A variant of a Dwyer–Kan theorem for model categories

Chorny, Boris; White, David

doi:10.2140/agt.2024.24.2185

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A variant of a Dwyer–Kan theorem for model categories

Boris Chorny and David White

Algebraic & Geometric Topology 24 (2024) 2185–2208

DOI: 10.2140/agt.2024.24.2185

Abstract

If all objects of a simplicial combinatorial model category $𝒜$ are cofibrant, we construct the homotopy model structure on the category of small functors $𝒮^{𝒜}$ , where the fibrant objects are the levelwise fibrant homotopy functors, ie functors preserving weak equivalences. When $𝒜$ fails to have all objects cofibrant, we construct the bifibrant-projective model structure on $𝒮^{𝒜}$ and prove that it is an adequate substitute for the homotopy model structure. Next, we generalize a theorem of Dwyer and Kan, characterizing which functors $f : 𝒜 \to ℬ$ induce a Quillen equivalence $𝒮^{𝒜} ⇆ 𝒮^{ℬ}$ with the model structures above. 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.

Keywords

small functors, model categories, infinity categories, fibrant projective, bifibrant projective

Mathematical Subject Classification

Primary: 18N40, 55P65

References

Publication

Received: 6 June 2022

Revised: 25 January 2023

Accepted: 3 April 2023

Published: 16 July 2024

Authors

Boris Chorny
Department of Mathematics, Physics and Computer Science University of Haifa at Oranim Tivon Israel

David White
Department of Mathematics and Computer Science Denison University Granville, OH United States

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Volume 24, issue 4 (2024)