Left and right Kan extensions are both "push-forwards" that share a certain property. I'd like to hear other, non-Kan, examples of such push-forwards, as well as perhaps a better way to think about them.

## Two properties of Kan extensions

Given a span of categories $g,f$ as shown to the left below $$ \begin{array}{ccc} A&\xrightarrow{\;f\;}&V\\ g\downarrow\\ B \end{array} \hspace{1in} \begin{array}{ccc} A&\xrightarrow{f}&V\\ g\downarrow\\ B&\underset{\mathrm{Ext}_g(f)}{- - \!\!\!\to}&V \end{array} $$ both the left and right Kan extension (assuming they always exist for $V$) return a functor $\mathrm{Ext}_g(f)\colon B\to V$. In each case the extension comes with a universal property, but I'm going to ignore it. Instead, I want to concentrate on some common corollaries of those properties.

**Property 1:** $\quad$ $\mathrm{Ext}_{-}(f)$ is functorial in the subscript: given any $h\colon B\to B'$ we have
$$
\mathrm{Ext}_h\left(\mathrm{Ext}_g(f)\right)
\cong
\mathrm{Ext}_{h\circ g}(f)
$$
and $\mathrm{Ext}_{\mathrm{id}}(f)=f$.

**Property 2:**$\quad$ $\text{Ext}$ satisfies a Beck-Chevalley condition: given a "pullback square of the right kind"
$$\begin{array}{ccc}
A'&\xrightarrow{\;p\;}&A\\
g'\downarrow~~~&&~\downarrow g\\
B'&\xrightarrow{q}&B
\end{array}
$$
one has $\mathrm{Ext}_g(f)\circ q\cong\mathrm{Ext}_{g'}(f\circ p)$.

For left Kan extensions, a "pullback square of the right kind" is a comma category $A'\cong(q\downarrow g)$). What I generally mean by "pullback squares" below is a construction 1. that you can apply to cospans to get spans, and 2. which 'pastes' in the usual way.

**Claim:** The functor $\mathrm{Ext}_{-}(V)$ satisfies Properties 1 and 2 iff the functions $\mathrm{Hom}(-,V)$ form the on-objects part of a functor $$\mathrm{Hom}(-,V)\colon\mathbf{Span}\mathbf{Cat}\to\mathbf{Set},$$
where compositions in $\mathbf{Span}\mathbf{Cat}$ are given by the "pullback squares" from Property 2.

## Other examples

I have a sense that this sort of extension structure is pretty common, though I only know one more example well. Below I'll give it and then two others I'm less-well acquainted with.

In the category $\mathbf{FinSet}$ the same pattern shows up whenever $V$ has the structure of a commutative monoid $(V,0,+)$. Given a span $B\xleftarrow{g} A\xrightarrow{f} V$, one obtains a function $\text{Ext}_g(f)\colon B\to V$ given by \begin{equation}\tag{1} \text{Ext}_g(f)(b):=\sum_{\{a\;\mid\; g(a)=b\}}f(a) \end{equation} This has the above two properties, where "pullback square" really means pullback square.

Another example where this sort of extension comes up is in TQFTs, in particular Dijkgraaf-Witten theory. I'm no expert, but see the answer to this mathoverflow question.

In some homotopy category of spaces, I have a vague recollection that by using pullbacks and gysin/shriek maps, one can define extensions as above, at least for maps into certain cohomology theories $V$.

## Questions

- How should I think of these extensions in general, and where might I read more about them?
- What are other examples where this shows up? "Bundle to function correspondence" comes to mind.
- What is the relation to "classifying spaces"? Often it seems $V$ comes with a bundle $E\to V$ and that bundle pullback---together with some sort of push-forward---gives another functor $\mathbf{Span}\to\mathbf{Set}$ equivalent to that defined by $\mathrm{Ext}$.