Lifting retracted diagrams with respect to projectable functors

2004/09/16 by Friedrich Wehrung, Wehrung, Friedrich
Mathematics · #06A12 #06D05 #08B25 #18A20 #18A25 #18A30 #18A35 #18A40 #19A49 #Category Theory (math.CT) #FOS: Mathematics #General Mathematics (math.GM) #math.CT #math.GM #msc:06A12 #msc:06D05 #msc:08B25 #msc:18A20 #msc:18A25 #msc:18A30 #msc:18A35 #msc:18A40 #msc:19A49

paper · pdf · doi:10.48550/arxiv.math/0409270

arxiv created 2004/09/16 · arxiv updated 2009/12/01

Abstract

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↦ V(R) functor on von Neumann regular rings.

Related

• From objects to diagrams for ranges of functors
• Liftings of diagrams of semilattices by diagrams of dimension groups
• Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction
• Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties
• Categories of partial algebras for critical points between varieties of algebras