Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories

TL;DR

This paper extends classical formulas for homotopy (co)limits to general combinatorial model categories, providing new proofs and generalizations of key formulas like Bousfield-Kan and fat totalization.

Contribution

It generalizes fundamental homotopy (co)limit formulas to non-simplicially enriched combinatorial model categories, broadening their applicability.

Findings

Proves Bousfield-Kan formula in general model categories

Establishes fat totalization formula without simplicial enrichment

Shows homotopy-final functors preserve homotopy limits in full generality

Abstract

We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield-Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.