The Operadic Theory of Convexity

In this paper, we present an operadic characterization of convexity utilizing a PROP which governs convex structures and derive several convex Grothendieck constructions. Our main focus is a Grothendieck construction which simultaneously captures convex structures and monoidal structures on categories. Our proof of this Grothendieck construction makes heavy use of both our operadic characterization of convexity and operads governing monoidal structures. We apply these new tools to two key concepts: entropy in information theory and quantum contextuality in quantum foundations. In the former, we explain that Baez, Fritz, and Leinster’s categorical characterization of entropy has a more natural formulation in terms of continuous, convex-monoidal functors out of convex Grothendieck constructions; and in the latter we show that certain convex monoids used to characterize contextual distributions naturally arise as convex Grothendieck constructions.