Lectures on Local Homotopy Theory
The links below are to pdf files for the lecture notes for my course on local homotopy theory. This is the homotopy theory of simplicial sheaves, simplicial presheaves and presheaves of spectra.
In addition to these notes, the basic source material for the course is the book Local Homotopy Theory, by J.F. Jardine (SpringerVerlag, 2015). Supplemental reading is given in the list of references.
If you have any comments or questions about these notes, please do not hesitate to contact me:
 J.F. Jardine


Email: jardine@uwo.ca

Lecture 01: Simplicial sets, Simplicial homotopy theory
 Section 1: Simplicial sets
 Section 2: The simplex category and realization
 Section 3: Model structure for simplicial sets

Lecture 02: Grothendieck toposes, geometric morphisms, Boolean localization
 Section 4: Grothendieck topologies
 Section 5: Exactness properties
 Section 6: Geometric morphisms
 Section 7: Points

Lecture 03: Rigidity

Lecture 04: Local weak equivalences, local fibrations
 Section 9: Local weak equivalences

Lecture 05: Model structures: simplicial presheaves and simplicial sheaves
 Section 10: Injective model structure
 Section 11: Other model structures

Lecture 06: Simplicial modules, derived category
 Section 12: Chain complexes
 Section 13: The derived category

Lecture 07: Cocycles, sheaf cohomology, descent spectral sequences
 Section 14: Cocycles
 Section 15: Sheaf cohomology
 Section 16: Descent spectral sequences

Lecture 08: Torsors, stacks and homotopy theory
 Section 17: Torsors for groups
 Section 18: Torsors for groupoids
 Section 19: Stacks and homotopy theory

Lecture 09: Verdier hypercovering theorem
 Section 20: The Verdier hypercovering theorem

Lecture 10: Localization for simplicial presheaves
 Section 21: Localization for simplicial presheaves

Lecture 11: Presheaves of spectra, stable category
 Section 22: Presheaves of spectra
 Section 23: The stable category: basic properties

Lecture 12: \(T\)spectra
 Section 24: \(T\)spectra

Lecture 13: BrownGersten and Nisnevich descent, motivic descent
 Section 25: Descent theorems

Lecture 14: Stable homotopy theory of \(T\)spectra
 Section 26: Stable homotopy theory of \(T\)spectra
 Section 27: \((S^{1} \wedge K)\)spectra