nLab reverse monoidal category

Definition

The reverse monoidal category of a monoidal category, $\mathcal{C}$, has the same underlying category and unit as $\mathcal{C}$ but reversed monoidal product, $X \otimes^{rev} Y = Y \otimes X$, and similarly for tensors of morphisms. The associator in the reverse category is $\alpha^{rev}_{X,Y,Z} = \alpha^{-1}_{Z,Y,X}$.

We have $\Sigma(V^{rev})\simeq(\Sigma V)^{op}$ where $\Sigma V$ is the delooping of $V$, i.e. $V$ viewed as a one-object bicategory where ${op}$ is the opposite on 1-cells.

Note that this notion is different from that of the opposite category, which involves reversing the order of the arrows.

Related pages

• opposite category
• opposite 2-category