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, ``same value of an invariant'') are defined by allowed transformations, of a morphism on its two sides or of a point of a set. We put them in one framework. A comparison datum produces a category, the comparison category, whose arrows are the allowed transformations, and two morphisms are related when they can be compared in both directions. Its normal form is a Grothendieck construction, so every such relation is mutual reachability for an action; Green's relations, orbit relations and, more generally, relations defined by an invariant 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 regularity of the transport. For completely functorial data we show that the relations obtained contain Green's relations and congruences, we study them at two opposite extremes: on finite groups (every morphism invertible) they are described through double cosets, and on nil monoids (only the unit invertible) we prove a collapse theorem. A short plain-language guide to the vocabulary (comparison categories and transport, Grothendieck construction and mutual reachability, functors, pseudofunctors and lax functors, enriched categories, colimits and stabilization) is included for readers who are not specialists.