Topological Grothendieck Construction

Question

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and $x\in F(c)$. An arrow from $(c,x)\rightarrow (c',x')$ is an arrow $f\colon c\rightarrow c'$ with $F(f)(x')=x$. This comes with a natural functor $F\wr C\rightarrow C$ forgetting the second coordinate.

In other words it is a category $F\wr C$ together with a functor $F\wr C\rightarrow C$, such that the diagram of nerves

$$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} N_1(F\wr C) & \ra{d_0} & N_o(F\wr C) \\ \da{} & & \da{} \\ N_1(C) & \ra{d_0} & N_o(C) \end{array} $$ is cartesian.

Now topologize the whole situation, i.e. let $C$ be a topological category (a category internal to $Top$) and $F\colon C^{op}\rightarrow Top$ a functor. The grothendieck construction of this situation should give me a topological category $F\wr C$ with a continuous functor $F\wr C\rightarrow C$, such that the diagram above is cartesian. However, I ran into difficulties making this precise. Taking the non-topological definition for the underlying sets $ob(F\wr C)$ and $mor(F\wr C)$, one can topologize the set of morphisms as subspace of $mor(C)$, but what is the right topology on $ob(F\wr C)$?

In some situations, the way to go is obvious :

• If $F$ takes values in subspaces of a fixed topological space $X$, one could topologize $ob(F \wr C)$ as subspace of $ob(C)\times X$.
• If $C$ has a discrete set of objects, $ob(F\wr C)$ can be topologized as a subspace of $\coprod\limits_{c\in ob(C)}\{c\}\times F(c)$.

Answer 1

This is a standard irritation. The issue is that $Top$ is not a category internal to $Top$, because it doesn't have a space of objects (and I don't mean for set-theoretic reasons), so what do you mean by a functor $F : C^{op} \to Top$?

One solution to this (which I learnt from Section 7 of S. Galatius, I. Madsen, U. Tillmann, M. Weiss, "The homotopy type of the cobordism category") is to define a "continuous functor" $F : C^{op} \to Top$ to be a topological category $F \wr C$ with a continuous functor to $C$ such that the appropriate square is cartesian.

Answer 2

There is a definition of an internal presheaf on an internal category in topological spaces. But this is not the same thing as a presheaf on the underlying enriched category (by forgetting the topology on the space of objects). An internal presheaf on an internal category $C$ is a space $F$ with a map $F\to ob(C)$ and an associative and unital action map $mor(C)\times_{ob(C)}F\to F$ (where the fiber product is taken along the target map $mor(C)\to ob(C)$). If you have such an object, you can form its Grothendieck construction. This is an internal category whose space of objects is $F$ and space of morphisms is $mor(C)\times_{ob(C)}F$. The source map $mor(C)\times_{ob(C)}F\to F$ is given by the projection and the target map $mor(C)\times_{ob(C)}F\to F$ is given by the action map. This definition gives you the catesian square you are requesting.

Answer 3

Let me expand a bit on what Oscar Randal-Williams is saying and mix it with what Geoffroy Horel is saying:

You can't naively speak about an "internal functor" $F:C^{op} \to Top.$ But lets do a small exercise:

Suppose that $C$ is an ordinary small category. Define a right $C$-set to be a set $X$ together with a map $$\mu:X \to C_0$$ (its moment map) together with an "action map" $$\rho:C_1 \times_{C_0} X \to X$$ where an object of $C_1 \times_{C_0} X$ consists of an arrow $f:D \to C$ and an object $x \in X$ such that $\mu\left(x\right)=C.$ The action map must satisfy the obvious axioms for a right action. There is an obvious notion of morphism of such $C$-sets (equivariant maps) and this category is caonically equivalent to $Set^{C^{op}}$. How does this work? Given a functor $F:C^{op} \to Set,$ we can let $X=\coprod_\limits{C \in C_0} F(C) \to C_0$ and if $f:D \to C$ and $x \in F(C)$ we let $\rho\left(f,x\right)=F\left(f\right)\left(x\right).$

Here's how to go the other way around:

If $X$ is a $C$-set, we can form the "action category" $X \rtimes C$. Its objects are $X$ and the arrows are the fibered product $C_1 \times_{C_0} X$ where a pair $\left(f,x\right)$ is an arrow from $F\left(f\right)\left(x\right)$ to $x$. There is a canonical functor $X \rtimes C \to C$ which is a discrete fibration, and since discrete fibrations (Grothendieck fibrations which encode a functor into sets rather than categories) are an equivalent category to presheaves of sets, we recover a presheaf of sets. More importantly however, notice that the action category $X \rtimes C$ applied to the $C$-set coming from a functor $F:C^{op} \to Set$ as above is exactly the Grothendieck construction $F \wr C$ of $F$.

Now, when $C$ is a small internal categeory to $Top$, then we can't define a presheaf as a functor $C^{op} \to Top$, but we CAN define it as a continuous right $C$-space, i.e. as a topological space $X$ together with a continuous moment map $\mu:X \to C_0,$ with a continuous right $C$-action... and if do, we can define the "grothendieck construction" of such an internal presheaf as the associated "action category" $X \rtimes C$ which carries a natural topology, which in particular, makes the canonical functor $X \rtimes C \to C$ continuous (and in fact is an internal split fibration in the $2$-category of categories internal to $Top$)

Answer 4

The construction is useful in the topological case and for topological groupoids was studied by Ehresmann. The construction is also related to the notion of covering morphism of groupoids. I do not have the Ehresmann reference to hand, but he wrote a substantial paper "Categories topologiques", and the notion of differentiable groupoid (now Lie groupoid) informs much of his work.

Before being fully aware of this work, we have this joint paper

R. Brown, G. Danesh-Naruie, and J.P.L. Hardy, `Topological groupoids II: covering morphisms and $G$-spaces'', Math. Nachr. 74 (1976) 143-156.

of which a copy is available here.

See also this paper:

A.C. Ehresmann, "How Charles Ehresmann's vision of geometry developed with time", in Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland) 8.05.2005-15.05.2005, Banach Centre Publications 76, Institute of Mathematics Polish Academy of Sciences, Warsaw, (2007).

The relation between actions of groupoids and covering morphisms of groupoids is given in Chapter 10 of Topology and Groupoids.