Lifting retracted diagrams with respect to projectable functors

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 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings, can be lifted, with respect to the Con_c functor, by a diagram of lattices, then so can every diagram, indexed by a lattice, of finite distributive 〈∨ ,0 〉-semilattices with 〈∨ ,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 our method to other functors, such as the $R \mapsto V(R)$ functor on von Neumann regular rings.