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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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    $\begingroup$ Hi! I think the definition you propose seems to be that of an "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 $\mathsf{Top}$). This MO question discusses this a bit. $\endgroup$
    – Emily
    Jan 20 at 13:27
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    $\begingroup$ (Incidentally, I think a second possible such definition would be via locally internal categories: one would take some nice category of spaces $\mathsf{Top}$ that is locally cartesian closed (I think compactly generated spaces aren't), and then view $\mathsf{Top}$ itself as a locally $\mathsf{Top}$-internal category, + $\endgroup$
    – Emily
    Jan 20 at 13:31
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    $\begingroup$ @Emily This helped, thanks! If you write your comments as an answer I will probably accept it. $\endgroup$ Jan 20 at 16:22
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    $\begingroup$ You probably know this already but in "The homotopy type of the cobordism category" GMTW define such functors by their "topological category of elements". (On page 37 here) $\endgroup$ Jan 20 at 20:35
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    $\begingroup$ Hi Greg, I have constructed a model category of internal functors over a category internal to simplicial sets in section 6 of my paper "A model structure on categories internal to simplcial sets". Pedro Boavida has compared this to an infinity-categorical version in his paper : "Segal objects and the Grothendieck construction". $\endgroup$ Jan 21 at 8:41

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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:


(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! :/)

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    $\begingroup$ The two definitions do agree, at least up to equivalence. As I mentioned in my answer to a followup question, there's a direct definition of a functor from an internal category to a locally internal one, which coincides exactly with the usual notion of internal diagram when the codomain is the self-internalized one. If you instead "complete" the domain internal category to a locally internal one and consider locally internal functors, there will be a lot of extra flab, but it will still be equivalent (B2.3.13 in Sketches of an Elephant). $\endgroup$ May 27 at 10:09
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    $\begingroup$ Also, you don't need to worry about ensuring local cartesian closure, since the definition works just as well for arbitrary Top-indexed categories (i.e. fibrations over Top). $\endgroup$ May 27 at 10:10
  • $\begingroup$ @MikeShulman Oh, this is great to know, thanks! $\endgroup$
    – Emily
    May 28 at 3:16

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