65-417: Topics in Category Theory for Geometry

Spring 2013

University of Hamburg, Mathematics

Monday 2-4, Geom 241

Lecturers:

Susama Agarwala, Geom 1513, susama.agarwala@math.uni-hamburg.de

Jeffrey Morton, Geom 1511, jeffrey.morton@math.uni-hamburg.de

Course information:

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:

Jeffrey Morton:

• 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

Susama Agarwala

• The category of correspondences

• Motivic cohomology

• Étale sheaves and Étale motivic cohomology

• The category of motives

Useful References:

MacLane & 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)

Lecture Notes:

(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