On monoids in the category of sets and relations

Abstract

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every homset $\mathbf{Rel}(A,B)$ is a poset with respect to inclusion. We examine the 2-category of monoids $\mathbf{RelMon}$ in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.

References in corpus (2)

• Categories of relations as models of quantum theory
• (Modular) Effect Algebras are Equivalent to (Frobenius) Antispecial Algebras

Cited by in corpus (2)

• A category of quantum posets
• Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories