koherentsets

Koherentsets are a formal framework for studying a set equipped with a coherent family of equivalence relations. A koherentset consists of a set S together with a directed family {R_i}_{i∈I} of equivalence relations on S, directed by a pre-order I, such that if i ≤ j then R_j refines R_i (x R_j y implies x R_i y). For each i, the quotient set S_i = S/R_i comes with a natural projection π_i: S → S_i. The collection of quotients and projections forms a projective system with bonding maps π_{j,i}: S_j → S_i defined by refinement. The projective limit lim← S_i, along with the canonical map φ: S → lim← S_i given by φ(x) = (π_i(x))_i, is central to the theory.

Koherentsets are studied through the relation between S and its projective limit. If φ is injective, the original set is faithfully represented by its coherent quotients; if φ is an isomorphism onto its image, the koherentset is called fully coherent. When each S_i is finite, the projective limit is a compact, totally disconnected space (a profinite space) with the inverse limit topology.

A standard example uses the additive group of integers: define R_n by x R_n y if x ≡ y mod n. The index set I is the natural numbers ordered by divisibility, so R_m ⊆ R_n when n|m. The corresponding projective limit is the profinite completion of Z, denoted Ẑ.

Variants include relaxing finiteness, considering noncommutative quotients, or replacing equivalence relations with congruence-like relations arising from a valuation or a topology. Koherentsets provide a language for describing coarse-to-fine refinements of indistinguishability or information content, with applications in topology, algebra, and theoretical computer science.

See also: inverse limit, profinite completion, equivalence relation, partition lattice, coherence.