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Free Doubly-Infinitary Distributive Categories are Cartesian Closed

Fernando Lucatelli Nunes, Matthijs Vákár

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.

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.
Paper Structure (22 sections, 14 theorems, 62 equations)

This paper contains 22 sections, 14 theorems, 62 equations.

Table of Contents

  1. Free completion under coproducts
  2. The Free Doubly-Infinitary Distributive Category
  3. Explicit description of the $\mathbf{Dist}$-construction
  4. Products and coproducts in $\mathbf{Dist}\left( {\mathbb{C}}\right)$
  5. Exponentials in $\mathbf{Dist}(\mathbb{C})$
  6. The Pseudomonad Structure of $\mathbf{Dist}$
  7. Free doubly-infinitary distributive categories on discrete categories
  8. Bicategorically semi-additive $2$-categories
  9. The construction
  10. Examples
  11. The $\mathbf{Fam}$ example
  12. The category of sets and beyond
  13. Posets
  14. Categories of categorical structures
  15. Category of topological spaces
  16. ...and 7 more sections

Key Result

Lemma 2.2

Coproducts exist in $\mathbf{Dist}\left( \mathbb{C}\right)$. More precisely, given a family of objects the coproduct is given by

Theorems & Definitions (45)

  • Definition 2.1: The Free Doubly-Infinitary Distributive Category
  • Lemma 2.2: Coproducts in $\mathbf{Dist}(\mathbb{C} )$
  • proof
  • Lemma 2.3: Products in $\mathbf{Dist}(\mathbb{C} )$
  • proof
  • Theorem 2.4
  • proof
  • proof
  • Remark 2.5: Exponential formulas
  • Remark 2.6: Inductive description of exponentials in $\mathbf{Dist} (\mathbb{C} )$
  • ...and 35 more