Pre-rigid Monoidal Categories

TL;DR

This paper investigates pre-rigid monoidal categories, exploring their properties, examples, and applications, particularly in relation to liftable pairs of adjoint functors and Turaev's Hopf group-(co)algebras.

Contribution

It introduces the concept of pre-rigid monoidal categories, analyzes their connection with closedness, and applies these ideas to construct liftable adjunctions without requiring right closedness.

Findings

Pre-rigid categories generalize vector spaces with non-finite-dimensional objects.

Pre-rigidity enables liftable adjunctions even without right closedness.

Examples include categories of vector spaces and Turaev's Hopf group-(co)algebras.

Abstract

Liftable pairs of adjoint functors between braided monoidal categories in the sense of \cite{GV-OnTheDuality} 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 $X^{\ast}$ and a nicely behaving evaluation map from $X^{\ast}\otimes X$ to the unit object exist. A prototypical example is the category of vector spaces over a field, where $X^{\ast}$ 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.