## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 19, Number 5 (2012), 769-820.

### Monoidal categories in, and linking, geometry and algebra

#### Abstract

This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, machinery such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might be, used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5 (2012), 769-820.

**Dates**

First available in Project Euclid: 27 November 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1354031551

**Digital Object Identifier**

doi:10.36045/bbms/1354031551

**Mathematical Reviews number (MathSciNet)**

MR3009017

**Zentralblatt MATH identifier**

1267.18006

**Subjects**

Primary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 18D20: Enriched categories (over closed or monoidal categories) 18D35: Structured objects in a category (group objects, etc.) 20C08: Hecke algebras and their representations 20C30: Representations of finite symmetric groups 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 81T45: Topological field theories [See also 57R56, 58Dxx] 20C33: Representations of finite groups of Lie type

**Keywords**

monoidal category enriched category braiding string diagram duoidal category Mackey functor Green functor manifold invariant topological quantum field theory Day convolution Joyal species finite general linear group link invariant cuspidal representation

#### Citation

Street, Ross. Monoidal categories in, and linking, geometry and algebra. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 5, 769--820. doi:10.36045/bbms/1354031551. https://projecteuclid.org/euclid.bbms/1354031551