nLab
reverse category

Context

Monoidal categories

monoidal categories

With symmetry

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 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 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}.