Reference for homotopy colimits

Question

In the first paragraph of the following paper by Heuts and Moerdijk they speak of

A well known construction of homotopy colimits $$h_{!}:sSet^A \rightarrow sSet/NA$$

$A$ is small category. $sSet$ is category of simplicial set. What is this construction?

I would be happy for any references.

Answer 1

It is literally explained two lines below.

It maps a diagram $F$ to the simplicial set $h_!(F)$ with as $n$-simplices the pairs $(x,a_0 \to \dots \to a_n)$ consisting of an $n$-simplex $a_0 \to \dots \to a_n$ in $NA$ and an $n$-simplex $x$ in $F(a_0)$.

It's probably instructive to try to compute this for the case of pushouts ($A = \bullet \gets \bullet \to \bullet$).

A possible reference is Chapter 14 in Model categories and their localizations by Hirschhorn. The thing you want is derived from what is denoted there by $B(- \downarrow A)$. It's a kind of bar construction. Proposition 14.8.9 says that $B(- \downarrow A)$ is a fibrant resolution of the constant diagram; it follows that homotopy colimits can be computed by the recipe given by Heuts and Moerdijk, which is the coend $\int^{a \in A} B(a \downarrow A) \times F(a)$ (this is probably explained somewhere in Hirschhorn's book, probably in a different chapter, but I can't track it down right now).