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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.

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    $\begingroup$ Re: question 3: it seems like the question is whether you only care about mapping spaces as weak homotopy types (in which case modeling them as simplicial sets is fine) or whether you actually care about them as topological spaces (in which case the singular simplicial set functor loses information about topological spaces). That is, is $\text{Top}$ here intended as a model of the $(\infty, 1)$-category of $\infty$-groupoids, or do you really care about $\text{Top}$? $\endgroup$ – Qiaochu Yuan Mar 6 '15 at 23:39
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    $\begingroup$ «Do you really care about Top?» is a great line :-) $\endgroup$ – Mariano Suárez-Álvarez Mar 7 '15 at 19:03
  • $\begingroup$ A related quote from "Cellular spaces, Null spaces and Homotopy Localizations" by E Dror-Farjoun (p.22): "A functor that preserves weak equivalences has been shown to be 'essentially continuous' by the work of Dwyer-Kan about computing the homotopy function complexes using only the model category structure via simplicial localizations" $\endgroup$ – Gustavo Granja Mar 12 '15 at 17:01

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