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A simplicial category $\mathcal{C}$ is an $\mathbf{sSet}$-enriched category which is bitensored, in that the functors $\mathbf{Map}(-,X)$ and $\mathbf{Map}(X,-)$ admit left adjoints for every $X\in \mathcal{C}$.

It is mentioned on page 114 in Model Categories that the category of topological spaces does not admit a simplicial model structure, with respect to the above definition. On the other hand, Example 9.1.45 in Model Categories and Their Localizations states that it does, and it defines the simplicial structure.

Which statement is correct?

I think the obstruction to the existence of such structure is the existence of exponentials $X^{\mid K\mid}$ for every $X\in \mathbf{Top}$ and $K\in \mathbf{sSet}$. So,

is there an example of a simplicial set whose geometric realisation is not locally compact?

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    $\begingroup$ See Notation 7.10.2 of Hirschorn: he is assuming that his topological spaces are nice. Hovey makes no such assumption, so there aren't even mapping spaces that behave correctly. Hovey even mentions that the problem goes away for k-spaces or compactly generated spaces. $\endgroup$
    – Dylan Wilson
    Commented Feb 11, 2017 at 0:36
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    $\begingroup$ (Hirschorn explains at length in that remark what assumptions are needed for which model categorical purposes. For example, Quillen got by with only tensoring and cotensoring for finite simplicial sets. That works for arbitrary spaces.) $\endgroup$
    – Dylan Wilson
    Commented Feb 11, 2017 at 0:38
  • $\begingroup$ @DylanWilson: Thank you! That answers my main question. $\endgroup$
    – user24453
    Commented Feb 11, 2017 at 2:41

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Yes, it does, if by Top you mean compactly generated spaces (as Dylan pointed out). Then, the internal hom is a space (with the compact-open topology), and its kification is a k-space satisfying hom-tensor duality. If you apply the Sing functor you get a simplicial set, making Top into a simplicial model category. In my work, I've never actually needed this. Most of the things that are true for simplicial model categories work just as well with topological mapping spaces (again, only in my work: I'm not trying to make a global claim). It might be worthwhile to see if you can get away with just using the topological mapping spaces, if that is more straight-forward.

A great reference for what can go wrong if Top includes all spaces is Gaunce Lewis's thesis (specifically, Appendix A), hosted by Peter May. The obstruction is not exponentials: it's hom tensor duality.

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  • $\begingroup$ Internal hom in compactly generated spaces is the k-ification of the compact-open topology: the compact-open topology is not generally compactly generated. $\endgroup$
    – Charles Rezk
    Commented Feb 11, 2017 at 17:33
  • $\begingroup$ Thanks Charles. Being a Hovey student, I picked up some of the terminology from his book, where "compactly generated" means compactly generated and weak Hausdorff, and "$k$-space" means compactly generated (but not weak Hausdorff). But perhaps it's best to avoid that particular nomenclature. $\endgroup$
    – David White
    Commented Feb 11, 2017 at 18:17
  • $\begingroup$ It has nothing to do with weak Hausdorff. The problem is that the compact-open topology on the mapping space is not generally a k-space (or at least so Gaunce Lewis says in his thesis, p. 161). $\endgroup$
    – Charles Rezk
    Commented Feb 11, 2017 at 19:00

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