Free Doubly-Infinitary Distributive Categories are Cartesian Closed

Abstract We study the composite free completion $$ \textsf{Dist}(\mathcal {C}):=\textsf{Fam}\!\bigl (\textsf{Fam}(\mathcal {C}^{op})^{op}\bigr ), $$ 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 $$\textsf{Dist}(\mathcal {C})$$ Dist ( C ) is the free doubly-infinitary distributive category on $$\mathcal {C}$$ C . Our main result is that $$\textsf{Dist}(\mathcal {C})$$ Dist ( C ) is cartesian closed . Finally, we compare doubly-infinitary distributivity with extensivity, ordinary infinitary distributivity, and cartesian closedness by means of separating examples.