category with duals (list of them)
dualizable object (what they have)
X\times Y + X\times Z \to X\times (Y+Z)
is an isomorphism. The canonical morphism is the unique morphism such that is , where is the coproduct injection, and dually for .
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
X\times 0 \to 0
is an isomorphism for any . 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.