# Math 603D: Homotopy theory and $\mathbb{A}^1$-homotopy theory

## Course Information

#### Instructor

The instructor for the course is me, Ben Williams.

#### Description

This course began as an attempt to teach a topics course in $\mathbb{A}^1$-homotopy theory (also known as "Motivic Homotopy Theory"). I attempted this before and I think it was not a great success, because the prerequisites are too hard. This is a modified version of that course. The main aim of the course is to construct, rigorously, the (unstable) $\mathbb{A}^1$-homotopy category. A secondary aim is to explain why one might want to do such a thing.

#### Prerequisites

The following are required:
• The theory of varieties.
• Homological algebra.
• A first course in algebraic topology.
The following are likely to be useful, and the more you know about them the better:
• The theory of schemes
• Grothendieck topologies
• Simplicial homotopy theory
To some extent, due to the expected small number of enrolled students, the course can accommodate deviations from the set plan.

#### Structure and Assessment

There will be three scheduled lectures each week, at times to be determined (the timetable is not set). Each week, two of these lectures will be by me, and the third will be by a registered student, on a topic outside of the main stream of the course development. These presentations will form the basis of your grade. I will augment this with four homework assignments over the course of the term.

#### Syllabus

A copy of the pdf class outline will be produced later.

The course will (I hope) be divided into three parts

• Abstract homotopy theory (4 weeks)
• Sheaves and local homotopy theory (4 weeks)
• $\mathbb{A}^1$ homotopy theory (4 weeks)

#### Sources

There is no general textbook reference for this course.

At one point I built a short guide to some relevant literature, but this is out of date and not comprehensive.

I am likely to draw on the following book sources

• Model Categories by Hovey.
• Model Categories and their Localizations by Hirschhorn.
• Simplicial Homotopy Theory by Goerss and Jardine.
• Sheaves in Geometry and Logic by MacLane and Moerdijk.
• Local Homotopy Theory by Jardine.
• "The Stacks Project" is not a book, but is a very useful guide to what is true in algebraic geometry. The references for basic algebraic geometry are generally to EGA.

#### Current Notes

At one stage, the notes require some fact about the category of compactly generated (not necessarily weak Hausdorff) spaces. This can be found in an appendix to Gaunce Lewis' Ph.D. Thesis, where they are called $k$-spaces.

#### Old Notes

The last time I taught this class, I made notes available. Beware that the order of things was different on that occasion. These became ragged towards the end of the term, as general fatigue set in. I meant to update them after the course, but never did. Here is the most recent version I have: Old Notes.

#### Homework Assignments

Homework 1 is due in class on Tuesday 11 February. Let me know in the event of any mistakes.

Homework 2 is due in class on Tuesday 25 February. Let me know in the event of any mistakes.

Homework 3 is due in class on Tuesday 10 March. Let me know in the event of any mistakes.

Homework 4 is due in class on Tuesday 24 March. Let me know in the event of any mistakes.

Homework 5 is due by Tuesday 24 March. Let me know in the event of any mistakes.