Pre-rigid monoidal categories

Abstract

Liftable pairs of adjoint functors between braided monoidal categories in the sense of [GV1] provide auto-adjunctions between the  associated categories of bialgebras. Motivated by finding interesting examples of such pairs, we study general pre-rigid monoidal  categories. Roughly speaking, these are monoidal categories in which for every object X, an object X ^∗ and a nicely behaving evaluation  map from X ^∗ ⊗ X to the unit object exist. A prototypical example is the category of vector spaces over a field, where X ^∗ is not a  categorical dual if X is not finite-dimensional. We explore the connection with related notions such as right closedness, and present meaningful examples. We also study the categorical frameworks for Turaev’s Hopf group-(co)algebras in the light of pre-rigidity and  closedness, filling some gaps in literature along the way. Finally, we show that braided pre-rigid monoidal categories indeed provide an  appropriate setting for liftability in the sense of loc. cit. and we present an application, varying on the theme of vector spaces, showing  how -in favorable cases- the notion of pre-rigidity allows to construct liftable pairs of adjoint functors when right closedness of the  category is not available.