Exact squares containing a cospan: what is known about this category?

Question

Suppose I have a cospan of categories $X\xrightarrow{F} Z\xleftarrow{G}Y$ and I'm looking for exact squares containing it. The category $Exact_{F,G}$ of such things has a terminal object, namely the comma category $(F\downarrow G)$. In fact, the comma category construction provides a section of the forgetful functor $U\colon Exact\to Cspn$ from the category of exact squares to the category of cospans.

But what else is known about this category $Exact_{F,G}$? I would like to know about completeness, cocompleteness, etc.

Are there cases (i.e. subcategories of special cospans $F,G$) for which something interesting can be said about $Exact$? For example, if we look at cospans for which $F$ is a discrete opfibration, there is another section of $U$, given by 1-categorical pullback.

My motivation comes from the case when all categories in sight are finite. Given a cospan, my real interest is in finding exact squares for which the upper-left-hand category is as small as possible.

Answer 1

Not a complete answer, but a few observations: $\newcommand{\Cat}{\mathbf{Cat}} \newcommand{\dn}{\downarrow} \newcommand{\Ex}{\mathrm{Exact}}$

• since it’s a full subcategory of the category of lax-commutative squares over $(F,G)$, it’s isomorphic to a full subcategory of $\Cat / (F \dn G)$. Also, $\Cat / (F \dn G)$ is easily seen to be complete and cocomplete, since $\Cat$ is.

• this subcategory is of the form “all objects which, under each of a class of functors $R_M : \Cat / (F \dn G) \to [M^Y, M^X]$, become isomorphic to the terminal object $1_{(F \dn G)}$”, where each $R_M$ is a right adjoint. It is therefore closed under limits in $(F \dn G)$, so it’s complete. Edit: this and the next point were mistaken; the precompose-then-Kan-extend functors are continuous (resp. co-continuous below) in their $M^Y$ (resp. $M^X$) arguments, but not in the $\Cat / (F \dn G)$ argument.

• it’s also of the form “all objects which, under each of a class of functors $L_M :\Cat / (F \dn G) \to [M^X, M^Y]$, become isomorphic to the terminal object”, where each $L_M$ is a left adjoint. So by the same argument, it’s co-complete.

• understanding $\mathrm{Exact}(F,G)$ in more detail reduces — again via factorisation through the comma category — to understanding it in the case where $F$ is an identity morphism, and (if desired) where we restrict to strictly commutative exact squares. Precisely, $\Ex(F,G) \cong \Ex_{\mathit{str}}(1_X,F^*G) \simeq \Ex(1_X,F^*G)$, where $F^*G$ is the projection from the comma category to $X$.

• one special case of this is (when $X \simeq 1$) the question of understanding the final subcategories of a given category, which I have a vague recollection of being told is a hard question in general — I’m not very sure of this memory, though.