Lectures on simplicial presheaves
The pdf files below are my lecture notes from a course on simplicial
presheaves, which was given at the University of Western Ontario during
the Winter term of 2005.
The notes are rough, skeletal, often ungrammatical, and are subject to
frequent change - the indicated dates are last updates. They are
intended as a first step in the formation of an online resource in
this subject area (see also my home page).
If you have comments or suggestions about these files, please do not
hesitate to contact me: email@example.com.
One comment: don't be misled by the titles. Typically, one of these
files is worth two or more hours of class time; in fact, the length of
the files tends to increase with the progress of the course.
- Lecture 001: Bisimplicial sets, homotopy colimits, projective model structures for diagram categories
- Lecture 002: Injective model structures for diagram categories
- Lecture 003: Homotopy inverse limits, homotopy colimits revisited
- Lecture 004: Cohomology and homology, Milnor exact sequence
- Lecture 005: Postnikov towers, descent spectral sequence
- Lecture 006: Presheaves and sheaves, Grothendieck toposes, Giraud's theorem
- Lecture 007: Geometric morphisms, frames and Boolean algebras, Boolean localization
- Lecture 008: Boolean localization: an elementary proof
- Lecture 009: Local weak equivalences, local fibrations
- Lecture 010: Model structures: simplicial presheaves and simplicial sheaves
- Lecture 011: Cocycles, sheaf cohomology, non-abelian cohomology, descent
- Lecture 012: Cubical sets
- Lecture 013: Intervals and localization
- Lecture 014: Localization: some details
- Lecture 015: Groupoids, presheaves of groupoids, stacks
- Some references