With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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 $\alpha^{rev}_{X,Y,Z} = \alpha^{-1}_{Z,Y,X}$.
$V^{rev}$ is equivalent to $(\Sigma V)^{op}$ 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.
Last revised on October 21, 2023 at 18:31:26. See the history of this page for a list of all contributions to it.