Duality in Monoidal Categories

TL;DR

This work clarifies the relationship between closedness and rigidity in monoidal categories by introducing tensor-representability and Grothendieck–Verdier duality as a unified framework. It proves that tensor-representability implies a form of duality (GV) but does not in general imply rigidity, providing a concrete counterexample in $sl_2$-crystals. The authors develop tools to detect GV structures in Day-convolution functor categories and show how these dualities transfer to Cauchy completions, yielding applications to representation-theoretic settings including Boolean algebras, Mackey functors, and crossed modules (2-groups). They demonstrate that rigidity in functor categories aligns with finitely-generated projectivity under suitable finiteness or semisimplicity assumptions, linking dualities to module-theoretic properties and yielding precise criteria across varied contexts. Overall, the paper generalizes duality concepts in monoidal categories, clarifies hierarchy among rigid, tensor-representable, GV, and closed structures, and connects these abstractions to concrete representation-theoretic phenomena with potential for further applications in algebra and quantum topology.

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.

Theorems & Definitions (95)

• Proposition 2.1
• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof