Let $\mathcal C=(\mathcal O, \mathcal M)$ be a category internal to topological spaces. Thus $\mathcal O$ and $\mathcal M$ are topological spaces: the space of objects and the space of morphisms respectively. These spaces come endowed with structure maps: $i \colon \mathcal O \to \mathcal M$, $s, t\colon \mathcal M\to \mathcal O$, and $c\colon \mathcal M \times_{\mathcal O} \mathcal M \to \mathcal M$, which satisfy well-known identities. Here $\mathcal M \times_{\mathcal O} \mathcal M$ is the pullback $\mathcal M\xrightarrow{s} \mathcal O \xleftarrow{t} \mathcal M$.

Let Top be the category of topological spaces. Feel free to use your favorite ``convenient'' category of topological spaces. My question is

What is the correct notion of a topological functor $F\colon \mathcal C\to $Top ?

Intuitively I feel that the following is a reasonable notion:

Proposed definition: A functor $F\colon \mathcal C\to $Top consists of a space over $\mathcal O$, that I will denote by $F\to \mathcal O$. This map is not necessarily a fibration. The fiber at a point $x\in \mathcal O$ corresponds to the value of $F$ at $x$. Furthermore, there has to be a structure map $$F\times_{\mathcal O} \mathcal M \to F$$ which satisfies certain more or less evident relations.

Is this notion of a topological functor in the literature? Does it have a name? Where can I read about it?

Disclosure: I have actually used this definition in a couple of papers, but in an ad hoc manner. I want to know if other people used it, and if it has been developed systematically.

The following question is not of immediate practical consequence to me, but it is presumably important for the general picture:

Is it possible to reinterpret this definition as an internal functor from $\mathcal C$ to Top, perhaps using the language of higher categories?

What I really want to know is whether people have studied the homotopy theory of such functors.

Is there a projective model structure on the category of functors $\mathcal C \to $Top, where weak equivalences/fibrations are fiber homotopy equivalences/fibrations over $\mathcal O$? Has anyone studied homotopical notions, such as homotopy limits and colimits, derived Kan extensions, and so forth, in this setting?

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