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I'm trying to understand why the following proposition is true:

Let $J$ be a small category and $F, G : J \to \textbf{Top}$ functors. If $\tau : F \Rightarrow G$ is a pointwise homotopy equivalence, then $\operatorname{hocolim}_J F \to \operatorname{hocolim}_J G$ is a homotopy equivalence.

This seems to be such a natural result that I'm surprised it's not mentioned at all in Riehl, Dugger or Hirschorn's texts on homotopy theory. Although I think all three mention a version of this result for weak homotopy equivalences.

For instance, Riehl has [Proposition 14.5.7, p. 259 of Categorical Homotopy Theory] that

If $X_\bullet \to Y_\bullet$ is a pointwise weak equivalence of split simplicial spaces, then $\vert X_\bullet \vert \to \vert Y_\bullet \vert$ is a weak equivalence.

The details of this are explained in Dugger [Theorem 3.5, p. 10]. Applying this to the $\operatorname{hocolim}$, after justifying a few points, one get's the desired result for weak homotopy equivalence [Dugger, Theorem 4.7, p. 17].

But this seems to be where the story ends and the fact that you actually get a homotopy equivalence doesn't seem to be that important. Can someone explain why this is the case?

Now, the only source I've found that states and proves this result is Munson and Volic's Cubical Homotopy Theory, [Theorem 8.3.7, p. 409], but it's a (very technical) ten page proof, that in turn references results all over the book.

So my main question : is there a simpler way to see why this is true? If so, could you explain or point me in the right direction?

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  • $\begingroup$ I think the main point for thinking about weak homotopy equivalences boils down to the preference of model structure on $\mathbf{Top}$, where weak equivalences are weak homotopy equivalences and fibrations are Serre fibrations. $\endgroup$ – Niall Apr 30 at 19:32
  • $\begingroup$ In that case, is there a model structure on $\textbf{Top}$ where the weak equivalences are the homotopy equivalences? Making it so that Riehl's result (which I think is model independent) implies the result I want? That it takes a pointwise homotopy equivalence to a homotopy equivalence? $\endgroup$ – Robert Cardona May 2 at 17:11

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