Simplicial effects and weakly associative partial groups

TL;DR

The paper develops a unified simplicial framework to study quantum-measurement-like structures by weakening associativity between effect algebras and related multi-object algebroids. It introduces four interrelated categories—partial monoids, weak partial monoids, partial unital magmas, and partial unital magmas with associativity data—and uses nerve constructions to connect them to 2-Segal and weak 2-Segal simplicial sets. A central contribution is the definition of simplicial effects as spiny, inverseless, weakly 2-Segal cyclic sets, along with a key nontrivial example that is a simplicial effect but not an effect algebroid; the work also analyzes invertibility, cyclic cohomology, and state spaces, including Gleason-type results in this generalized setting. The framework broadens the algebraic toolkit for quantum foundations, enabling new insights into simplicial distributions, contextuality, and the structure of quantum measurements beyond traditional effect algebras. Overall, the paper bridges combinatorial, categorical, and operator-algebraic approaches to provide a flexible, higher-categorical lens on effects and their simplicial manifestations.

Abstract

In this paper, we introduce a new category of simplicial effects that extends the categories of effect algebras and their multi-object counterpart, effect algebroids. Our approach is based on relaxing the associativity condition satisfied by effect algebras and, more generally, partial monoids. Within this framework, simplicial effects and weakly associative partial groups arise as two extreme cases in the category of weak partial monoids. Our motivation is to capture simplicial structures from the theory of simplicial distributions and measurements that behave like effects.

Figures (4)

• Figure 1: The action of the cyclic automorphism $\tau_2$ determined by the central element $0\in \mathds{Z}_{/3}$ on 2-simplices of $P_{\EuScript{H}}(N(\mathds{Z}_{/3}))$.
• Figure 2: Triangulations of the planar $4$-gon with ordered vertices $P_4$. We sometimes denote the former triangulation by $\mathop{\mathrm{}}\nolimits$ and the latter by $\mathop{\mathrm{}}\nolimits$.
• Figure 3: The poset $\EuScript{I}_3$.
• Figure 4: The three face maps of a 2-simplex in $P_{\EuScript{H}}(N(\mathds{Z}/3))$. Projectors inside the simplex highlighted in the same color are summed to yield the projector outside the simplex highlighted in that color.

Theorems & Definitions (27)

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