# Gluings and collages along profunctors

I'm dealing with this construction, available whenever you have two categories $\cal C,D$ and a profunctor $\varphi\colon {\cal C}\leadsto{\cal D}$ between them.

Define a category ${\cal C}\boxtimes_\varphi{\cal D}$ (the gluing of $\cal C,D$) having objects the triples $(C,D,\alpha)$, with $\alpha\in \varphi(C,D)$, and morphisms the triples $(u,v,\gamma)$ such that the square

is commutative, if we denote as a dotted arrow the "fake" (or "generalized") arrow $\alpha\in \varphi(C,D)$ and $\beta\in \varphi(C', D')$; profunctors carries a natural left-right action of hom-sets, so that "$\beta \cdot u$ and $v\cdot \alpha$ have meaning. The square above "commutes" if these two elements coincide in the same $\gamma\in \varphi(C, D')$.

Now, I have shown these interesting (albeit almost tautological) properties:

1. ${\cal C}\boxtimes_\varphi{\cal D}$ is isomorphic to the category of elements of $\varphi$;
2. There is an isomorphism $${\bf Cat}(\mathcal A, {\cal C}\boxtimes_\varphi{\cal D})\cong {\bf Cat}(\mathcal A, \mathcal C)\boxtimes_{\Phi}{\bf Cat}(\mathcal A, \mathcal D)$$ where $\Phi = {\bf Cat}(\mathcal A, \varphi)$ is the profunctor $(F, G)\mapsto \int_{A\in\cal A}\varphi(FA, GA)$, so that there is the suggestive formula $$\textstyle {\bf Cat}(\mathcal A, \text{Elts }\varphi)\cong \text{Elts }{\bf Cat}(\mathcal A, \varphi)$$
3. If $\varphi$ is the identity profunctor $\hom\colon \mathcal C^\text{op}\times\mathcal C\to \bf Sets$, then $\text{Elts }\varphi$ is the arrow category ${\bf Cat}(\Delta[1], \mathcal C)$.

Have you ever seen this construction in other places, exhibiting any other interesting property? Does it have a special name? Any generalization to the enriched setting (where the category of elements may fail to exist, but there the bare definition can be verbatim given)?

I wonder if there a relation between the cograph of $\varphi$ and its category of elements? In some contexts it seems the answer is yes (more on this if necessary, but it's not my priority at the moment).

• I would consider this just as the category of elements of $\varphi$, unless I’m missing something. How you would you see this as differing from any other construction of the category of elements? Commented Dec 25, 2014 at 0:53
• My question was more on the side of "has anybody used the same construction under a different name?". The answer is yes, in at least a case: arxiv.org/abs/1212.6170 page 25. I wanted to see if there were others. And I'm also interested in the enriched extension, where Elts(F) does not exist, but I can give the same definition. Is it well-suited to cover all the other interesting properties of the category of elements? On a separate note, merry Xmas! Commented Dec 25, 2014 at 10:23
• I would not think of this as a glueing: Consider my answer to your other question (mathoverflow.net/questions/190492/…). The Category of elements behaves more like a fibre product over a distributor A distributor also induces a category structure on the disjoint union of the objects $\mathrm{Ob}X\amalg \mathrm{Ob} Y$ and this one should be called the glueing as it exhibits the required adjointness property. Commented Jan 17, 2015 at 10:22