Realizable homotopy colimits

Abstract

We show that the composition of a homotopically meaningful `geometric realization' (or simple functor) with the simplicial replacement produces all homotopy colimits and Kan extensions in a relative category which is closed under coproducts. Examples (and its duals) include model categories, Δ-closed classes and other concrete examples such as complexes on (AB4) abelian categories, (filtered) commutative dg algebras and mixed Hodge complexes. The resulting homotopy colimits satisfy the expected properties as cofinality and Fubini, and are moreover colimits in a suitable 2- category of relative categories. Conversely, the existence of homotopy colimits satisfying these properties guarantees that hocolimΔ° is a simple functor.