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 tensored by 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.

Pre-rigid monoidal categories

Ardizzoni A.;
2023-01-01

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 tensored by 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.
2023
46
8
1503
1546
https://arxiv.org/pdf/2201.03952.pdf
Pre-rigid, closed, Turaev, Zunino and braided monoidal categories, liftable pairs of functors, bialgebras
Ardizzoni A.; Goyvaerts I.; Menini C.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1887485
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