Effect Algebras as Omega-categories
Lorenzo Perticone, Robin Adams
TL;DR
This work establishes a bridge between quantum logic and higher category theory by showing that any effect algebra $X$ can be identified with a $\mathbf{gaEA}^*$-enriched strict $\omega$-category via a hierarchical family of constructions $\mathbb{B}^{(n)}X$ and $\mathbb{B}^\infty X$. Morphisms between objects become intervals $[x,y]$, themselves effect algebras, enabling enrichment and iteration to higher categorical levels; explicit descriptions of $k$-cells, sources, targets, identities, and composition are provided. The authors give concrete quantum-theoretic and logical examples, including $\omega$-categories of subspaces and effects on a Hilbert space, as well as connections to partitions of unity and discrete POVMs, highlighting the relationship to Boolean algebras and effectuses. The work opens directions for generalizations to broader algebraic structures and for exploring the higher-categorical properties and potential semantic applications in quantum logic and type theory.
Abstract
We show how an effect algebra $\mathcal{X}$ can be regarded as a category, where the morphisms $x \rightarrow y$ are the elements $f$ such that $x \leq f \leq y$. This gives an embedding $\mathbf{EA} \rightarrow \mathbf{Cat}$. The interval $[x,y]$ proves to be an effect algebra in its own right, so $\mathcal{X}$ is an $\mathbf{EA}$-enriched category. The construction can therefore be repeated, meaning that every effect algebra can be identified with a strict $ω$-category. We describe explicitly the strict $ω$-category structure for two classes of operators on a Hilbert space.