Reference request for realizing a simplicial set as the homotopy colimit of its simplices

Question

I know that $$X\simeq hocolim_{Simp(X)}\Delta^n,$$ where $Simp(X)$ is the category of simplices of $X$, I know this for example because of proposition 7.5 of the nLab's page for homotopy limits. Though I have seen it appear here and there in various proofs.

However I cannot seem to find a textbook source for this, which I would like just because it feels better to cite a text book than the nLab. According to exercise 3.7 of this exercise sheet, this result can be found in Hirschhorn's book on model categories, but didn't find it there. I did my best to search in Riehl's book on abstract homotopy theory and in the book about simplicial homotopy theory in Jardine and Goerss. But was unsuccessful.

I am not saying this result cannot necessarily be found there, just after and hour of scrolling through those three pdf, and not really knowing where else to look, I taught I might turn to you guys.

Any help is much appreciated, thank you so much.

Answer 1

In Hirschhorn, this is Proposition 15.10.4(1) and Theorem 15.10.8(2). To see this, recall that one can compute the homotopy colimit of a functor $F\colon I\to\mathcal{M}$, where $I$ is a Reedy category and $\mathcal{M}$ is a model category, by computing a Reedy cofibrant replacement of $F$ and then applying the usual colimit functor to this replacement. Hence if we know that the diagram $\mathrm{Simp}(X)\to\mathrm{sSet}, (\sigma\in X_n)\mapsto \Delta^n$ is Reedy cofibrant, then the above two statements give you that $\mathrm{hocolim}_{\mathrm{Simp}(X)}\,\Delta^\bullet$ is already given up to weak equivalence by $\mathrm{colim}_{\mathrm{Simp}(X)}\,\Delta^\bullet\cong X$. Finally, to see that $\mathrm{Simp}(X)\to\mathrm{sSet}, (\sigma\in X_n)\mapsto \Delta^n$ is Reedy cofibrant (Definition 15.3.3(2)), we compute the latching map at a simplex $\sigma\in X_n$, which is the boundary inclusion $\partial\Delta^n\hookrightarrow\Delta^n$ and therefore a cofibration in $\mathrm{sSet}$.