Free Doubly-Infinitary Distributive Categories are Cartesian Closed
For category theorists, this provides a new natural condition and construction, but the result is primarily theoretical and incremental.
The paper introduces doubly-infinitary distributive categories, which have small products and coproducts with distribution, and shows that the free such category on any category is cartesian closed.
We study the composite free completion Dist(C) := Fam(Fam(C^op)^op), obtained by first freely adjoining small products and then freely adjoining small coproducts. A natural pseudodistributive law equips this endo-pseudofunctor with a composite pseudomonad structure. Its pseudoalgebras are precisely the categories with small products and small coproducts in which small products distribute over small coproducts. We call such categories doubly-infinitary distributive. This condition is natural, but does not seem to have been systematically isolated in the literature. Thus Dist(C) is the free doubly-infinitary distributive category on C. Our main result is that Dist(C) is cartesian closed. Finally, we compare doubly-infinitary distributivity with extensivity, ordinary infinitary distributivity, and cartesian closedness by means of separating examples.