From objects to diagrams for ranges of functors

Pierre Gillibert (1) , Friedrich Wehrung (2)

1 MFF-UK - Department of Algebra
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 Grätzer-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.

Mots clés: Category; functor; larder; lifter; condensate; Löwenheim-Skolem Theorem; weakly presented; Armature Lemma; Buttress Lemma; Condensate Lifting Lemma; Kuratowski's Free Set Theorem; Erdös cardinal; product; colimit; epimorphism; monomorphism; section; retraction; retract; projection; projectable; pseudo join-semilattice; almost join-semilattice; projectability witness; semilattice; lattice; congruence; distributive; modular

Domaines: Mathématiques générales [math.GM]; Catégories et ensembles [math.CT]; Logique [math.LO]; Anneaux et algèbres [math.RA]

Pierre Gillibert, Friedrich Wehrung. From objects to diagrams for ranges of functors. 2010. ⟨hal-00462941v1⟩