For the non-cartesian case see at distributive monoidal 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 category $C$ with finite products $(-)\times(-)$ and finite coproducts $(-) + (-)$ is called (finitary) distributive if for any $X,Y,Z\in C$ the canonical distributivity morphism
is an isomorphism. The canonical morphism is the unique morphism such that $X\times Y \to X\times (Y+Z)$ is $X\times i$, where $i\colon Y\to Y +Z$ is the coproduct injection, and dually for $X\times Z \to X\times (Y+Z)$.
Equivalently, a category $C$ with finite products and coproducts is distributive if for any $X \in C$, the functor $X \times -$ preserves finite coproducts.
This notion is part of a hierarchy of distributivity for monoidal structures, and generalizes to distributive monoidal categories and rig categories. A linearly distributive category is not distributive in this sense.
This axiom on binary coproducts easily implies the analogous $n$-ary result for $n\gt 2$. In fact it also implies the analogous 0-ary statement that the projection
is an isomorphism for any $X$ (see Proposition below). Moreover, for a category with finite products and coproducts to be distributive, it actually suffices for there to be any natural family of isomorphisms $X\times Y + X\times Z \cong X\times (Y+Z)$, not necessarily the canonical ones (Lack).
A category $C$ with finite products and all small coproducts is infinitary distributive if the statement applies to all small coproducts. One can also consider $\kappa$-distributivity for a cardinal number $\kappa$, meaning the statement applies to coproducts of cardinality $\lt\kappa$.
Any extensive category with finite products is distributive, but the converse is not true.
In a category with products and coproducts, if products distribute over binary coproducts, then coproduct coprojections are monic.
Let $i_B: B \to B + C$ be a coproduct coprojection, and suppose given maps $f, g: A \to B$ such that $i_B f = i_B g$. We observe that the coprojection
is monic because it has a retraction $(1_{A \times B}, \phi): A \times B + A \times C \to A \times B$. (All we need here is the existence of a map $\phi: A \times C \to A \times B$, for example the composite $A \times C \stackrel{\pi_A}{\to} A \stackrel{\langle 1_A, f \rangle}{\to} A \times B$.)
The composite of the coprojection $i$ with the canonical isomorphism $A \times B + A \times C \cong A \times (B + C)$, namely $1_A \times i_B: A \times B \to A \times (B + C)$, is therefore also monic. Given that $\langle 1_A, i_B f \rangle = \langle 1_A, i_B g \rangle: A \to A \times (B + C)$, we conclude
whence $\langle 1_A, f\rangle = \langle 1_A, g\rangle: A \to A \times B$ since $1_A \times i_B$ is monic. It follows that $f = g$, as was to be shown.
If products distribute over binary coproducts, then products distribute over nullary coproducts (i.e., the projection $X \times 0 \to 0$ is an isomorphism for all objects $X$).
We show that $X \times 0$ is initial. Clearly $\hom(X \times 0, Y)$ is inhabited by $X \times 0 \to 0 \to Y$ for any object $Y$. On the other hand, since the two coprojections $0 \to 0 + 0$ coincide, the same holds for the two coprojections $X \times 0 \to (X \times 0) + (X \times 0)$, by applying the distributivity isomorphism $X \times (0 + 0) \cong (X \times 0) + (X \times 0)$. This is enough to show that any two maps $X \times 0 \to Y$ coincide, since given maps $f, g : X \times 0 \to Y$, we have $f = [f, g] \circ i_1 = [f, g] \circ i_2 = g$.
In a distributive category, the initial object is strict.
Given an arrow $f: A \to 0$, we have that $\pi_A: A \times 0 \to A$ is a retraction of $\langle 1, f \rangle: A \to A \times 0$, so that $A$ is a retract of $A \times 0 \cong 0$. But retracts of initial objects are initial.
Every bicartesian closed category is distributive since in that case the cartesian product $X \times -$ is a left adjoint and then the distributivity requirement is an instance of the preservation of colimits by left adjoints. For example,
But closedness is far from necessary, for example
is a distributive category.
These categories have in common that they are extensive. An example^{1} of a distributive category that is not extensive is given by
Since for posets viewed as categories, finite products and coproducts are given by meets and joins we find:
A further non-example:
The free distributive category on a category $\mathcal{C}$ with finite products is given by the finite free coproduct completion: $FinCoprod(\mathcal{C})$. That is, $FinCoprod(\mathcal{C})$ is distributive, and induces a bijection between distributive functors $FinCoprod(\mathcal{C})\to \mathcal{D}$ and product preserving functors $\mathcal{C}\to \mathcal{D}$, natural in distributive categories $\mathcal{D}$.
The free distributive category on a category $\mathcal{C}$ is the finite free coproduct completion of the finite free product completion: $FinCoprod(FinProd(\mathcal{C}))$. That is, $FinCoprod(FinProd(\mathcal{C}))$ is distributive, and induces a bijection between distributive functors $FinCoprod(FinProd(\mathcal{C}))\to \mathcal{D}$ and functors $\mathcal{C}\to \mathcal{D}$, natural in distributive categories $\mathcal{D}$.
In particular, then, the free distributive category, which is the free distributive category on the empty category, is the category of finite sets, up to equivalence.
Aurelio Carboni, Stephen Lack, R. F. C. Walters, Introduction to extensive and distributive categories, JPAA 84 (1993) pp. 145-158 [doi:10.1016/0022-4049(93)90035-R]
Robin Cockett, Introduction to distributive categories, Mathematical Structures in Computer Science 3 (2009) 277-307 [doi:10.101/S0960129500000232]
Stephen Lack, Non-canonical isomorphisms, Journal of Pure and Applied Algebra 216 3 (2012) 593-597 [arXiv:0912.2126, doi:10.1016/j.jpaa.2011.07.012]
See also
Freyd, Lack, Lawvere et al: CATLIST discussion on extensive categories 1996
Wikipedia, Distributive category
For more see at distributive monoidal category.
Pointed out by Peter Freyd in this discussion. ↩
Last revised on November 27, 2023 at 23:46:28. See the history of this page for a list of all contributions to it.