WiSe 24/25: Topology and Topoi
Georg Lehner
Comments
There are various dualities in mathematics that share formal similarities. One is given by Galois theory: For a given field, the poset of Galois extensions and the poset of subgroups of its absolute Galois group are dual to another. Another example is covering space theory: For a given topological space, there is a duality between coverings and subgroups of its fundamental group. We will also discuss Stone dualities, as well as various incarnations of these.
These dualities are special cases of a very general phenomenon that can be expressed by looking at Grothendieck Topoi. These Topoi are categories of sheaves on a site, and can also be thought of as generalized topological spaces. Any topos has a pro-finite homotopy group and there is an abstract Galois theory that generalizes both classical Galois theory and covering space theory.
Towards the end of the lecture series we will look at shape theory. Shape theory allows one to do homotopy theory even with wild topological spaces. To any higher topos, one can associate its shape. We will attempt to prove the result that for a locally contractible topos, its sub-topos of locally contractible objects is equivalent to the category of local systems over its shape.
closeSuggested reading
Johnstone - Stone Spaces
MacLane, Moerdijk - Sheaves in Geometry and Logic
Hoyois - Higher Galois Theory
16 Class schedule
Regular appointments