The Condensate Lifting Lemma (CLL)

Abstract

In this chapter we shall finalize the approach to this work’s main result, the Condensate Lifting Lemma (CLL). The statement of CLL involves a “condensate” $F(X)\otimes \vec{A}$. General condensates are defined in Sect. 3.1. The statement of CLL also involves categories A, B, S with functors $\Phi :\mathcal{A} \rightarrow \mathcal{S}$ and $\Psi :\mathcal{B} \rightarrow \mathcal{S}$ For further applications of our work, such as Gillibert (The possible values of critical points between varieties of lattices, preprint 2010. Available online at http://hal.archives-ouvertes.fr/ hal-00468048), we need to divide CLL into two parts: the Armature Lemma (Lemma 3.2.2), which deals with the functor $\Phi :\mathcal{A} \rightarrow \mathcal{S}$, and the Buttress Lemma (Lemma 3.3.2), which deals with the functor $\Phi :\mathcal{B} \rightarrow \mathcal{S}$. The Buttress Lemma, which are objects of poset-theoretical nature with a set-theoretical slant, will be defined in Sect. 3.2. In Sect. 3.4 we shall put together the various assumptions surrounding A, B, S, Φ, and Ψ in the definition of a larder (Definition 3.4.1), and we shall state and prove CLL. In Sect. 3.5 we shall relate the poset-theoretical assumptions from CLL with infinite combinatorics, proving in particular that the shapes of the diagrams involved (the posets P) are almost join-semilat- tices satisfying certain infinite combinatorial statements (cf. Corollary 3.5.8). In Sect. 3.7 we shall weaken both the assumptions and the conclusion from CLL, making it possible to consider diagrams indexed by almost join-semilat- tices P for which there is no lifter. In Sect. 3.8 we shall split up the definition of a -larder between left larder and right λ-larder, making it possible to write a large part of this work as a toolbox, in particular stating the right larderhood of many structures.