# Bundle-to-function correspondence

To a morphism of sets $f\colon E\to B$ with finite fibers, one may assign a function $$|f^{-1}|\colon B\to{\mathbb N}$$ sending an element $b\in B$ to the cardinality of the fiber $f^{-1}(b)$.

To a proper morphism of manifolds imbedded in Euclidean space $f\colon E\to B$ one may assign a function $$|f^{-1}|\colon B\to{\mathbb R}$$ by sending an element $b\in B$ to the volume of the fiber $f^{-1}(b)$. ... Or we could assign the dimension of the fiber (valuing in ${\mathbb N}$), or we could assign the number of points in the fiber (valuing in ${\mathbb N}\cup${$\infty$}).

If $M$ is a commutative monoid (thought of as a set with addition operation), let $Set_{/M}$ denote the category of sets equipped with a map to $M$. Then to a morphism $f\colon E\to B$ in $Set{/M}$, where $f$ has finite fibers, one may assign a function $$|f^{-1}|\colon B\to M$$ sending an element $b\in B$ to the sum of the elements in the fiber. ... Or we could use a possibly non-commutative monoid $M$ but replace Sets-over-$M$ with Sequences-over-$M$.

To a discrete op-fibration of categories $f\colon E\to B$ one can assign a functor $$|f^{-1}|\colon B\to Set$$ sending $b$ to its fiber.

Question: What do all these have in common? More specifically, where can I find some category-theoretic way to understand situations of this type? The "type" here seems to be something like: a "finite type" morphism in a concrete category with "valuation" can be converted into a "valuation" on the base.

One might call it "integration along the fiber" or "gysin" or "sheaf-to-function correspondence." What is the generalized setup? What is the notion of "valuation" or "measure" supposed to be?

• Do you have in mind a specific common property? Are you looking for some general result that would allow you to relate the "valuations" at different fibers? Aug 22 '10 at 15:11
• Well, that might lead me to what I want, so let's say "yes." For example, we need certain restrictions on a map $f\colon E\to B$ of manifolds to guarantee that the induced map $|f^{-1}|\colon B\to{\mathbb R}$ is continuous, $C^n$, or smooth. Do you know these conditions? But really I'm looking for the conditions under which a map $E\to B$ in a category $C$ can be converted into a map $B\to X$ for some monoid $X$, and how that $X$ correlates to $C$. Aug 22 '10 at 15:22
• You might change the second one to use the compactly supported Euler characteristic of the fiber, in which case you don't have to ask that the map be proper. Then I think it would be the characteristic 0 version of the first one in characteristic p. Hopefully Kevin McGerty will comment. Aug 22 '10 at 20:37
• They are all either reversed examples of pulling back a universal object along a map from B, or else fail to be an example of this in an interesting (interesting-to-me at least) way. Aug 22 '10 at 21:10
• To follow up on Allen -- on a variety you have the space of constructible functions, which is the algebra generated by the characteristic functions of closed subvarieties. For such a function $\pi$, you can "push-forward" by integrating their value over the fiber of a map $f \colon X\to Y$ using the Euler characteristic with compact supports. This doesn't need $f$ to be compact to obtain a function, but you need some finiteness for $f_*(\pi)$ to be constructible. I think that $f$ being algebraic is enough, but in the (sub)analytic cases you then still need $f$ to be proper. Aug 26 '10 at 11:07

This answer comes by private correspondence from Mathieu Anel. I record it here, with some minor clean-up, because it's exactly what I was looking for. --David Spivak

We suppose that $f:E\to B$ is a (kind of) fibration.

If there exists a moduli object $M$ for fibrations, $f$ is classified by a function $[f]:B\to M$ (I've changed your notation).

Now I assume you want the map $f\mapsto [f]$ compatible with addition and products of fibers like $[f\times_Bg]=[f][g]$.

Remark : $M$ has a monoidal (or even a rig) structure iff the fibrations have a monoidal (or rig) structure. for example, discrete and Grothendieck fibrations are stable by disjoint sums and pull-backs and their moduli objects ($Sets$ and $Cat$) are rigs. other example, the direct sum and tensor product of vector bundles. in each case $f\mapsto [f]$ is always compatible with the operations (it's tautological).

Now compose with any rig morphism $g:M\to R$ where $R$ is any rig (e.g. a ring) and $g[f]$ is a "measure." And, $[f]$ is the "universal measure."

Examples:

1. $M$ = moduli of (finite) sets $M\to {\mathbb N}$ = cardinality

2. $M$ = moduli for (finite dimensional) vector bundles $M\to {\mathbb N}$ = dimension (this one is compatible only with tensor product)

3. $M$ = moduli of compact riemannian manifolds $M\to {\mathbb R}$ = volume

4. K-theory : $M$ is anything that is a rig, and $g:M\to \Pi_0(M)^+$ (where $^+$ is the additive group completion)

5. $M$ = moduli for l-adic sheaves, $M\to R$ = trace of the frobenius operator