Replacing functors by topologically or simplicially enriched functors

Question

I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, perhaps continuous on morphisms in the sense that when a set map $K\to Top(X,Y)$ corresponds to a continuous map $K\times X\to Y$ then the resulting set map $K\to Map(F(X),F(Y))$ corresponds to a continuous map $K\times F(X)\to F(Y)$ when $K$ is a cell.

Question 1: Is this true? If so, what is a reference?

There are also simplicial analogues of this: instead of seeking to replace $F$ by a continuous functor, one could seek to replace it by a simplicial functor, i.e. a map of simplicially enriched categories from $Top$ to $Top$ (so that in addition to assigning an object $F(X)$ to each object $X$ it also assigns a map of ``simplicial hom-sets'' to each pair of objects). I believe this can be done by realizing the simplicial object $$n\mapsto F(X^{\Delta^n}).$$ Perhaps another way is to use the cosimplicial object $$n\mapsto F(X\times \Delta^n).$$ Here $F$ might be a functor from simplicial sets to simplicial sets rather than from spaces to spaces.

Question 2: Again, references please.

Question 3: Can a simplicially enriched replacement for $F$ be used to make a continuous replacement? I get a bit muddled here by the thought that while continuity is a property simplicial enrichment is a structure.

ADDED LATER:

To clarify: I'm quite sure that I see how to replace a functor (that preserves weak equivalences) by a weakly equivalent one that is simplicially enriched, in both the topological and the simplicial setting. I think that this is "well known", and I would be grateful for a good reference.

I would be even more interested in a method of replacing such a functor by a weakly equivalent one that is topologically enriched (in the topological setting, of course). I think that somebody told me years ago that this can be done.

And to address a possible point of confusion: Although realizations of "simplicial hom-sets" can be used as "hom-spaces", that's not quite the same as putting a topology on the ordinary hom-set.

Answers to any of this would help both me and a student of mine with things that we are writing.

Answer 1

Regarding Question 2, it seems to me that Proposition 6.4 in the paper Simplicial structures on model categories and functors by Rezk, Schwede and Shipley pretty much gives you what you want (in a somewhat more general setting of simplicial model categories), using the same idea that you outlined.

For Question 3, can't we use derived Kan extensions to pass from simplicial enrichment to topological one? To avoid set-theoretic issues you may want take Top to be a skeletally small category of CW complexes of some bounded cardinality. Having fixed a category of spaces Top, let Top$^s$ be the category with the same objects as Top, equipped with the topological enrichment that is the geometric realization of the standard simplicial enrichment. I.e., $$\mbox{Top}^s(X, Y)=|S_*\mbox{map}(X, Y)|$$ where $S_*$ is the singular set, and $\mbox{map}(X, Y)$ is the usual mapping space. There is a canonical functor $\mbox{Top}^s\to$ Top, that is a weak equivalence in the sense that it is a bijection between sets of objects, and a weak equivalence on each mapping space.

Suppose $F\colon \mbox{Top}\to \mbox{Top}$ is a simplicially enriched functor. Then $F$ can be regarded as a continuous functor from Top$^s$ to itself. From here we easily get a continuous functor $F\colon$ Top$^s\to $Top (which I continue denoting by $F$). Finally to get $F$ to be a continuous functor from Top to Top, replace $F$ with the homotopy left Kan extension of the functor $F\colon \mbox{Top}^s\to \mbox{Top}$ along the canonical functor $\mbox{Top}^s\to \mbox{Top}$. Explicitly, you can construct the Kan extension as the realization of the bar construction $$ \tilde F(Y)\leftarrow\coprod_{X_0\in \mbox{Top}} F(X_0)\times \mbox{map}(X_0, Y)\Leftarrow \coprod_{X_0, X_1\in \mbox{Top}} F(X_0)\times |S_*\mbox{map}(X_0, X_1)| \times \mbox{map}(X_1, Y)\cdots $$ This new functor $\tilde F$ is enriched over Top. That $\tilde F$ is equivalent to $F$ follows from the fact that the canonical functor $\mbox{Top}^s\to \mbox{Top}$ is a weak equivalence.

Answer 2

The recent preprint Replacing functors with enriched ones contains a possible answer. Theorem A in the paper says that if $\mathcal C$ is a small topological category that admits either tensors or cotensors by the unit interval, and $\mathcal N$ is a ``good'' topological model category, then an unenriched functor $F\colon \mathcal C \to \mathcal N$ is equivalent to a topological functor if and only if it sends homotopy equivalences to weak equivalences. The statement applies, for example, to functors from finite CW complexes to spaces.

Furthermore, the paper establishes a Quillen equivalence between a category of continuous functors and a category of topological functors (Theorem 3.15).