Algebra/Topology Seminar

Speaker: Louis Martini

Title: Properness in topology and higher topos theory

Abstract: The notion of properness of a map of spaces is supposed to encode a relative analogue of compactness. In his book on higher topos theory, Jacob Lurie proposed a notion of properness for higher topoi, but it remained unclear in what sense his definition can be understood as a relativised version of toposic compactness. In my talk, I will report on joint work with Sebastian Wolf in which we formalise a notion of relative compactness for maps of higher topoi and show that this is equivalent to properness of these maps. I will furthermore explain how one can use this result to show that every proper and separated map of topological spaces gives rise to a proper morphism between the associated higher sheaf topoi.