A CATEGORICAL APPROACH TO ALGEBRAS AND COALGEBRAS

Year 2018, , 153 - 173, 05.07.2018

Robert Wisbauer

https://doi.org/10.24330/ieja.440239

Cited By: 1

Abstract

Algebraic and coalgebraic structures are often handled independently.

In this survey we want to show that they both show up naturally when

approaching them from a categorical point of view. Azumaya, Frobenius, separable,

and Hopf algebras are obtained when both notions are combined. The

starting point and guiding lines for this approach are given by adjoint pairs of

functors and their elementary properties.

Keywords

(Co)algebras, (co)monads, Frobenius algebras, separable algebras, Hopf algebras

References

• J. Beck, Distributive laws, in Seminar on Triples and Categorical Homology Theory, B. Eckmann (ed.), Springer, Berlin, 80 (1969), 119-140.
• G. Bohm, T. Brzezinski and R. Wisbauer, Monads and comonads on module categories, J. Algebra, 322(5) (2009), 1719-1747.
• T. Brzezinski and R. Wisbauer, Corings and Comodules, London Mathematical Society Lecture Note Series, 309, Cambridge University Press, Cambridge, 2003.
• J. Clark and R. Wisbauer, Idempotent monads and ?-functors, J. Pure Appl. Algebra, 215(2) (2011), 145-153.
• S. Eilenberg and S. Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc., 58(2) (1945), 231-294.
• S. Eilenberg and J. C. Moore, Adjoint functors and triples, Illinois. J. Math., 9 (1965), 381-398.
• F. Frobenius, Theorie der hypercomplexen Groben, Sitz. Kon. Preuss. Akad. Wiss., (1903), 504-537; Gesammelte Abhandlungen, art. 70, 284-317.
• H. Hopf,  Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, Ann. of Math., 42(2) (1941), 22-52.
• S. O. Ivanov, Nakayama functors and Eilenberg-Watts theorems, J. Math. Sci., 183(5) (2012), 675-680.
• D. M. Kan, Adjoint functors, Trans. Amer. Math. Soc., 87 (1958), 294-329.
• H. Kleisli, Every standard construction is induced by a pair of adjoint functors, Proc. Amer. Math. Soc., 16 (1965), 544-546.
• S. Mac Lane, Categories for the Working Mathematician, 2nd edn, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.
• R. Marczinzik, A bocs theoretic characterization of gendo-symmetric algebras, J. Algebra, 470 (2017), 160-171.
• B. Mesablishvili, Entwining structures in monoidal categories, J. Algebra, 319(6) (2008), 2496-2517.
• B. Mesablishvili and R. Wisbauer, Galois functors and entwining structures, J. Algebra, 324(3) (2010), 464-506.
• B. Mesablishvili and R. Wisbauer, Bimonads and Hopf monads on categories, J. K-Theory, 7(2) (2011), 349-388.
• B. Mesablishvili and R. Wisbauer, Notes on bimonads and Hopf monads, Theory Appl. Categ., 26(10) (2012), 281-303.
• B. Mesablishvili and R. Wisbauer, QF functors and (co)monads, J. Algebra, 376 (2013), 101-122.
• B. Mesablishvili and R. Wisbauer, The fundamental theorem for weak braided bimonads, J. Algebra, 490 (2017), 55-103.
• J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math., 81(2) (1965), 211-264.
• K. Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 6 (1958), 83-142.
• A. V. Roiter, Matrix problems and representations of BOCSs, Representation Theory I, Lecture Notes in Math., 831, Springer, Berlin-New York, (1980), 288-324.
• M. Sato, Fuller's theorem on equivalences, J. Algebra, 52(1) (1978), 274-284.
• M. Sato, On equivalences between module subcategories, J. Algebra, 59(2) (1979), 412-420.
• R. Street, Frobenius monads and pseudomonoids, J. Math. Phys., 45(10) (2004), 3930-3948.
• D. Turi and G. Plotkin, Towards a mathematical operational semantics, Proceedings 12th Ann. IEEE Symp. on Logic in Computer Science, LICS'97, Warsaw, Poland, (1997).
• R. Wisbauer, Foundations of Module and Ring Theory, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
• R. Wisbauer, Tilting in module categories, Abelian groups, module theory, and topology (Padua, 1997), Lecture Notes in Pure and Appl. Math., 201, Dekker, New York, (1998), 421-444.
• R. Wisbauer, Static modules and equivalences, Interactions between ring theory and representations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., 210, Dekker, New York, (2000), 423-449.
• R. Wisbauer, Algebra versus coalgebras, Appl. Categ. Structures, 16 (2008), 255-295.
• R. Wisbauer, Comodules and contramodules, Glasg. Math. J., 52(A) (2010), 151-162.
• R. Wisbauer, Regular pairings of functors and weak (co)monads, Algebra Discrete Math., 15(1) (2013), 127-154.
• R. Wisbauer, Weak Frobenius monads and Frobenius bimodules, Algebra Discrete Math., 21(2) (2016), 287-308.
• R. Wisbauer, Separability in algebra and category theory, Proc. Aligarh, (2016).
• J. Worthington, A bialgebraic approach to automata and formal language theory, Ann. Pure Appl. Logic, 163(7) (2012), 745-762.