Monoidal categories in, and linking, geometry and algebra.(English)Zbl 1267.18006

This paper is a survey of some recent developments in the theory of monoidal categories with applications and connections to various topics, including knot theory, representation theory, 3-manifold invariants, and conformal field theories. In the first section, the author provides basic background information on monoidal categories, string diagrams, duals, braidings, and tangles. Section 2 introduces promonoidal structures, Day convolution, and Hecke algebroids, and gives applications to the representation theory of the finite general linear group and Hall algebras. Sections 3 and 4 are concerned with monoidal centres, Mackey functors, and duoidal categories, motivated in part by the Reshetikhin-Turaev and Turaev-Viro invariants. The paper also contains an extensive list of references.

MSC:

 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 18D20 Enriched categories (over closed or monoidal categories) 18D35 Structured objects in a category (MSC2010) 20C08 Hecke algebras and their representations 20C30 Representations of finite symmetric groups 57M25 Knots and links in the $$3$$-sphere (MSC2010) 81T45 Topological field theories in quantum mechanics 20C33 Representations of finite groups of Lie type
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