Mathieu Anel, CRM

Introduction to sheaves and stacks

The purpose of these lectures is to introduce Grothendieck's approach to geometry where a space is thought of as the quotient of all spaces mapping into it, i.e. as the colimit of its functor of points $Hom_{Top}(-,X):Top^{op}\to Sets$. This viewpoint leads to important generalizations of the notion of space (e.g. sheaves and stacks), convenient for example to deal with bad quotients and moduli problems.
The course will be divided into four parts, each emphasizing examples (mostly coming from algebraic geometry).

1. sheaves: topology via glueing, Grothendieck topology, sheaves, spectra of rings.
   examples: R/Q, R/Q^{dis}, G/G^{dis},T_a = R/Z+aZ...
2. 1-stacks: categories and moduli problems, 1-descent, geometricity, tangent complex.
   examples: Vect, BG, QCoh...
3. higher stacks: higher categories, homotopy colimits and descent, Quillen presheaves.
   examples: Perf, Cat, Ab...
4. motivations for derived stacks: derived intersections, Quillen-Illusie tangent complex.
   examples: Perf, dgCat, dgAb...