Simplicial Perspectives on (Pseudo) Effect Algebras
Dominik Lachman
TL;DR
This work develops a combinatorial-topological framework for (pseudo) effect algebras by embedding Frobenius algebras in $\mathbf{Rel}$ into the world of $\epsilon$-simplicial sets via the nerve $N$. It proves that the universal group of an effect algebra $E$ is its first homology $H_1(N(E))$, and shows that the mapping spaces $[N(E),N(F)]$ realize a natural enrichment of the category of effect algebras over $\mathbf{RelFA}$. The analysis extends to pseudo effect algebras, establishing a Kan-fibration property for the map $ [N(E),N(F)] \to [N(\underline{1}),N(F)]$ and tying conjugations to edges in the mapping space. Through detailed simplicial-combinatorial machinery, including horn fillings and box-product arguments, the paper connects orthogonality, braiding, and associativity to explicit lifting properties, offering a robust topological lens on quantum-logical structures with potential links to higher-categorical analogues in quasi-categories.
Abstract
The study of Frobenius algebras in the category $\mathbf{Rel}$ via their nerve functor into simplicial sets has been introduced recently. In this article, we focus on the particular case of effect algebras and pseudo effect algebras and investigate these using tools from combinatorial topology. To each (pseudo) effect algebra $E$, we associate a simplicial set with edge marking $N(E)$, called an $ε$-simplicial set, and analyze its structural properties. In particular, we provide certain characterizations of effect algebras, orthoalgebras, and orthomodular posets among Frobenius algebras in $\mathbf{Rel}$. We show that the universal group $\mathrm{Gr}(E)$ of an effect algebra $E$ coincides with the first homology group of $N(E)$. For a pair of effect algebras $E,F$, we study the mapping space $[N(E),N(F)]$ and prove that the category of effect algebras can be enriched over the category of Frobenius algebras in $\mathbf{Rel}$. We extend this result to the category of pseudo effect algebras. Given pseudo effect algebras $E,F$, for the initial pseudo effect algebra $\underline{1}$, we show that the unique morphism $\underline{1}\to E$ induces a Kan fibration \( [N(E),N(F)] \;\longrightarrow\; [N(\underline{1}),N(F)]. \) We discuss how this result captures several structural features of conjugations in the theory of pseudo effect algebras.