nLab distributivity for monoidal structures

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.

• 2-fold monoidal categories and duoidal categories have two monoidal structures, but rather than a “distributivity” morphism as above, they have transformations $(X\star Y)\odot (Z\star W) \to (X\odot Z) \star (Y\odot W)$.

• A linearly distributive category also has two monoidal structures, but its comparison morphisms have the form $X\otimes (Y\bullet Z) \to (X\otimes Y)\bullet Z$.

• A 2-rig can mean a lot of different things, perhaps a distributive monoidal category but perhaps also including additivity or cocompleteness.

Related entries

• distributive law

• distributivity for monoidal structures

  • distributive monoidal category

  • bipermutative category

  • linearly distributive category

  • rig category