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On Hopf Adjunctions, Hopf Monads and Frobenius-Type Properties

Adriana Balan
2016 Applied Categorical Structures  
In particular, if T I is a Frobenius monoid in the monoidal category of T-algebras and T is of descent type, then T is a Frobenius monad and a Frobenius monoidal functor.  ...  We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad T on a monoidal category has a right adjoint which is also a Hopf comonad, if the object T I is  ...  Then tensoring with A yields the Hopf monad T = A⊗− on the (braided) autonomous category C.  ... 
doi:10.1007/s10485-016-9428-0 fatcat:mjpkiglehbhtratxb3btk7lzdu

Hopf comonads on naturally Frobenius map-monoidales [article]

Gabriella Böhm, Stephen Lack
2014 arXiv   pre-print
algebras, and of Hopf monads in the sense of Bruguières and Virelizier.  ...  We study monoidal comonads on a naturally Frobenius map-monoidale M in a monoidal bicategory M.  ...  We wish to express our thanks to Ross Street and Ignacio López  ... 
arXiv:1411.5788v1 fatcat:awovrv26wbh6fn3knhfzrxk7wy

Hopf polyads, Hopf categories and Hopf group monoids viewed as Hopf monads [article]

Gabriella Böhm
2017 arXiv   pre-print
Hopf group monoids and Hopf categories are Hopf monads on a distinguished type of monoidales fitting the framework studied recently by Böhm and Lack.  ...  spirit of Turaev and Zunino), and Hopf categories over V (by Batista, Caenepeel and Vercruysse) both turn out to be Hopf monads in Span| V.  ...  Hopf group monoids and Hopf categories are even Hopf monads on naturally Frobenius opmap monoidales; explaining e.g. the existence and the properties of their antipodes.  ... 
arXiv:1611.05157v2 fatcat:qb7ht6ahtfbg3jrmxfsasr6tum

Hopf Algebras—Variant Notions and Reconstruction Theorems [chapter]

Joost Vercruysse
2013 Quantum Physics and Linguistics  
Hopf algebras are closely related to monoidal categories.  ...  This result is known as the Tannaka reconstruction theorem (for Hopf algebras).  ...  The author would like to thank Gabi Böhm for useful comments on an earlier version of the paper and Stef Caenepeel for discussions.  ... 
doi:10.1093/acprof:oso/9780199646296.003.0005 fatcat:opi73fqwbvh2bcnrwtx4l6gcha

Hopf algebras---Variant notions and reconstruction theorems [article]

Joost Vercruysse
2012 arXiv   pre-print
Hopf algebras are closely related to monoidal categories.  ...  This result is known as the Tannaka reconstruction theorem (for Hopf algebras).  ...  The author would like to thank Gabi Böhm for useful comments on an earlier version of the paper and Stef Caenepeel for discussions.  ... 
arXiv:1202.3613v1 fatcat:qi5ljdiykbgffozzfosehdwyda

Coalgebraic structures in module theory

Robert Wisbauer
2012 Linear and multilinear algebra  
For this we recall the basic categorical notions and then apply them to linear algebra and module theory.  ...  Although coalgebras and coalgebraic structures are well-known for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory.  ...  Hopf monads.  ... 
doi:10.1080/03081087.2012.675334 fatcat:icqmcbv22vhopmnv3o4luzs6ci

Hopf Algebroids [article]

Gabriella Böhm
2008 arXiv   pre-print
She is grateful for their helpful comments to Dénes Bajnok, Imre Bálint, Tomasz Brzeziński and Kornél Szlachányi.  ...  Acknowledgement The author's work is supported by the Bolyai János Fellowship and the Hungarian Scientific Research Fund OTKA F67910.  ...  Hopf algebroids provide us with results of both types, ones which extend known results about Hopf algebras and also ones which are conceptually new.  ... 
arXiv:0805.3806v2 fatcat:7nyixdf35vhbfo44lngmel7324

The Fundamental Theorem for weak braided bimonads [article]

Bachuki Mesablishvili, Robert Wisbauer
2016 arXiv   pre-print
The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A, employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras.  ...  This subsumes the theory of braided Hopf algebras (based on weak Yang-Baxter operators) as considered by Alonso Álvarez and others.  ...  The first author gratefully acknowledges the support by the Shota Rustaveli National Science Foundation Grants DI/18/5-113/13 and FR/189/5-113/14.  ... 
arXiv:1604.05887v1 fatcat:zk624uogvvdqrltuedkrvxzley

Maschke type theorems for Hopf monoids [article]

Gabriella Böhm
2020 arXiv   pre-print
It covers the examples provided by Hopf monoids in braided monoidal categories, weak Hopf algebras, Hopf algebroids over central base algebras, Hopf monads on autonomous monoidal categories and Hopf categories  ...  We prove two Maschke type theorems, relating the separability of the underlying monoid and comonoid, respectively, to the existence of normalized integrals.  ...  , Hopf monads on autonomous monoidal categories and Hopf categories.  ... 
arXiv:2004.11519v1 fatcat:pckgykqv65ee3lbjj4udwap4ie

On unimodular finite tensor categories [article]

Kenichi Shimizu
2015 arXiv   pre-print
We show that the following conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3) L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and (6) R(1) is  ...  As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.  ...  We call the Hopf monad Z the central Hopf monad on C. For later use, we recall from [10] and [6] the definition of the central Hopf monad and the construction of the isomorphism Z C ∼ = Z(C).  ... 
arXiv:1402.3482v4 fatcat:yerwn5ssubclxhknogl56yvqaq

A Larson-Sweedler Theorem for Hopf V-Categories [article]

Mitchell Buckley, Timmy Fieremans, Christina Vasilakopoulou, Joost Vercruysse
2020 arXiv   pre-print
To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories.  ...  Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.  ...  JV wants to thank Paolo Saracco for interesting and motivating discussions on the interaction between Hopf and Frobenius properties.  ... 
arXiv:1908.02049v3 fatcat:36yezwqosffzrdrtjbcikdigb4

Pivotal Objects in Monoidal Categories and Their Hopf Monads [article]

Aryan Ghobadi
2020 arXiv   pre-print
If C has suitable colimits we show that C(P,Q) is monadic and thereby construct a family of Hopf monads on arbitrary closed monoidal categories C.  ...  Given such an object and a choice of dual Q, we construct the category C(P,Q), of objects which intertwine with P and Q in a compatible manner.  ...  The adjunction F ⊣ U is monadic and the monad (T, U θ F , ν) is a Hopf monad. Proof. By Lemma 4.7 and Beck's Theorem 2.1, the adjunction is monadic.  ... 
arXiv:2005.07183v2 fatcat:gwgk5g2kq5a45chi55477sodiy

Non-degeneracy conditions for braided finite tensor categories [article]

Kenichi Shimizu
2016 arXiv   pre-print
As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in C is non-degenerate if and only if C is.  ...  In this paper, we show that C is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map  ...  A Hopf monad on C is a bimonad on C admitting an antipode; see [BV07, BV12, BLV11] for the basic theory of Hopf monads.  ... 
arXiv:1602.06534v2 fatcat:werw5uo32zfz3po2xxjtqy6rhm

The logarithmic Cardy case: Boundary states and annuli

Jürgen Fuchs, Terry Gannon, Gregor Schaumann, Christoph Schweigert
2018 Nuclear Physics B  
Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.  ...  for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data.  ...  Acknowledgements We are grateful to Ingo Runkel and Azat Gainutdinov for discussions and helpful comments on the manuscript.  ... 
doi:10.1016/j.nuclphysb.2018.03.005 fatcat:nbzk2hiyejhftfct5jfhnfeaam

Nakayama functor for monads on finite abelian categories [article]

Kenichi Shimizu
2022 arXiv   pre-print
If ℳ is a finite abelian category and 𝐓 is a linear right exact monad on ℳ, then the category 𝐓 of 𝐓-modules is a finite abelian category.  ...  Some examples from the Hopf algebra theory are also discussed.  ...  The above formula and its variant also give a categorical analogue of Radford's S 4 -formula [ENO04] and a Frobenius type property of tensor functors between finite tensor categories [Shi17c] .  ... 
arXiv:2208.08203v1 fatcat:bt47xjvrerfqloj2q75trsgixq
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