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 (Springer-Verlag, 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

E-mail: 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
  • Section 8: 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: Brown-Gersten 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

References