Comparison categories: the normal form and the regularity of comparison data

Abstract

Many equivalence relations of mathematics (Green's relations on a semigroup, orbit relations of group actions, conjugacy) are defined by transforming a morphism on its two sides. We put them in one framework. A \emph{comparison datum} produces a category, the \emph{comparison category}, whose arrows are the allowed transformations, and two morphisms are related when they can be compared in both directions. Its \emph{normal form} is a Grothendieck construction, so every such relation is mutual reachability for an action; Green's relations and orbit relations are worked out in detail from it. If the way morphisms are transported is an arbitrary function, every relation occurs, so the real constraint is the \emph{regularity} of the transport. For completely functorial data we show that the relations obtained contain Green's relations, congruences, conjugacy and twisted conjugacy, we describe them on finite groups through double cosets, and we prove a dichotomy on nil monoids.

arXiv commentAI-generated from an initial draft and supervision by the author. 11 pages