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Introduction to monoidal and tensor categories and their applications in modern mathematics. Beginning with a
review of basic category theory, we then explore the theory of abelian categories, monoidal categories, rigid
monoidal categories, tensor categories, and module categories. Moreover, we develop the dual theory of
coalgebras, bialgebras, Hopf algebras, and quantum groups, with a strong emphasis on their categorical
structures.
Last update: Šmíd Dalibor, Mgr., Ph.D. (11.06.2025)
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Oral exam. Last update: Šmíd Dalibor, Mgr., Ph.D. (11.06.2025)
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1) T. Leinster: Basic Category Theory, Cambridge Studies in Advanced Mathematics 143, Cam- bridge University Press, 2014. Available at arXiv:1612.09375.
2) S. MacLane: Categories for the Working Mathematician, Springer Verlag, Berlin, 1971
3) J. Adamek, H. Herrlich, G. Strecker: Abstract and Concrete Categories , John Wiley, New York, 1990
4) Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015). ISBN: 978-1-4704-3441-0. Last update: Šmíd Dalibor, Mgr., Ph.D. (11.06.2025)
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Review: Categories, functors, functor categories, adjunctions, (co)limits, abelian categories.
Monoidal categories: Definition and basic results, monoidal functors, MacLane’s theorems, rigid objects.
Tensor/multitensor categories: Unit objects, Grothendieck rings, fiber functors.
Coalgebras, bialgebras, Hopf algebras; quantum groups, pointed tensor categories.
Quasi-(bi)algebras, quasi-Hopf algebras, quantum traces; pivotal and spherical categories.
Basics of Module categories: definitions and basic results of module categories, exact module categories, algebras in categories, internal homs, categories of module functors, dual categories. Last update: Šmíd Dalibor, Mgr., Ph.D. (11.06.2025)
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