Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The [Grothendieck construction](http://ncatlab.org/nlab/show/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 $\coprod\limits\_{c\in ob(C)}\{c\}\times F(c)$.



Topological Grothendieck Construction
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.


