Algebra Seminar: Duality in Monoidal Categories

Abstract: 

Dualities are an important tool in the study of monoidal categories and their applications. For example, underlying the construction of Tor and Ext functors is the tensor–hom adjunction in the category of bimodules over a unital ring—this is referred to as a closed monoidal structure.

A stronger concept, rigidity, models the behaviour of finite-dimensional vector spaces; that is, the existence of evaluation and coevaluation morphisms, implementing a notion of dual basis. Under delooping, this corresponds to the concept of an adjunction in a bicategory, with coevaluation as unit and evaluation as counit. Grothendieck–Verdier duality, also called *-autonomy, lies between the strict confinements of rigidity, and the generality of monoidal closedness. It is closely linked to linearly distributive categories with negation.

An immediate consequence of rigidity is that the internal-hom functor is tensor representable. That is, a dualising functor sending any object to its dual exists, and tensoring with the object is left adjoint to tensoring with its dual.

This raises a naive question: Is a monoidal category with tensor representable internal-hom automatically rigid? 

While true in many algebraic contexts, it is expected that the result does not hold in general. We provide a counterexample in the form of a "free" construction. Additionally, we will link tensor representability to Grothendieck–Verdier duality, and study tensor representable functor categories.

This talk is based on joint work with Sebastian Halbig.

This is a seminar in our algebra seminar series.