Local club condensation and L-likeness

We present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model.We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ⁺ is consistent with ¬__κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω₂ is regular, CC(κ⁺) - Chang’s Conjecture at κ⁺ - is consistent with Local Club Condensation at κ⁺, both under suitable large cardinal consistency assumptions.