This lecture deals with the notion of geometry following A. Grothendieck's homotopy categorical approach in terms of sheaves on a category with respect to a given topology. Starting with the example of topological varieties, then the category of schemes, the goal of this lecture is to show how this approach gives access to various classical geometrical contexts in algebraic geometry that finally leads to the category of algebraic stacks.
The notion of stacks can be seen as an enhancement of the notion of schemes by allowing to deal with ``bad'' group action on spaces, and by providing fine answers to moduli classification problems. Typical examples encountered in algebraic geometry are given by moduli spaces of curves, Hurwitz spaces of covers or vectors bundles.
While some familiarity with algebraic geometry and schemes is assumed -- e.g. as given in Algebraische Geometrie (Prof. Dr I. Bauer WS 17/18) -- the required reminders will be given following the background of the participants. This advanced lecture is aimed at students willing to broaden their knowledge on the last recent advances in homotopical algebraic geometry.