$\infty$-topos 的真态射
Proper morphisms of $\infty$-topoi
AI总结：
本文用相对化紧性刻画 $\infty$-topos 真态射，证明其等价于保持目标内蕴滤过指标余极限，并应用于真且分离的拓扑映射以推广 Lurie 的结果。
摘要：
我们通过一种相对化的紧性概念刻画了 $\infty$-topos 的真态射：证明了 $\infty$-topos 之间的几何态射为真，当且仅当它与目标中以内蕴滤过 $\infty$-范畴为指标的余极限交换。特别地，结果表明对任意 $\infty$-topos，整体截面函子为真，当且仅当它保持滤过余极限。作为应用，我们证明拓扑空间之间的每个真且分离映射都会在相关的层 $\infty$-topos 之间产生真态射，从而推广了 Lurie 的结果。在此过程中，我们发展了内蕴于某个 $\infty$-topos 的 locale 式高阶 topos 理论的若干方面，这可能具有独立意义。
英文摘要：
We characterise proper morphisms of $\infty$-topoi in terms of a relativised notion of compactness: we show that a geometric morphism of $\infty$-topoi is proper if and only if it commutes with colimits indexed by filtered internal $\infty$-categories in the target. In particular, our result implies that for any $\infty$-topos, the global sections functor is proper if and only if it preserves filtered colimits. As an application, we show that every proper and separated map of topological spaces gives rise to a proper morphism between the associated sheaf $\infty$-topoi, generalising a result of Lurie. Along the way, we develop some aspects of the theory of localic higher topoi internal to an $\infty$-topos, which might be of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
