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.