TITLE: Topological Grothendieck Construction
QUESTION [10 upvotes]: 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 $\coprod\limits_{c\in ob(C)}\{c\}\times F(c)$.

REPLY [4 votes]: 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 $Top$)