Duality in Monoidal Categories

TL;DR

This paper explores the relationship between closed and rigid monoidal categories, providing counterexamples and characterizations, and applies these concepts to areas like algebra, functor categories, and group representations.

Contribution

It offers new insights into when closed monoidal categories are rigid, including counterexamples and characterizations, and applies these results to various mathematical structures.

Findings

Counterexample in the category of sl2-crystals showing not all closed categories are rigid.

Characterizations of Grothendieck-Verdier duality and rigidity in functor categories with Day convolution.

Applications to QF-2 algebras, Mackey functors, and group-graded representations.

Abstract

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal hom is tensor-representable? We provide a counterexample in terms of the category of sl2-crystals. As a byproduct, we obtain characterisations of the Grothendieck-Verdier duality and rigidity of functor categories endowed with Day convolution as their tensor product. This has various applications, three of which we study in detail: generalisations of quasi-Frobenius algebras, called QF-2 algebras; Mackey functors, where we prove that, as expected due to work of Bouc, an object being rigidly dualisable is equivalent to it being finitely-generated projective; and crossed modules of finite groups, where we associate to each of these objects a Grothendieck-Verdier category of group-graded representations.