Colored DG-operads and homotopy adjunction for DG-categories

TL;DR

This paper develops a homotopy theory framework for DG-categories using colored DG-operads, enabling the study of homotopy monads and adjunctions in a enriched categorical setting.

Contribution

It introduces a relaxed notion of categories enriched in DG-categories and constructs model structures for colored DG-operads and DGCat-enriched categories.

Findings

Constructed model structures on colored DG-operads and DGCat-enriched categories.

Defined strong homotopy monads and adjunctions in DG-categories.

Enabled analysis of homotopy maps in enriched categorical contexts.

Abstract

Generalizing the approach to pseudo monoidal DG-categories as certain colored non-symmetric DG-operads, we introduce a certain relaxed notion of a category enriched in DG-categories. We construct model structures on the category of colored non-symmetric DG-operads and on the category of DGCat-enriched categories with a fixed set of objects. This allows us to talk about strong homotopy maps in both settings. We discuss the notion of a strong homotopy monad in a DG-category and a notion of strong homotopy adjunction data for two DG-functors.