Distributive category explained

A,B,C , the canonical map[ id _{A} x \iota _{1,} id _{A} x \iota _{2]} : A x B + A x C \to A x (B + C)

is an isomorphism, and for all objects

A , the canonical map 0 \to A x 0 is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object A the endofunctor A x - defined by B\mapsto A x B preserves coproducts up to isomorphisms f .^[1] It follows that f and aforementioned canonical maps are equal for each choice of objects. A x - 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,, and be sets. Then

\begin{align} A x (B\amalg C) & = \{(a,d) \mid a \in A and d \in B \amalg C\} \\ &\cong \{(a,d) \mid a \in A and d \in B\} \amalg \{(a,d) \mid a \in A and d \in C\} \\ & = (A x B) \amalg (A x C) \end{align} where \amalg denotes the coproduct in Set, namely the disjoint union, and \cong denotes a bijection. In the case where,, and are finite sets, this result reflects the distributive property: the above sets each have cardinality |A| ⋅ (|B| + |C|) = |A| ⋅ |B| + |A| ⋅ |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.^[2]

Further reading

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

Notes and References

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