From objects to diagrams for ranges of functors
Résumé
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
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