Homotopy theory of non-symmetric operads

TL;DR

This paper develops model structures for categories of non-symmetric operads and their algebras, broadening the theoretical framework for operad theory in enriched non-symmetric monoidal model categories.

Contribution

It introduces natural model structures on categories of non-symmetric operads without restrictions, under standard model category assumptions.

Findings

Established model structures for non-symmetric operads

Extended results to categories of algebras over these operads

Applied to enriched non-symmetric monoidal model categories

Abstract

We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.