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Stream: theory: category theory

Topic: A "change of variables" for the Grothendieck construction

Amar Hadzihasanovic (Jan 24 2025 at 19:12):

I convinced myself of this fact, which I have not seen before.
If $\mathsf{F}: C^\mathrm{op} \to \mathbf{Cat}$ is a functor with $C$ small, the Grothendieck construction $\int \mathsf{F}$ can be exhibited as the coend

$$\int^{c \in C} \mathsf{F}c \times (C/c),$$

where $C/c$ is the slice category of $C$ over $c$, which is covariantly functorial over $C$.
Then I think that this formula admits the following “change of variables”, analogous to the measure-theoretic one: if $\mathsf{G}: D \to C$ is a functor of small categories, then

$$\int^{d \in D} \mathsf{FG}d \times (D/d) \simeq \int^{c \in C} \mathsf{F}c \times (\mathsf{G}/c)$$

are both expressions for the Grothendieck construction of $\mathsf{FG}$; here $\mathsf{G}/c$ is the comma of $\mathsf{G}$ over (the constant functor at) $c$, which is also covariantly functorial in $c$.
Another way in which the Grothendieck construction formally behaves like an integral?

Mike Shulman (Jan 24 2025 at 19:57):

Is this equivalent to the statement that the Grothendieck construction is a lax colimit, using the expression of a lax colimit as a weighted colimit and the computation of a weighted colimit as a coend?

Amar Hadzihasanovic (Jan 24 2025 at 20:28):

I talked about it with @fosco the other day and he seemed to think that it can be recovered this way, yes; the "slice over c" functor being a kind of "lax resolution" of the functor that is constantly the terminal category...

Amar Hadzihasanovic (Jan 24 2025 at 20:29):

The particular case of the category of elements of a presheaf is in the Coend calculus book

fosco (Jan 25 2025 at 09:38):

There was a discussion on MathOverflow some time ago about change of variable for coends. https://mathoverflow.net/questions/424774/change-of-coordinates-for-coends/ I am convinced, with a little additional work, something on the lines of that post can be made precise.

Clémence Chanavat (Jan 25 2025 at 11:51):

Using enriched Yoneda lemma, Fubini and the fact that $- \times -$ is biclosed,

$$\int^{d : D} FGd \times D / d \cong \int^{d : D} \int^{c : C} Fc \times \hom_C(Gd, c) \times D / d \cong \int^{c : C}Fc \times \int^{d : D} \hom_C(Gd, c) \times D / d.$$

Now, $\int^{d : D} \hom_C(Gd, c) \times D / d$ is the formula you gave for the Grothendieck construction of the functor

$$\hom_C(G-, c) \colon D^{op} \to \mathbf{Set},$$

which is $G / c$, so that indeed

$$\int^{d : D} FGd \times D / d \cong \int^{c : C}Fc \times \int^{d : D} \hom_C(Gd, c) \times D / d \cong \int^{c : C}Fc \times G / c.$$

Amar Hadzihasanovic (Jan 27 2025 at 10:56):

Given that it plays the formal role of $\mathrm{d}c$ in the “integral” $\int \mathsf{F}(c) \, \mathrm{d}c$, there is something very suggestive about the slice $C/c$ modelling something like “displacements ending at $c$” that become “infinitesimal” as they approach the terminal object $\mathrm{id}_c$.

Amar Hadzihasanovic (Jan 27 2025 at 11:02):

Going for wild speculation, makes me wonder if, in some über-abstracted measure theory on enriched higher categories qua generalised spaces, (enriched, higher) Grothendieck constructions may play the role of “universal integrals” in encoding “the structure considered in computing an integral on that space”.

fosco (Jan 27 2025 at 13:54):

I wonder if someone thought about smooth fundamental groupoids of a smooth space (objects: points, morphisms: smooth homotopy equivalence classes of smooth paths), fibrations $F : {\cal E} \to {\cal B}$ among them, and the coend $\int^C {\cal E}_B \times {\cal B}/B$ describing... something?

John Baez (Jan 27 2025 at 17:10):

Amar Hadzihasanovic said:

Going for wild speculation, makes me wonder if, in some über-abstracted measure theory on enriched higher categories qua generalised spaces, (enriched, higher) Grothendieck constructions may play the role of “universal integrals” in encoding “the structure considered in computing an integral on that space”.

Since we're in the realm of wild speculation, I won't feel ashamed to point out that in this paper, Theorem 7 proves an equation that can, without lying at all, be summarized

$|\int F| = \int |F|$

where at left the integral sign is the Grothendieck construction applied to a finite-set-valued functor and the absolute value sign is groupoid cardinality, while at right the absolute value sign takes the cardinality of this finite-set-valued functor to get a natural-number-valued function, and the integral sign is an actual integral.

So, it shows that under some quite specific assumptions the Grothendieck construction categorifies an integral.

Amar Hadzihasanovic (Jan 27 2025 at 17:18):

Nice!