2023/11/14 by Louis Martini, Martini, Louis, Sebastián Wolf +1
Mathematics · Medicine · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.2311.08051
openalex publication_date 2023/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We characterise proper morphisms of ∞-topoi in terms of a relativised notion of compactness: we show that a geometric morphism of ∞-topoi is proper if and only if it commutes with colimits indexed by filtered internal ∞-categories in the target. In particular, our result implies that for any ∞-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 ∞-topoi, generalising a result of Lurie. Along the way, we develop some aspects of the theory of localic higher topoi internal to an ∞-topos, which might be of independent interest.