(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, morphsisms = 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 e1*e_2.
Stage 3: direct sum over all triples of composable edges
(e_1, e_2, e_3) ...