Derived equivalences of actions of a category

TL;DR

This paper explores derived equivalences of oplax functors from a category to small $Bbbk$-categories, generalizing group actions, and establishes a Morita type theorem to facilitate their classification.

Contribution

It introduces a framework for derived equivalences of oplax functors from a category, extending the understanding of Morita theory in this context.

Findings

Established a Morita type theorem for oplax functors.

Provided a foundation for studying derived equivalences of Grothendieck constructions.

Generalized group actions to category actions in the derived setting.

Abstract

Let $\Bbbk$ be a commutative ring and $I$ a category. As a generalization of a $\Bbbk$-category with a (pseudo) action of a group we consider a family of $\Bbbk$-categories with a (pseudo, lax, or oplax) action of $I$, namely an oplax functor from $I$ to the 2-category of small $\Bbbk$-categories. We investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors.