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.


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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