Simon wrote:

> One can think (sometimes usefully, sometimes less usefully) of a feasability relation (or profunctor) between preorders as giving rise to a new preorder, known as the **collage**. The underlying set of the collage is the disjoint union \\(X \sqcup Y\\) and on both \\(X\\) and \\(Y\\) the new preorder restricts to the preorders you started with, but for all \\(x \in X\\) and \\(y \in Y\\) you have \\(y\not\le x\\) whilst \\(x\le y\, \iff\ \Phi(x,y)\\).

I think it's really great how one can glom together two preorders into a "collage" this way using a feasibility relation... or more generally, glom together two categories using profunctor. But the preorder case is so easy to visualize and like!



I got interested in collages when thinking about the [10-fold way](https://golem.ph.utexas.edu/category/2014/07/the_tenfold_way_part_2.html) in condensed matter physics. I noticed that it arose from taking the 2-element super-Brauer group of the complex numbers and the 8-element super-Brauer group of the real numbers and glomming them into a single structure. It's easy to check that a field homomorphism like \\(\mathbb{R} \hookrightarrow \mathbb{C}\\) gives rise to a homomorphism between their super-Brauer groups (whatever those are). But how do you combine the Brauer groups into a single thing?

Well, if you have a group homomorphism \\(f : G \to H\\) you can a make the disjoint union \\(G \sqcup H\\) into a monoid in a pretty obvious way. This is not quite the collage of the underlying categories of these groups, since it has one object rather than two. But it's related.