A duality formalism in the spirit of Grothendieck and Verdier

TL;DR

This paper introduces Grothendieck-Verdier categories, a weakening of rigidity in monoidal categories, and explores their properties and analogues of pivotal and ribbon categories, with applications to derived categories of sheaves.

Contribution

It formalizes Grothendieck-Verdier categories and establishes their relation to pivotal and ribbon categories, extending known structures to a broader class of monoidal categories.

Findings

Grothendieck-Verdier categories generalize rigid monoidal categories.

Analogues of pivotal and ribbon structures are developed for these categories.

Applications include derived categories of constructible sheaves and equivariant sheaves.

Abstract

We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples include the derived category of constructible sheaves on a scheme (with respect to tensor product) as well as the derived and equivariant derived categories of constructible sheaves on an algebraic group (with respect to convolution). We show that the notions of pivotal category and ribbon category, which are well known in the setting of rigid monoidal categories, as well as certain standard results associated with these notions, have natural analogues in the world of Grothendieck-Verdier categories.