Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

TL;DR

This paper develops and compares projective and Reedy model category structures for bimodules over operads, providing explicit replacements and analyzing their properties to advance homotopical algebra in operad theory.

Contribution

It introduces explicit model structures for bimodules and infinitesimal bimodules over operads, establishing their properties and equivalence of homotopy categories.

Findings

Both model structures yield the same homotopy categories.

Explicit cofibrant and fibrant replacements are constructed.

The categories are right proper and sometimes left proper.

Abstract

We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.