Lifting retracted diagrams with respect to projectable functors

Friedrich Wehrung (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean (v,0)-semilattices with (v,0)-embeddings, can be lifted with respect to the $\Conc$ functor on lattices, then so can every diagram, indexed by a lattice, of finite distributive (v,0)-semilattices with (v,0-embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of the method to other functors, such as the $R\mapsto V(R)$ functor on von Neumann regular rings.

Mots clés: Category; functor; diagram; epic; section; retraction; retract; projection; projectable; product; colimit; unfolding; semilattice; lattice; congruence; distributive; Boolean; ring; von Neumann regular

Domaines: Mathématiques générales [math.GM]; Catégories et ensembles [math.CT]

Friedrich Wehrung. Lifting retracted diagrams with respect to projectable functors. Algebra Universalis, 2005, 54 (3), pp.349--371. ⟨10.1007/s00012-005-1951-z⟩. ⟨hal-00002855⟩