nLab bicartesian category

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

Category theory

Contents

Definition

A category is bicartesian if it is both cartesian and cocartesian, that is if both it and its opposite may be made into cartesian monoidal categories. Note this is different than a semiadditive category, i.e. a category with coinciding products and coproducts. Thus every semiadditive category is bicartesian, but not every bicartesian category is semiadditive.

A bicartesian category which is also cartesian closed is a bicartesian closed category. Bicartesian closed categories are usually not cocartesian closed.

Last revised on April 11, 2025 at 08:24:12. See the history of this page for a list of all contributions to it.