# nLab distributivity for monoidal structures

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## Distributivity for monoidal structures

There are many variations on what it means for one monoidal structure on a category to distribute over another. Here we collect a list of them and remark on their relationships. Note that our terminology is by no means universal.

The following notions of distributivity exist in a linear hierarchy of less to more general.

• A distributive category has finite products and coproducts (hence is both cartesian and cocartesian monoidal), and the former distribute over the latter, in that the canonical morphism $(X\times Y) + (X\times Z) \to X\times (Y+Z)$ is an isomorphism.

• A distributive monoidal category is a monoidal category with coproducts whose tensor product preserves coproducts in each variable separately. If it is cartesian monoidal, then it is exactly a distributive category as above.

• A rig category (also called a bimonoidal category) is a category with two monoidal structures, say $\otimes$ and $\oplus$, together with coherent natural distributivity isomorphisms such as $X\otimes (Y\oplus Z) \cong (X\otimes Y) \oplus (X\otimes Z)$. Generally one requires $\oplus$ to be symmetric. If $\oplus$ is a cocartesian structure, then it is exactly a distributive monoidal category as above.

• A bipermutative category is a “semistrict” symmetric rig category: both $\otimes$ and $\oplus$ are permutative categories (symmetric strict monoidal categories) and one of the distributivity isomorphisms is an identity.
• A colax-distributive rig category is like a rig category, but the distributivity morphisms are not assumed to be invertible.

There are also the following related notions which are not comparable in generality.

Last revised on March 25, 2015 at 10:33:18. See the history of this page for a list of all contributions to it.