What is the right notion of a functor from an internal topological category to a topologically enriched category? Let $\mathcal{C}$ be a category internal to (some convenient model for) topological spaces (which I will denote by $\mathsf{Top}$). In the question Greg Arone asks:

What is the correct notion of a topological functor $\mathcal{C} \to \mathsf{Top}$?

In particular, Greg wanted to know about the homotopy theory of such functors.  My questions can be seen as somewhat of a follow-up question. Let $\mathcal{M}$ be a topologically enriched category (feel free to assume $\mathcal{M}$ is (co)tensored over $\mathsf{Top}$ if that aids in an answer).

What is the correct notion of a topological functor $\mathcal{C} \to \mathcal{M}$?

Particular examples I am interested in include (some convenient model for) based topological spaces and spectra.
I am also interested in understanding the homotopy theory of such functors.

If, in addition, $\mathcal{M}$ is a topological model category, what do we know about the homotopy theory of topological functors $\mathcal{C} \to \mathcal{M}$?


I am aware of one definition of such functors in the literature. In "Derivatives of embedding functors I: the stable case" Arone makes the following definition (Definition 3.1), which I will state only for based spaces, but Arone also defines for spectra.
A functor from a small topological category $\mathcal{C}$ to the category of based spaces consists of the following data:

*

*A ex-space $F$ over the space $ob(\mathcal{C})$ of objects of $\mathcal{C}$, the fiber over $c \in \mathcal{C}$ is what we would usually call $F(c)$.

*A fiberwise map of objects over the space $mor(\mathcal{C})$ of morphisms of $\mathcal{C}$
$$
\alpha: s^* (F) \to t^*(F),
$$
where $s^*(F)$ and $t^*(F)$ are the pullbacks of $F$ from $ob(\mathcal{C})$ to $mor(\mathcal{C})$ along the source and the larget maps respectively.

This data is subject to unicity and composition law conditions, which I leave to the reference. The idea is that when $\mathcal{C}$ is discrete, this precisely recovers the standard definitions.
I'm hopeful that there is some 'slicker' way to define such functors analogous to that of a $\mathsf{Top}$-internal diagram as in Emily's answer to Greg's original question.
 A: I don't believe it is possible to recover the "correct" notion of "functor $\mathcal{C}\to \rm Top$", as described at the other question you linked to, by viewing $\rm Top$ only as a topologically enriched category.  But it is possible if you view $\rm Top$ as a more richly structured object called a locally internal category.  The examples of based spaces and spectra that you mention can also be enhanced to locally internal categories over $\rm Top$, and we thereby obtain notions of functor from an internal category $\mathcal{C}$ to these locally internal categories.
Intuitively, a locally internal category over $\rm Top$ consists of a $({\rm Top}/X)$-enriched category $\mathcal{M}_X$ for each $X\in \rm Top$, such that each map $f:X\to Y$ in $\rm Top$ induces a functor $f^*:\mathcal{M}/Y \to \mathcal{M}/X$, in a coherent way.  (To be precise, this $f^*$ doesn't typecheck, since its domain and codomain are enriched in different categories, so we apply $f^* : {\rm Top}/Y \to {\rm Top}/X$ homwise to its domain; see the nLab link and the references cited therein.)
For instance, $\rm Top$ itself underlies such a locally internal category, where ${\rm Top}_X = {\rm Top}/X$.  The category $\rm Ex$ of based spaces also underlies a locally internal category, where ${\rm Ex}_X$ is the category of pointed objects in ${\rm Top}/X$, i.e. the category of sectioned spaces or "ex-spaces".  There is also a locally internal category $\rm Sp$ of spectra, where ${\rm Sp}_X$ is the category of parametrized spectra over $X$ (e.g. in the May-Sigurdsson sense).
Now if $\mathcal{M}$ is a locally internal category and $\mathcal{C} = (\mathcal{C}_1 \rightrightarrows \mathcal{C}_0)$ is an internal category, a functor $D:\mathcal{C} \to \mathcal{M}$ consists of:

*

*An object $D\in \mathcal{M}_{\mathcal{C}_0}$

*A morphism $s^* D \to t^* D$ in $\mathcal{M}_{\mathcal{C}_1}$

*An associativity condition in $\mathcal{M}_{\mathcal{C}_1 \times_{\mathcal{C}_0} \mathcal{C}_1}$ and a unit condition in $\mathcal{M}_{\mathcal{C}_0}$.

If you interpret this in the locally internal category $\rm Top$, you get the notion of functor $\mathcal{C} \to \rm Top$ mentioned in the other answer.  Specializing to $\rm Ex$ and $\rm Sp$ will then answer your question.
Of course, nothing is special about $\rm Top$ here; any category with finite limits will work to define locally internal categories, although it must be locally cartesian closed in order for itself to be an example.  Locally internal categories can also be identified with fibrations or indexed categories that satisfy an internalized "local smallness" condition.
I can't resist also mentioning my paper Enriched indexed categories, which studies a generalization (independently discovered by a number of people) of the notion of locally internal category to a notion of category that is simultaneously indexed by (or internal to) a base category and enriched in another category (which itself is indexed over the same base).  In this way one can define and study categories that are "internal to $\rm Top$", with a topology on their set of objects, but also enriched over a (monoidal) locally internal category like $\rm Ex$ or $\rm Sp$, as well as functors from such categories to their enriching categories, or other categories enriched over those.
