Abstract
We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in Agore (Proc Am Math Soc 139:855–863, 2011) on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.
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References
Agore A.L.: Limits of Coalgebras, Bialgebras and Hopf Algebras. Proc. Amer. Math. Soc. 139, 855–863 (2011)
Agore A.L.: Categorical Constructions for Hopf Algebras. Comm. Algebra 39, 1476–1481 (2011)
B. Bakalov and A. Kirillov Jr., Lectures on Tensor Categories and Modular Functor, AMS University Lecture Series, 2001, 221 pp
Barr M.: Coalgebras Over a Commutative Ring. J. Algebra 32, 600–610 (1974)
Block R.E., Leroux P.: Generalized Dual Coalgebras of Algebras, with Applications to Cofree Coalgebras. J. Pure Appl. Algebra 36, 15–21 (1985)
Brzezinski T.: The Structure of Corings. Induction Functor, Maschke-Type Theorem and Frobenius and Galois-Type Properties. Algebr. Represent. Theory 5, 389–410 (2002)
T. Brzezinski and R. Wisbauer, Corings and comodules, London Mathematical Society Lecture Note Series Vol 309, Cambridge University Press, 2003
Cohn P.M.: On the Free Product of Associative Rings. Math. Z. 71, 380–398 (1959)
S. Dascalescu, C. Nastasescu and S. Raianu, Hopf Algebras: An Introduction., Marcel Dekker, Inc. 2001
T.F. Fox, The Construction of Cofree Coalgebras, J. Pure Appl. Algebra 84, 1993
Hazewinkel M.: Cofree Coalgebras and Multivariable Recursiveness. J. Pure Appl. Algebra 183, 61–103 (2003)
S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics vol. 5, 2nd ed., Springer, 1978
Molnar R.K.: Semi-direct products of Hopf algebras. J. Algebra 47, 29–51 (1977)
Pareigis B., Sweedler M.E.: On Generators and Cogenerators. Manuscripta Math. 2, 49–66 (1970)
Porst H.-E.: On corings and comodules. Arch. Math. (Brno) 42, 419–425 (2006)
Porst H.-E.: Fundamental constructions for coalgebras, corings, and comodules. Appl. Categ. Structures 16, 223–238 (2008)
Porst H.-E.: On Categories of Monoids, Comonoids, and Bimonoids. Quaestiones Math. 31, 127–139 (2008)
Porst H.-E.: Universal Constructions for Hopf Algebras. J. Pure Appl. Algebra 212, 2547–2554 (2008)
D.E. Radford, Hopf Algebras, World Scientific Publishing, 2012
M.E. Sweedler, The Predual Theorem to the Jacobson-Bourbaki Theorem, Trans. Amer. Math. Soc. 213, 1975
R. Wisbauer, Foundations of Module and Ring Theory: A Handbook for Study and Research., Springer-Verlag. 1991
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Abdulwahid, A.H., Iovanov, M.C. Generators for comonoids and universal constructions. Arch. Math. 106, 21–33 (2016). https://doi.org/10.1007/s00013-015-0826-6
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DOI: https://doi.org/10.1007/s00013-015-0826-6