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Monoidal categories and monoidal 2-categories

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1765)

Abstract

In this chapter we provide the definitions of various algebraic structures that will be the building blocks for our construction of TQFT functors. In particular, we will review braided abelian tensor categories (BTQ, the properties of Hopf algebras in such BTC’s, as well as the construction of a symmetric monoidal 2-category of abelian categories. Our discussion will also include a number of new lemmas that will significantly simplify the proof of topological invariance in Chapter 6.

More specifically, in the discussion of braided tensor categories we give a detailed discussion of rigidity in BTC’s, the various ribbon and balancing elements, and their relations. As a result, we obtain that any BTC is equivalent to one, which is strictly rigid, i.e., we have X = X vv and the canonical balancing is just the identity.

Keywords

  • Hopf Algebra
  • Natural Transformation
  • Monoidal Category
  • Abelian Category
  • Tensor Category

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Monoidal categories and monoidal 2-categories. In: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Lecture Notes in Mathematics, vol 1765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44625-7_5

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  • DOI: https://doi.org/10.1007/3-540-44625-7_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42416-1

  • Online ISBN: 978-3-540-44625-5

  • eBook Packages: Springer Book Archive