Misplaced Pages

This article may require cleanup to meet Misplaced Pages's quality standards. The specific problem is: there's more than one proposed notion under this name, see last ref in further reading. Please help improve this article if you can. (July 2014) (Learn how and when to remove this message)

In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects ${\displaystyle A,B,C}$, the canonical map

${\displaystyle :A\!\times \!B\,+A\!\times \!C\to A\!\times \!(B+C)}$

is an isomorphism, and for all objects ${\displaystyle A}$, the canonical map ${\displaystyle 0\to A\times 0}$ is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object ${\displaystyle A}$ the endofunctor ${\displaystyle A\times -}$ defined by ${\displaystyle B\mapsto A\times B}$ preserves coproducts up to isomorphisms ${\displaystyle f}$. It follows that ${\displaystyle f}$ and aforementioned canonical maps are equal for each choice of objects.

In particular, if the functor ${\displaystyle A\times -}$ has a right adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any bicartesian closed category) is distributive.

Example

The category of sets is distributive. Let A, B, and C be sets. Then

${\displaystyle {\begin{aligned}A\times (B\amalg C)&=\{(a,d)\mid a\in A{\text{ and }}d\in B\amalg C\}\\&\cong \{(a,d)\mid a\in A{\text{ and }}d\in B\}\amalg \{(a,d)\mid a\in A{\text{ and }}d\in C\}\\&=(A\times B)\amalg (A\times C)\end{aligned}}}$

where ${\displaystyle \amalg }$ denotes the coproduct in Set, namely the disjoint union, and ${\displaystyle \cong }$ denotes a bijection. In the case where A, B, and C are finite sets, this result reflects the distributive property: the above sets each have cardinality ${\displaystyle |A|\cdot (|B|+|C|)=|A|\cdot |B|+|A|\cdot |C|}$.

The categories Grp and Ab are not distributive, even though they have both products and coproducts.

An even simpler category that has both products and coproducts but is not distributive is the category of pointed sets.

References

1. Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 275.
2. F. W. Lawvere; Stephen Hoel Schanuel (2009). Conceptual Mathematics: A First Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296–298. ISBN 978-0-521-89485-2.

Further reading

• Cockett, J. R. B. (1993). "Introduction to distributive categories". Mathematical Structures in Computer Science. 3 (3): 277–307. doi:10.1017/S0960129500000232.
• Carboni, Aurelio (1993). "Introduction to extensive and distributive categories". Journal of Pure and Applied Algebra. 84 (2): 145–158. doi:10.1016/0022-4049(93)90035-R.

This category theory-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Category theory
• Category theory stubs