category with duals (list of them)
dualizable object (what they have)
is an isomorphism. The canonical morphism is the unique morphism such that is , where is the coproduct injection, and dually for .
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 -ary result for . In fact it also implies the analogous 0-ary statement that the projection
is an isomorphism for any (see Proposition 2 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 , not necessarily the canonical ones; see the paper of Lack referenced below.
A category with finite products and all small coproducts is infinitary distributive if the statement applies to all small coproducts. One can also consider -distributivity for a cardinal number , meaning the statement applies to coproducts of cardinality .
Any extensive category is distributive, but the converse is not true.
Let be a coproduct coprojection, and suppose given maps such that . We observe that the coprojection
is monic because it has a retraction . (All we need here is the existence of a map , for example the composite .)
The composite of the coprojection with the canonical isomorphism , namely , is therefore also monic. Given that , we conclude
whence since is monic. It follows that , as was to be shown.
If products distribute over binary coproducts, then products distribute over nullary coproducts (i.e., the projection is an isomorphism for all objects ).
Clearly is inhabited by for any object . On the other hand, since the two coprojections coincide, the same holds for the two coprojections , by applying the distributivity isomorphism . This is enough to show that any two maps coincide.
Given an arrow , we have that is a retraction of , so that is a retract of . But retracts of initial objects are initial.