A variant of a Dwyer–Kan theorem for model categories

Chorny, Boris¹ ; White, David²

¹ Department of Mathematics, Physics and Computer Science, University of Haifa at Oranim, Tivon, Israel
² Department of Mathematics and Computer Science, Denison University, Granville, OH, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 4, pp. 2185-2208

Résumé

If all objects of a simplicial combinatorial model category 𝒜 are cofibrant, we construct the homotopy model structure on the category of small functors 𝒮𝒜, 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 𝒮𝒜 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→ℬ induce a Quillen equivalence 𝒮𝒜⇆𝒮ℬ 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.

Détail

DOI : 10.2140/agt.2024.24.2185

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