From objects to diagrams for ranges of functors

TL;DR

This paper develops a framework for transferring object-level isomorphisms to diagram categories indexed by specific posets, extending classical theorems and analyzing properties of algebraic structures and their congruences.

Contribution

It generalizes the Grätzer-Schmidt Theorem to diagram categories and explores the properties of congruence lattices and critical points in algebraic structures.

Findings

Extended Grätzer-Schmidt Theorem to finite poset-indexed diagrams

Established bounds on the relative critical point between quasivarieties

Demonstrated limitations on congruence-permutable extensions of certain lattices

Abstract

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for "many" objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general framework under which it is possible to transfer this statement to diagrams of A. These diagrams are all indexed by posets in which every principal ideal is a join-semilattice and the set of all upper bounds of any finite subset is a finitely generated upper subset. Various consequences follow, in particular: (1) The Gr\"atzer-Schmidt Theorem, which states that every algebraic lattice is isomorphic to the congruence lattice of some algebra, can be extended to finite poset-indexed diagrams of algebraic lattices and compactness-preserving complete join-homomorphisms (and no finiteness restriction if there are large enough cardinals). (2) In a host of situations, the relative critical point between two locally finite quasivarieties is either less than aleph omega or equal to infinity. (3) A lattice of cardinality aleph 1 may not have any congruence-permutable, congruence-preserving extension.