Centers and homotopy centers in enriched monoidal categories

TL;DR

This paper develops a theory of centers and homotopy centers for monoids in monoidal categories enriched over duoidal categories, extending classical notions to more complex categorical structures.

Contribution

It introduces a new framework for centers and homotopy centers in enriched monoidal categories, including examples like 2-categories and dg-categories.

Findings

Centers include non-classical examples like the 2-category of categories.

Homotopy centers generalize Hochschild complexes to higher categories.

Provides a unified approach to centers in enriched monoidal settings.

Abstract

We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. Duoidal categories (introduced by Aguillar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not `classical.' In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include the Gray-category Gray of 2-categories, 2-functors and pseudonatural transformations and Tamarkin's homotopy 2-category of dg-categories, dg-functors and coherent dg-transformations.