Simplicial Perspectives on (Pseudo) Effect Algebras

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 $\epsilon$-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.