Operads, Algebras and Modules in General Model Categories

TL;DR

This paper develops a comprehensive homotopy-theoretic framework for operads, algebras, and modules within cofibrantly generated symmetric monoidal model categories, extending existing theories and establishing new model structures.

Contribution

It introduces J-semi model structures for operads and algebras, and develops the theory of S-modules for a broader homotopy theory of commutative algebras.

Findings

Established J-semi model structures for operads and algebras

Developed the theory of S-modules for homotopy theory of commutative algebras

Proved base change and projection formula results

Abstract

In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.