Abstract
A level j : E j → E of a topos E is said to have monic skeleta if, for every X in E , the counit j ! ( j ⁎ X ) → X is monic. For instance, the centre of a hyperconnected geometric morphism is such a level. We establish two related sufficient conditions for an adjunction to extend to a level with monic skeleta. As an application, we characterize the hyperconnected geometric morphisms that are local providing an interesting expression for the associated centres that suggests a generalization of open subtoposes. As a corollary, we obtain that a hyperconnected p : E → S is pre-cohesive if and only if p ⁎ : E → S preserves coequalizers and p ⁎ : S → E is cartesian closed.
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