Realizable homotopy colimits

Published 4 Apr 2011 in math.AG, math.AT, math.CT, and math.KT | (1104.0646v3)

Abstract: In this paper we prove that for any model category, the Bousfield-Kan construction of the homotopy colimit is the absolute left derived functor of the colimit. This is achieved by showing that the Bousfield-Kan homotopy colimit is moreover a realizable homotopy colimit, defined by means of a suitable 2-category of relative categories. In addition, in the case of exact coproducts, we characterize the realizable homotopy colimits that satisfy a cofinality property as those given by a formula following the pattern of Bousfield-Kan construction: they are the composition of a "geometric realization" with the simplicial replacement.