# Lectures on Local Homotopy Theory

## J.F. Jardine

The links below are to pdf files for my lecture notes for a course on local homotopy theory, or the homotopy theory of simplicial sheaves 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.

The syllabus for the course is the list of file names below.

The course is given as a set of lectures at the University of Western Ontario, and is available by video conference to students from other universities. Students from other sites can participate, from either traditional video conference rooms or by using suitably equipped personal computers. Please contact me if you wish to do so.

The grades for students registered in the course at Western will be based on lectures that they present in class. For students at another site, it is convenient that a local instructor set up a reading course, which would be graded by using a method of his or her choice.

If you have any comments or questions about the course, please do not hesitate to contact me at jardine@uwo.ca.

**NB**: This is not a "live" site, in that there are no direct links from this page to updated course material. Please see the Course home page for current content, including revised lecture notes and recordings of lectures.

**Western students: **Please see the** Course OWL site ** for the official course outline, and for information about course books.

Lecture 01: **Simplicial sets, Simplicial homotopy theory**

Lecture 02: **Grothendieck toposes, geometric morphisms, Boolean localization**

Lecture 03: **Rigidity**

Lecture 04: **Local weak equivalences, local fibrations**

Lecture 05: **Model structures: simplicial presheaves and simplicial sheaves**

Lecture 06: **Simplicial modules, derived category**

Lecture 07: **Cocycles, sheaf cohomology, descent spectral sequences**

Lecture 08: **Torsors for groups and groupoids, stacks and homotopy theory**

Lecture 09: **The Verdier hypercovering theorem**

Lecture 10: **Localization for simplicial presheaves**

Lecture 11: **Presheaves of spectra, the stable category**

Lecture 12: **T-spectra**

Lecture 13: **Brown-Gersten and Nisnevich descent, motivic descent**

Lecture 14: **Stable homotopy theory of T-spectra**

### References