**Definitions**

**Transfors between 2-categories**

**Morphisms in 2-categories**

**Structures in 2-categories**

**Limits in 2-categories**

**Structures on 2-categories**

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).

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.

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

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

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)

Last revised on October 7, 2022 at 13:03:32. See the history of this page for a list of all contributions to it.