Adjunctions in monoidal categories

Lindner, Harald

doi:10.1007/bf01167969

Cited by 8 publications

(6 citation statements)

References 7 publications

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“…It follows that they are not just right adjoint to − ⊗ B, but also left adjoint. Now it is a small step to so-called compact dagger categories [33,27], which we make here for the sake of simplicity.…”

Section: Dagger Categoriesmentioning

confidence: 99%

Reversible Monadic Computing

Heunen

Karvonen

2015

Electronic Notes in Theoretical Computer Science

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We extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity. We demonstrate that Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley's theorem; by proving that effectful computations (Kleisli morphisms) are reversible precisely when the monad is Frobenius; by characterizing the largest reversible subcategory of Eilenberg-Moore algebras; and by identifying the latter algebras as measurements in our leading example of quantum computing. Strong Frobenius monads are characterized internally by Frobenius monoids. and so T (I) is special.

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Section: Dagger Categoriesmentioning

confidence: 99%

Reversible Monadic Computing

Heunen

Karvonen

2015

Electronic Notes in Theoretical Computer Science

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“…• Strong monoidal functors between right rigid categories preserve duals, see e.g. [Li,Proposition 3].…”

Section: Introductionmentioning

confidence: 99%

Pre-rigid Monoidal Categories

Ardizzoni¹,

Goyvaerts²,

Menini³

2022

Preprint

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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.

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“…PROOF Strong monoidal functors preserve compact objects [158,Proposition 3], and Lemma 2.6.7 shows that the compact objects in Mod R are precisely the finitely projective R-modules.…”

Section: Compact Objectsmentioning

confidence: 99%

“…Two compact objects in a symmetric monoidal category are dual to each other precisely when they are both left and right adjoint to each other as morphisms in the induced one-object bicategory. Under this identification, η and ε are the unit and counit of the adjunction [65,158].…”

Section: 64mentioning

confidence: 99%

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Categorical Quantum Models and Logics

Heunen¹

2009

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Adjunctions in monoidal categories

Cited by 8 publications

References 7 publications

Reversible Monadic Computing

Reversible Monadic Computing

Pre-rigid Monoidal Categories

Categorical Quantum Models and Logics

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