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Free Doubly-Infinitary Distributive Categories are Cartesian Closed (2403.10447v6)

Published 15 Mar 2024 in math.CT, cs.LO, cs.PL, and math.LO

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.

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