# Replacing functors by topologically or simplicially enriched functors

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.

• 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 $\text{Top}$? – Qiaochu Yuan Mar 6 '15 at 23:39