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

There are many types of equivalence relations in mathematics, each one enjoying its own defnition, but many of them can also be defined by transformations of morphisms: Green's relations, orbit relations, conjugacy, equality of orbit closures, equivalence of Cauchy sequences, stable isomorphism. We unify them through the comparison category of a comparison datum, read at levels $-1,0,1,2,\dots$ (comparisons in both directions, connected components, homotopy groups). Its normal form is a full subcategory of a Grothendieck construction $\int_VH$ cut out by a map $s$; the classical relations are traced back to it by restriction along $s$, enrichment by a cost (limit relations) and filtered colimits (stabilization). If the map $σ$ transporting morphisms into a $2$-category is arbitrary, every relation occurs, so the constraint is the regularity of $σ$. We grade data along seven regularity classes. The rigid and diagonal classes are classified completely (on finite groups, by the double coset relations); completely functorial data contain conjugacy and obey a dichotomy on nil monoids. Whether every relation on a monoid is level $0$ of a completely functorial datum is open.