Abstract. We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ = κ + , another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that H V κ + ⊆ HOD W . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Abstract. Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal κ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2 κ supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author. Introduction and preliminariesIt is well known that if κ is 2 κ supercompact, then κ is quite large in both size and consistency strength. As an example of the former, if κ is 2 κ supercompact, then κ has a normal measure concentrating on measurable cardinals. The key to the proof of this fact and many other similar ones is the existence of an elementary embedding j : V → M with critical point κ so that M 2 κ ⊆ M . Thus, if 2 κ > κ + , one can ask whether κ must be large in size if κ is merely δ supercompact for some κ < δ < 2 κ . A natural question of the above venue to ask is whether a cardinal κ can be both the least measurable cardinal and δ supercompact for some κ < δ < 2 κ if 2 κ > κ + . Indeed, the first author posed this very question to Woodin in the spring of 1983. In response, using Radin forcing, Woodin (see [CW]) proved the following Theorem. Suppose V |= "ZFC + GCH + κ < λ are such that κ is λ + supercompact and λ is regular". There is then a generic extensionThe purpose of this paper is to extend the techniques of [AS] and show that they can be used to demonstrate that Menas' result of [Me] which says that the least measurable cardinal κ which is a limit of supercompact or strongly compact cardinals is strongly compact but not 2 κ supercompact is best possible. Along the way, we generalize and strengthen Woodin's result above, and we also produce a model in which, on a proper class, the notions of measurability, δ supercompactness,

Abstract. We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension|= "κ is λ supercompact", except possibly if κ is a measurable limit of cardinals which are λ supercompact. Introduction and PreliminariesIt is a well known fact that the notion of strongly compact cardinal represents a singularity in the hierarchy of large cardinals. The work of Magidor [Ma1] shows that the least strongly compact cardinal and the least supercompact cardinal can coincide, but also, the least strongly compact cardinal and the least measurable cardinal can coincide. The work of Kimchi and Magidor [KiM] generalizes this, showing that the class of strongly compact cardinals and the class of supercompact cardinals can coincide (except by results of Menas [Me] and [A] at certain measurable limits of supercompact cardinals), and the first n strongly compact cardinals (for n a natural number) and the first n measurable cardinals can coincide. Thus, the precise identity of certain members of the class of strongly compact cardinals cannot be ascertained visà vis the class of measurable cardinals or the class of supercompact cardinals.An interesting aspect of the proofs of both [Ma1] and [KiM] is that in each result, all "bad" instances of strong compactness are not obliterated. Specifically, in each model, since the strategy employed in destroying strongly compact cardinals which aren't also supercompact is to make them non-strongly compact after a certain point either by adding a Prikry sequence or a non-reflecting stationary set of ordinals of the appropriate cofinality, there may be cardinals κ and λ so that κ is λ strongly

Abstract. Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.Two important but apparently unrelated results occupy the large cardinal literature. On the one hand, Laver [Lav78] famously proved that any supercompact cardinal κ can be made indestructible by <κ-directed closed forcing. On the other hand, Apter and Shelah [AS97] *