Two partial monoid structures on a set

TL;DR

This paper introduces a new algebraic framework for double categories using two partial monoid structures, providing insights into their symmetries and extending to n-fold categories.

Contribution

It formulates partial monoids with $s$ and $t$ maps, linking double categories to algebraic structures and exploring their symmetries and higher-dimensional generalizations.

Findings

Double categories are equivalent to sets with two partial monoid structures.

Partial monoid homomorphisms involve $s$ and $t$ maps instead of identities.

The approach extends to algebraic formulations of n-fold categories.

Abstract

It is well-known that small categories have equivalent descriptions as partial monoids. We provide a formulation of partial monoid and partial monoid homomorphism involving $s$ and $t$ instead of identities and then following a recent investigation into involutive double categories, we prove that a double category is equivalent to a set equipped with two partial monoid structures in which all structure maps $(s,t,\circ)$ are partial monoid homomorphisms. We discuss the light this purely algebraic perspective sheds on the symmetries of these structures and its applications. Iteration of the procedure leads also to a pure algebraic formulation of the notion of $n$-fold category.