With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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 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.
Last revised on March 25, 2015 at 10:33:18. See the history of this page for a list of all contributions to it.