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The operadic theory of convexity

Redi Haderi, Cihan Okay, Walker H. Stern

Abstract

In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of $\scr{O}$-monoidal categories, we state and prove a Grothendieck construction for lax $\scr{O}$-monoidal functors into convex sets. We apply this construction to the categorical characterization of entropy of Baez, Fritz, and Leinster, and to the study of quantum contextuality in the framework of simplicial distributions.

The operadic theory of convexity

Abstract

In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of -monoidal categories, we state and prove a Grothendieck construction for lax -monoidal functors into convex sets. We apply this construction to the categorical characterization of entropy of Baez, Fritz, and Leinster, and to the study of quantum contextuality in the framework of simplicial distributions.
Paper Structure (19 sections, 32 theorems, 86 equations)

This paper contains 19 sections, 32 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The convex tensor product
  3. $\EuScript{O}$-monoidal categories and the Grothendieck construction
  4. Structure of the paper
  5. Acknowledgements
  6. Convex sets
  7. The convexity monad and first properties
  8. The join
  9. The convex tensor product
  10. PROPs and operads for convexity
  11. The $\operatorname{Mat}_R$ and $\operatorname{Conv}_R$ PROPs
  12. Convex objects
  13. The convex Grothendieck construction
  14. The naïve convex Grothendieck construction
  15. $\operatorname{QConv}_R$-monoidal categories
  16. ...and 4 more sections

Key Result

Proposition 1

The Grothendieck construction induces an equivalence of categories \begin{tikzcd} \cint_\scr{C}:&[-3em]\Fun(\scr{C},\CSet) \arrow[r] & \Conv(\sf{DFib}(\scr{C})). \end{tikzcd}

Theorems & Definitions (113)

  • Proposition
  • Theorem
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 103 more