nLab reverse monoidal category

Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Definition

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

V revV^{rev} is equivalent to (ΣV) op(\Sigma V)^{op} where ΣV\Sigma V is the delooping of VV, i.e. VV viewed as a one-object bicategory where op{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.