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arXiv · 2403.10447

Free Doubly-Infinitary Distributive Categories are Cartesian Closed

Fernando Lucatelli Nunes·Matthijs Vákár
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Abstract

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.

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Fernando Lucatelli Nunes, Matthijs Vákár. 2024-03-15. Free Doubly-Infinitary Distributive Categories are Cartesian Closed. https://doi.org/10.1007/s10485-026-09887-7

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