henno Brandsma Sep 09 '18 at 10:21
  • @goblin I'm curious, where does your definition come from ? Is it just by some analogy "$2^Y \approx \mathbf{Sh}(Y)$", or is there something more to it ? – Maxime Ramzi Sep 09 '18 at 10:28
  • @Max, I've been interested in this kind of thing for awhile. It seems that most definitions don't really work; here's an example from the smooth world. One idea would be replace relations between spaces with spans, but this doesn't seem to help. Also, there's connections between sheaves and multivalued functions. – goblin GONE Sep 10 '18 at 00:42
  • In short, I dunno why I'm interested in this notion. I took a wild stab in the dark because I'm at a dead-end. When you're stuck, sometimes injecting some carefully organized chaos into your viewpoint helps :) – goblin GONE Sep 10 '18 at 00:44
  • @HennoBrandsma, there's good sense in this viewpoint on relations - I didn't just make it up in order to be different :) Peter Freyd's work on allegories provides context, and offers a variety of useful tools in abstract algebra. Unfortunately, the relational viewpoint seems not to work very well in the lands of topology and/or real analysis. Hence the question. – goblin GONE Sep 10 '18 at 00:48
  • @MaliceVidrine, I don't think it gives a monad on $\mathbf{Top}$ because what would the endofunctor be? But it might give rise to a monad on a larger category that includes $\mathbf{Top}$. We could look at the category of sites, for example, and something good might come of it. – goblin GONE Sep 10 '18 at 00:55
  • Mostly I was wondering if there was an obvious right adjoint. My thinking was that the thing that makes the $A\to PB$ version of relations work is that $\mathbf{Rel}$ is the Kleisli category of the covariant powerset monad, so the the first thing I'd wonder is if my proposed "relations" are the Kleisli category for any natural monad. Not that that's a necessarily productive thing to ask, but it was where I was going :P – Malice Vidrine Sep 10 '18 at 01:09

1 Answers1

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Yes. I'll take the definition to be a sheaf on $X \times Y$, although I think $\text{Sh}(Y)$-valued sheaf on $X$ is equivalent. If $F \in \text{Sh}(X \times Y)$ is a sheaf and $G \in \text{Sh}(Y \times Z)$ is a sheaf, then their composition $F \circ G \in \text{Sh}(X \times Z)$ is given by

  1. taking the external product $F \boxtimes G \in \text{Sh}(X \times Y \times Y \times Z)$,
  2. pulling back along the diagonal map $\Delta : Y \to Y \times Y$, then
  3. pushing forward along the map $Y \to \text{pt}$.

This is a categorification of a similar recipe for composing linear transformations $V \to W$, thought of as elements of $V^{\ast} \otimes W$, by first taking external tensor products and then taking a trace inside. Composition of relations can also be understood in this way.

The general keyword for constructions of this form is "integral transforms on sheaves" although it is usually done for sheaves on schemes (as in the Fourier-Mukai transform); see, for example, the nLab.

Qiaochu Yuan
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