Examples and references for Kan-like extensions?

Question

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.

1. 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.

2. 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.

3. 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 $\mathrm{Ext}$.

Answer 1

Here is an example of Kan extensions as pushforwards which I find is helpful to keep in mind. Perhaps this example is already implicit in your question, but it might be worthwhile to spell it out.

Let $X$ be a topological space, and $\Pi(X)$ its fundamental groupoid - or better, its fundamental $\infty$-groupoid. A local system on $X$ with values in some ($\infty$-)category $\mathcal C$ is, by definition, a functor: $$ \rho: \Pi(X) \to \mathcal C$$ For example, a common choice for $\mathcal C$ is the ($\infty$-)category of (dg-)vector spaces. Let $Loc(X) = Fun(\Pi(X), \mathcal C)$ denote the category of local systems on $X$.

Given a map $f:X\to Y$ of spaces, there is an associated map $f:\Pi(X) \to \Pi(Y)$ of groupoids. One can define the pullback functor $$f^\ast :Loc(Y) \to Loc(X)$$ by just composition: $$\begin{array}{ccc} \Pi(X)& \xrightarrow{f} & \Pi(Y)\\ &\searrow_{f^\ast(\rho)}& \downarrow_{\rho}\\ & & \mathcal C \end{array}$$.

If the category $\mathcal C$ has enough limits and colimits, then $f^\ast$ has a left adjoint $f_!$ given by left Kan extension, and a right adjoint $f_\ast$ given by right Kan extension. In the case when $\mathcal C$ is dg vector spaces, the pointwise formula for Kan extensions tells you that $f_\ast$ and $f_!$ are just computing the homology or cohomology of your local system along the (homotopy) fibers of $f$.

In this language, I believe (though I haven't looked carefully at what you wrote) property 1 is telling you that $f_! g_! = (fg)_!$, and property 2 is telling you that there should also be a base-change property for cartesian diagrams of spaces: $q^\ast g_! = g'_! p^\ast$. [Perhaps I was a little hasty in relating that to your Property 2...]

In any case, the base change property should be telling you that the assignment $X \mapsto Loc(X)$ defines a functor (of the appropriate kind) out of the category of spans of spaces (whose objects are spaces and morphisms are spans) given by the pull-push formula.

I don't know if this answers any of your questions, but hopefully it is of interest to someone. Feel free to downvote. :)

Answer 2

This will not really be an answer to your explicit question but when I used to teach Kan extensions in category theory courses at Master's level, I used to explore the case in which $A$ and $B$ are groups (which you can take to be finite if you like, and which are, of course, to be considered as groupoids with a single object in the usual 'delooping' way) and $V$ to be the category of sets or of vector spaces. The interpretation of $f:A\to V$ is then a set/vector space with an $A$-action. I leave you to flesh out what the two Kan extensions are, but note the ubiquity if this sort of situation and the insights that it gives you for the general case if you are wanting examples to understand your 'claim' better.

I hope that helps.