nLab symmetric bicategory

Definition

Definition

A symmetric bicategory (May & Sigurdsson 2006, Def. 16.2.1) is a bicategory $B$ equipped with a bijective-on-objects biequivalence $t \colon B \simeq B^{op}$ to its opposite 2-category.

A closed symmetric bicategory is a symmetric bicategory that is closed (May & Sigurdsson 2006, Def. 16.3.1).

Remark

The notion of symmetric bicategory is a horizontal categorification of that of a symmetric monoidal category. It is not to be confused with the notion of a symmetric monoidal bicategory, which is instead a vertical categorification of the same concept.

Note, however, that single-object symmetric bicategories are more expressive than symmetric monoidal categories: a symmetric bicategory with a single object, $Obj(B) \simeq \{\ast\}$, is a symmetric monoidal category if and only if the component functor of $t$ is the identity functor on the single hom-category $B(\ast,\ast)$.

A symmetric bicategory is a categorified dagger-category.

Examples

Example

Span is a symmetric bicategory, whose involution is given by reversing each span.

Example

Prof is a closed symmetric bicategory, whose involution is induced by the duality involution on Cat.

Related concepts

• closed bicategory

• duality involution

• dagger category

References

The notion of symmetric bicategories was introduced in Definition 16.2.1 (of the published pdf-version, not in the arXiv version, of):

• Peter May, Johann Sigurdsson, Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, pdf, pdf, arXiv:math/0411656)