For the cartesian case see at distributive category.
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
A distributive monoidal category is a monoidal category whose tensor product distributes over coproducts.
A distributive monoidal category (this is not entirely standard terminology) is a monoidal category with coproducts whose tensor product preserves coproducts in each variable: i.e. such that the canonical morphisms
are isomorphisms.
Depending on the arity of the coproducts in question, we may speak of a finitary or infinitary distributive monoidal category.
The special case of distributive cartesian monoidal categories is known simply as distributive categories. Conversely, distributive monoidal categories are a special case of rig categories.
See also at distributivity for monoidal structures.
A more abstract way to say this, due to Weber and Batanin, is that if is the free monoidal category monad and is the structure map of a monoidal category , then is distributive if it admits left Kan extensions along functors between discrete categories (of some size), and moreover if
is such a Kan extension, then so is
Beyond distributive categories, examples of distributive monoidal categories include the following:
Ab, the category of abelian groups equipped with the tensor product of abelian groups
Mod, the category of modules over a commutative ring , equipped with the tensor product of modules
Vect = Mod, the category of vector space over some field , equipped with the tensor product of vector spaces,
Vect(X), the category of (topological) vector bundles over some (topological) space, equipped with the tensor product of vector bundles,
In all these cases the coproduct is the respective direct sum (e.g. direct sum of vector bundles in the case of vector bundles).
Also:
the category of pointed sets with respect to forming smash product and wedge sum,
more generally, the category of pointed topological spaces with respect to forming smash product and wedge sum (e.g. Hatcher, Section 4.F).
VectBund, the category of (topological) vector bundles over any (convenient) topological spaces, equipped with the external tensor product of vector bundles (more on this below).
We spell out in much detail the example of the category of vector bundles over arbitrary base spaces and equipped with the external tensor product of vector bundles — for the simple special case that the base spaces are discrete topological spaces, i.e. plain sets:
For any ground field, write for the category of indexed sets of vector spaces.
We may and will present objects of as pairs consisting of a set and a function (really a functor on the discrete category on ) to Vect:
The coproduct in is given by disjoint union of bundles:
It is immediate to check the universal property characterizing the coproduct.
The external tensor product (Def. ) distributes over the coproduct (Prop. ):
and hence gives a distributive monoidal category:
Unwinding the above definitions and using that Set is a distributive category, we have the following sequence of natural isomorphisms:
A monoidal category is finitary distributive if its tensor product preserves binary coproducts in each variable and the monoidal unit is weakly terminal (e.g., if there is a morphism out of a terminal object).
There is a canonical isomorphism
and thus a canonical isomorphism
whose restriction along the coproduct inclusion is the identity . Let be the restriction of along the other coproduct inclusion. Then induces an evident bijection
Since is inhabited for all (with the help of some map , there is some map ), this forces to be a singleton for any , so that is initial.
A distinguishing feature of (infinitary) distributive monoidal categories is that the monad for monoid objects in such a category has a particularly simple expression:
The same is true for the monad on enriched graphs whose algebras are categories enriched over such a monoidal category. This also generalizes to lax monoidal categories, a.k.a. “multitensors”; see Weber 13.
Hans-Joachim Baues, Mamuka Jibladze, Andy Tonks, first page of: Cohomology of monoids in monoidal categories, in: Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202, AMS (1997) 137-166 [doi:10.1090/conm/202, preprint:pdf]
Anna Labella, Categories with sums and right distributive tensor product, Journal of Pure and Applied Algebra 178 3 (2003) 273-296 [doi:10.1016/S0022-4049(02)00169-X]
Mark Weber, Multitensors and monads on categories of enriched graphs, TAC 28 26 (2013) [tac:28-26]
Last revised on October 15, 2024 at 21:25:25. See the history of this page for a list of all contributions to it.