Dr. Fritz Hörmann:
Vorstellungsvortrag: Descent
Zeit und Ort
Donnerstag, 7.7.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Zusammenfassung
The notion of descent is ubiquitous in mathematics. An object satisfies descent\nwhenever its nature is determined by local conditions, for instance:\n\n1. (sheaf condition) A function on a topological space can be given locally on an open\ncovering. The local functions can be glued to a global function if they agree on\noverlaps.\n\n2. (descent) A vector bundle on a topological space can - by definition - be given\nlocally on an open covering. The local bundles are glued to a global one by means of\nglueing data on overlaps that satisfy a compatibility condition on "overlaps of\noverlaps".\nThis comes in many flavors such as descent for modules over rings, families of\nvarieties, etc.\n\n3. (cohomological descent) Any type of cohomology of a topological space or\nalgebraic variety can be recovered (in a certain sense) from the cohomology of an\nopen cover. The "glueing data" in this case is much more complicated and carries the\nessential information.\nWe will explain in this talk how all these instances of descent (and many more) are\nunified by adopting a higher-categorical point of view, the examples above becoming\ndescent for set-like objects, 1-category like objects, or infinity-categorical objects. As\nmodel for "infinity-categorical" questions of descent, we present the theory of fibered\nderivators, the topic of the habilitation thesis of the speaker. Our main motivation has\nbeen descent for Grothendieck six-functor formalisms (encoding Serre duality,\nVerdier duality, etc.).