65-417: Topics in Category Theory for Geometry
University of Hamburg, Mathematics
Monday 2-4, Geom 241
Susama Agarwala, Geom 1513, firstname.lastname@example.org
Jeffrey Morton, Geom 1511, email@example.com
This coutse is divided into 2 parts. The first set of lectures (given mostly by Dr. Morton) give a categorical understanding of certain objects that appear in geometry: sheaves, localization, derived functors. The second set of lectures (given mostly by Dr. Agarwala) works towards understanding of the category of mixed Tate motives, using the framework developed in part 1.
The following is a rough outline of the topics to be covered in class:
Elementary toposes as categories with special properties
Grothendieck's example: toposes of sheaves on sites
Derived functors, such as direct and inverse images associated to maps of underlying spaces
Localization and derived categories
The category of correspondences
Étale sheaves and Étale motivic cohomology
The category of motives
& Moerdijk, "Sheaves
in Geometry and Logic: A First Introduction to Topos Theory"
Dimca, "Sheaves in Topology"
Borceux, "Handbook of Categorical Algebra 3: Categories of Sheaves"
Kashiwara & Schapira: "Categories and Sheaves"
Eisenbud & Harris “Geometry of Schemes”
Hartshorne “Algebraic Geometry”
Vakil “Foundations of Algebraic Geometry” (Most recent version can be found from a link on this page)
Mazza, Voevodsky, Weibel “Lecture Notes on Motivic Cohomology”
Levine “Mixed Tate Motives” (A very detailed, but dense, set of lecture notes. A good reference for understaing technical details left out in class)
(Advisory note! These scans are of notes we have written for ourselves, made available by request. They are not meant to replace an edited textbook. They may contain errors and incompleteness. CAVEAT LECTOR.)
Lecture 1: Toposes and Categories
Lecture 2: Sheaves and Schemes
Lecture 3: Presheaves to Sheaves
Lecture 4: Sites (The proof that the Zariski site forms a basis omitted.)
Lecture 5: Grothendieck Toposes (Associated sheaf functor, geometric morphisms, and proof that sheaf categories are toposes)
Lecture 6: Sheaves as Generalized Spaces
Lecture 7: Sheaves of Abelian Groups, Simplicial Sets and Complexes: the Dold-Puppe Theorem