Free Doubly-Infinitary Distributive Categories are Cartesian Closed
Abstract
We investigate categories in which products distribute over coproducts, a structure we call doubly-infinitary distributive categories. Through a range of examples, we explore how this notion relates to established concepts such as extensivity, infinitary distributivity, and cartesian closedness. We show that doubly-infinitary distributivity strictly strengthens the classical notion of infinitary distributivity. Moreover, we prove that free doubly-infinitary distributive categories are cartesian closed, unlike free distributive categories. The paper concludes with observations on non-canonical isomorphisms, alongside open questions and directions for future research.
Cite
@article{arxiv.2403.10447,
title = {Free Doubly-Infinitary Distributive Categories are Cartesian Closed},
author = {Fernando Lucatelli Nunes and Matthijs Vákár},
journal= {arXiv preprint arXiv:2403.10447},
year = {2025}
}
Comments
17 pages, fixed notation and typos, and adapted some notation for the audience. Moreover, added the example of pseudo-topological spaces (to be coherent with the introduction of the corresponding PhD thesis)