A variant of a Dwyer-Kan theorem for model categories

TL;DR

This paper generalizes the Dwyer-Kan theorem within model categories, exploring homotopy model structures and their applications to Goodwillie calculus, providing new Quillen equivalences between categories of functors.

Contribution

It introduces a new approach to homotopy model structures for small functors and extends the Dwyer-Kan theorem to broader contexts in model categories.

Findings

Existence of homotopy model structure when all objects are cofibrant.

Bifibrant-projective model structure as an alternative.

Quillen equivalence between categories of linear functors.

Abstract

If all objects of a simplicial combinatorial model category \cat A are cofibrant, then there exists the homotopy model structure on the category of small functors $\sS^{\cat A}$, where the fibrant objects are homotopy functors, i.e., functors preserving weak equivalences. Otherwise, we argue that the bifibrant-projective model structure is an adequate substitution of the homotopy model structure. Next, we use this concept to generalize the Dwyer-Kan theorem about the Quillen equivalence of the categories of homotopy functors. We include an application to Goodwillie calculus, and we prove that the category of small linear functors from simplicial sets to simplicial sets is Quillen equivalent to the category of small linear functors from topological spaces to simplicial sets.