Spherical Categories

TL;DR

This paper introduces spherical categories, a new class of monoidal categories with duals, and explores their properties, examples, and applications in topological field theories and representation theory of Hopf algebras.

Contribution

It defines spherical categories, proves a coherence theorem, and constructs examples including spherical Hopf algebras, expanding the framework for topological and algebraic applications.

Findings

Spherical categories generalize monoidal categories with duals.

The natural quotient of a spherical category is semisimple.

Examples include involutory and ribbon Hopf algebras.

Abstract

This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the third section we define spherical Hopf algebras so that the category of representations is spherical. Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural quotient in these cases and show it is semisimple.