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- W4226129816 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." @default.
- W4226129816 created "2022-05-05" @default.
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- W4226129816 date "2022-01-11" @default.
- W4226129816 modified "2023-10-17" @default.
- W4226129816 title "Pre-rigid Monoidal Categories" @default.
- W4226129816 doi "https://doi.org/10.48550/arxiv.2201.03952" @default.
- W4226129816 hasPublicationYear "2022" @default.
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