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Topological spaces form a simplicial category, where the hom-simplicial set between the spaces $X,Y$ is roughly $$\hom(X,Y)_n =\text{Top}(|\Delta^n|\times X,Y)$$

Show that $\text{Top}$ is locally Kan, i.e. that the $\hom(X,Y)_\bullet$ are Kan complexes.

If we restrict to a convenient category of spaces, say compactly generated, then we have an adjunction $$ \text{Top}(Z\times X,Y)\cong \text{Top}(Z,\text{Map}(X,Y)). $$

So that $$\hom(X,Y)_n\cong \text{Top}(|\Delta^n|,\text{Map}(X,Y)) \cong \text{Sing}(\text{Map}(X,Y))_n.$$

We figure that $\hom(X,Y)_\bullet$ is a Kan complex because $\text{Sing}(\text{Map}(X,Y))_\bullet$ always is.

How to show that the simplicial enrichment of the category $\text{Top}$ of all spaces is a simplicial category?

This is Exercise 2.4.1.10 at Kerodon.

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    $\begingroup$ I think this reduces to proving that $- \times X$ preserves finite colimits of $\Delta^n$'s in order to show that maps $S \rightarrow \text{hom}_{\bullet}(X,Y)$ are in bijection with maps $\mid S \mid \times \ X \rightarrow Y$ when $S$ is finite (finitely many non degenerate simplices). Specifically, $S = \Lambda^n_k$. $\endgroup$ Feb 8, 2021 at 14:58
  • $\begingroup$ I'm surprised that kerodon is not using compactly-generated spaces. I agree with Noel Lundstrom -- if you're taking a product with a compact Hausdorff space, I don't think it should matter whether you're using compactly-generated or all topological spaces. $\endgroup$
    – tcamps
    Feb 15, 2021 at 18:55

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