Skip to main content Accessibility help

Bimonads and Hopf monads on categories

  • Bachuki Mesablishvili (a1) and Robert Wisbauer (a2)


The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on : we define a bimonad on as an endofunctor B which is a monad and a comonad with an entwining λ : BBBB satisfying certain conditions. This λ is also employed to define the category of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ : BBBB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining λ and lead to an extension of the theory of braided Hopf algebras.

An antipode is defined as a natural transformation S : BB with special properties. For categories with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between and the category of B-bimodules. This is a general form of the Fundamental Theorem of Hopf algebras.

Finally we observe a nice symmetry: If B is an endofunctor with a right adjoint R, then B is a (Hopf) bimonad if and only if R is a (Hopf) bimonad. Thus a k-vector space H is a Hopf algebra if and only if Homk(H,−) is a Hopf bimonad. This provides a rich source for Hopf monads not defined by tensor products and generalises the well-known fact that a finite dimensional k-vector space H is a Hopf algebra if and only if its dual H* = Homk(H,k) is a Hopf algebra. Moreover, we obtain that any set G is a group if and only if the functor Map(G,−) is a Hopf monad on the category of sets.



Hide All
1.Barr, M., Composite cotriples and derived functors, in: Sem. Triples Categor. Homology Theory, Springer LN Math. 80, 336356 (1969)
2.Beck, J., Distributive laws, in: Seminar on Triples and Categorical Homology Theory, Eckmann, B. (ed.), Springer LNM 80, 119140 (1969)
3.Bespalov, Y. and Drabant, B., Hopf (bi-)modules and crossed modules in braided monoidal categories, J. Pure Appl. Algebra 123(1-3), 105129 (1998)
4.Bespalov, Y., Kerler, Th., Lyubashenko, V. and Turaev, V., Integrals for braided Hopf algebras, J. Pure Appl. Algebra 148(2), 113164 (2000)
5.Böhm, G., Brzeziński, T. and Wisbauer, R., Monads and comonads in module categories, J. Algebra 322, 17191747 (2009)
6.Borceux, F. and Dejean, D., Cauchy completion in category theory, Cah. Topol. Géom. Différ. Catégoriques 27, 133146 (1986)
7.Bruguières, A. and Virelizier, A., Hopf monads, Adv. Math. 215(2), 679733 (2007)
8.Brzeziński, T. and Wisbauer, R., Corings and Comodules, London Math. Soc. Lecture Note Series 309, Cambridge University Press (2003)
9.Day, B., McCrudden, P. and Street, R., Dualizations and antipodes, Appl. Categ. Struct. 11(3), 229260 (2003)
10.Dubuc, E., Kan extensions in enriched category theory, Lecture Notes in Mathematics 145, Berlin-Heidelberg-New York: Springer-Verlag (1970)
11.Eilenberg, S. and Moore, J.C., Adjoint functors and triples, Ill. J. Math. 9, 381398 (1965)
12.Gumm, H.P., Universelle Coalgebra, in: Allgemeine Algebra, Ihringer, Th., Berliner Stud. zur Math., Band 10, 155207, Heldermann Verlag (2003)
13.Kelly, G.M. and Street, R., Review of the elements of 2-categories, Category Sem., Proc., Sydney 1972/1973, Lect. Notes Math. 420, 75103 (1974)
14.Lack, S. and Street, R., The formal theory of monads II, J. Pure Appl. Algebra 175(1-3), 243265 (2002)
15.Loday, J.-L., Generalized bialgebras and triples of operads, Astérisque 320 (2008), arXiv:math/0611885
16.López Franco, I., Formal Hopf algebra theory I : Hopf modules for pseudomonoids, J. Pure Appl. Algebra 213, 10461063 (2009)
17.McCrudden, P., Opmonoidal monads, Theory Appl. Categ. 10, 469485 (2002)
18.Mesablishvili, B., Descent in categories of (co)algebras, Homology, Homotopy and Applications 7, 18 (2005)
19.Mesablishvili, B., Monads of effective descent type and comonadicity, Theory Appl. Categ. 16, 145 (2006)
20.Mesablishvili, B., Entwining Structures in Monoidal Categories, J. Algebra 319(6), 24962517 (2008)
21.Moerdijk, I., Monads on tensor categories, J. Pure Appl. Algebra 168(2-3), 189208 (2002
22.Power, J. and Watanabe, H., Combining a monad and a comonad, Theor. Comput. Sci. 280(1-2), 137162 (2002)
23.Schubert, H., Categories, Berlin-Heidelberg-New York, Springer-Verlag (1972)
24.Street, R., Frobenius monads and pseudomonoids, J. Math. Phys. 45(10), 39303948 (2004)
25.Szlachányi, K., Adjointable monoidal functors and quantum groupoids, Caenepeel, S. (ed.) et al., Hopf algebras in noncommutative geometry and physics, Proc. conf. on Hopf algebras and quantum groups, Brussels 2002, Marcel Dekker LN PAM 239, 291307 (2005)
26.Takeuchi, M., Survey of braided Hopf algebras, in: New trends in Hopf algebra theory, Proc. Coll. Quantum Groups and Hopf Algebras, La Falda, Argentina 1999, Andruskiewitsch, N. et al. (ed.), Providence, RI: American Math. Soc., Contemp. Math. 267, 301323 (2000)
27.Turi, D. and Plotkin, G., Towards a mathematical operational semantics, Proceedings 12th Ann. IEEE Symp. on Logic in Computer Science, LICS'97, Warsaw, Poland (1997)
28.Wisbauer, R., On Galois comodules, Commun. Algebra 34(7), 26832711 (2006)
29.Wisbauer, R., Algebras versus coalgebras, Appl. Categor. Struct. 16(1-2), 255295 (2008)
30.Wolff, H., V-Localizations and V-monads, J. Algebra 24, 405438 (1973)



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed