Core Concepts
Characterizing convexity through algebras over a PROP and exploring tensor-product-like structures.
Abstract
The article delves into the operadic theory of convexity, introducing the concept of algebras over a PROP to characterize convexity. It establishes a tensor-product-like symmetric monoidal structure on convex sets, exploring the Grothendieck construction for lax O-monoidal functors. The paper applies this theory to entropy characterization and quantum contextuality within simplicial distributions.
- Introduction
- Convexity's fundamental role in various fields.
- Convex Sets
- Introduction to the convexity monad and properties.
- PROPs and Operads for Convexity
- Definition of colored PROPs and the MatR and ConvR PROPs.
- Convex Grothendieck Construction
- Generalizations enabled by the operadic approach.
- Examples and Applications
- Application of the theory to entropy and quantum contextuality.
Stats
The Grothendieck construction induces an equivalence of categories C Z C: Fun(C, CSet) Conv(DFib(C)).
The Grothendieck construction induces an equivalence of categories C Z O I: FunO,lax((I, ⊙), (CSet, ⊗)) OCFibI.
Quotes
"Convexity's role in probability theory, optimization, and beyond reveals great depths hidden behind the simple definition."
"The categorical machinery is itself, in its application to convexity, concrete, tangible, and comprehensible."