A cardinal κ is nearly θ-supercompact if for every , there exists a transitive closed under <κ sequences with , a transitive N, and an elementary embedding with critical point κ such that and .2 This concept strictly refines the θ-supercompactness hierarchy as every θ-supercompact cardinal is nearly θ-supercompact, and every nearly -supercompact cardinal κ is θ-supercompact. Moreover, if κ is a θ-supercompact cardinal for some θ such that , we can move to a forcing extension preserving all cardinals below where κ remains θ-supercompact but is not nearly -supercompact. We will also show that if κ is nearly θ-supercompact for some such that , then there exists a forcing extension preserving all cardinals at or above κ where κ is nearly θ-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result. A forcing poset is <κ-directed closed if it is γ-directed closed for all in the sense of Jech (2003) [13, Def. 21.6]. We will prove that if κ is nearly θ-supercompact for some such that , then there is a forcing extension where its near θ-supercompactness is preserved and indestructible by any further <κ-directed closed θ-c.c. forcing of size at most θ. Finally, these cardinals have high consistency strength. Specifically, we will show that if κ is nearly θ-supercompact for some for which , then AD holds in . In particular, if κ is nearly -supercompact and , then AD holds in .
Here, we use to mean the theory of ZFC without the Powerset axiom, but where ZFC is understood to be axiomatized with Collection instead of Replacement.