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.

中文翻译：

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本地的超连接地图

摘要 A 级 j : E j → E 的拓扑 E 被称为具有单调骨架，如果对于 E 中的每个 X，单位 j ！( j ⁎ X ) → X 是单数。例如，超连通几何态射的中心就是这样的水平。我们建立了两个相关的充分条件，使附属扩展到与 monic skeleta 的水平。作为一个应用，我们描述了局部超连通几何态射，为相关中心提供了一个有趣的表达，表明开放子拓扑的泛化。作为推论，我们得到超连通 p : E → S 是预内聚的，当且仅当 p ⁎ : E → S 保留协均衡器并且 p ⁎ : S → E 是笛卡尔闭的。