category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 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 and , together with coherent natural distributivity isomorphisms such as . Generally one requires to be symmetric. If 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 .
A linearly distributive category also has two monoidal structures, but its comparison morphisms have the form .
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.