nLab spherical category

Idea

A spherical category is a monoidal category with duals that behaves as if its morphisms can be drawn and moved around on a 2-sphere.

Definition

A spherical category is a pivotal category $\mathcal{C}$ where the left and right trace operations coincide on all objects.

If $\mathcal{C}$ is in addition a tensor category so that this trace may be interpreted as an element of the ground field, then the trace $tr\big( id_X\big)$ is called the quantum dimension of the object $X$ (e.g. FRS02 (2.17))

Related concepts

• Turaev-Viro model

• fusion category

• pivotal category

• spherical category

References

The definition is originally due to:

• John Barrett, Bruce Westbury, Spherical Categories (arXiv:hep-th/9310164)

Review:

• Michael Müger, section 2.3 of: From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories (arXiv:0111204)

• Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Section 2.1 of: TFT construction of RCFT correlators I: Partition functions, Nucl. Phys. B 646 (2002) 353-497 $[$arXiv:hep-th/0204148$]$

More is in:

• Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, On braided fusion categories I (arXiv:0906.0620)