(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will think of as a category and also as a directed graph (objects = vertices, morphisms = edges). Assume we have a functor $F$ from the graph into (say) chain complexes. We will construct a big chain complex (the homotopy colimit) in stages.
Stage 0: direct sum over all vertices $v$ of $F(v)$
Stage 1: direct sum over all edges $e$ of the mapping cylinder of $F(e)$, with the ends of the mapping cylinder identified with the appropriate parts of stage 0.
Stage 2: direct sum over all pairs of composable edges $(e_1, e_2)$ of a higher order mapping cylinder, with appropriate identifications to parts of stage 1. This implements a relation between the three stage 1 mapping cylinders corresponding to $e_1, e_2$ and $e_1*e_2$.
Stage 3: direct sum over all triples of composable edges $(e_1, e_2, e_3) \dots$