What is the right notion of a functor from an internal topological category to topological spaces?

Question

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?

Answer 1

The definition you propose is that of a $\mathsf{Top}$-internal diagram in $\mathcal{C}$. It comes from viewing categories as many-object monoids, and functors/presheaves on categories as the many-object generalisation of left or right modules over a monoid (and then internalising these notions to $\mathsf{Top}$). Similarly, the bimodule version of this is called an "internal profunctor".

You can find more about internal diagrams in the following references:

• Andrade, From manifolds to invariants of Eₙ-algebras, Section 2.6.
• Borceux, Handbook of Categorical Algebra, Volume I, Section 8.2.
• Jacobs, Categorical Logic and Type Theory, Section 7.4.
• Johnstone, Sketches of an Elephant, Volume 2, Section B.2.3.
• Johnstone, Topos Theory, Section 2.2.
• Tomašić, A topos-theoretic view of difference algebra, Section I.4.
• Wong, The Grothendieck Construction in Enriched, Internal and ∞-Category Theory, Section 4.4. (This one treats the non-Cartesian case).
• Wong, Smash Products for Non-cartesian Internal Prestacks, paper version of the above.
• MO 199237 and MO 263927.

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(Incidentally, a second possible (non-standard) such definition would be via locally internal categories, the "locally small version" of internal categories: we could pick a nice category of spaces $\mathsf{Spc}$ (one that is locally Cartesian closed; in particular that of compactly generated Hausdorff spaces isn't; see also this nLab page), and then view it as a locally $\mathsf{Spc}$-internal category, via self-internalisation. Then a topological functor from $\mathcal{C}$ to $\mathsf{Spc}$ would be a locally $\mathsf{Spc}$-internal functor from the externalisation of $\mathcal{C}$ (i.e. $\mathcal{C}$ viewed as a locally internal category) to this locally $\mathsf{Spc}$-internal version of $\mathsf{Spc}$. I don't think these two definitions agree (Edit: They actually do! See the comments below), though in any case this second definition is definitely a hassle to work with! :/)