Realizable homotopy colimits

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.

💡 Research Summary

The paper “Realizable homotopy colimits” resolves a long‑standing gap between concrete constructions of homotopy colimits and their abstract universal characterizations. The author works in the setting of relative categories—categories equipped with a distinguished class of weak equivalences—and endows these with a 2‑category structure (RelCat) where 1‑morphisms are weak‑equivalence‑preserving functors and 2‑morphisms are natural transformations localized at pointwise weak equivalences.

A central notion introduced is that of a realizable homotopy colimit. Given a relative category (C,W) that is closed under (finite) coproducts, a realizable homotopy colimit is defined as a functor \