nLab opposite 2-category

Definition

A 2-category $K$ has three different opposites (two independent):

• The $1$-cell dual $K^{op}$, in which the $1$-cells but not the $2$-cells are reversed.
• The $2$-cell dual $K^{co}$, in which the $2$-cells but not the $1$-cells are reversed.
• The $1$- and $2$-cell dual $K^{co op}$ in which both $1$-cells and $2$-cells are reversed.

Interaction with 2-functor 2-categories

The category 2-Cat is cartesian closed, with the exponential $B^A$ being given by the 2-functor 2-category $[A, B]$ whose objects are (strict) 2-functors from $A$ to $B$, whose 1-cells are 2-natural transformations, and whose 2-cells are modifications.

• $[A, B]^{op} = [A^{op}, B^{op}]$
• $[A, B]^{co} = [A^{co}, B^{co}]$
• $[A, B]^{co op} = [A^{co op}, B^{co op}]$

Related pages

• opposite category

• contravariant 2-functor

• 3-category with contravariance

• opposite adjunction

• reverse monoidal category