Categories and Topology
This course is taught by Ieke Moerdijk and myself on Mondays, 3.15-5pm, live through Zoom, starting September 7. If you wish to join, please let one of us know by email so that we can include you in the mailing list. Any updates on course proceedings will also appear here.
Course summary
Constructions of topological spaces from categories play a central role in many parts of mathematics, notably in algebraic K-theory and in algebraic topology itself. These constructions make use of the theory of simplicial sets, which form "combinatorial models" for topological spaces. The purpose of this course is to provide an introduction to the theory of simplicial sets together with its relation to the homotopy theory of topological spaces, and to study some of the constructions used in higher algebraic K-theory. The topics we intend to cover include the following:
Basics of simplicial sets
Geometric realization of simplicial sets
Classifying spaces of groups and of categories
Kan complexes and Kan fibrations
Homotopy (co)limits
Simplicial sets and algebraic K-theory (after Quillen and Waldhausen)
The course will be mostly self-contained, requiring not much more than basic knowledge of topological spaces. Some experience with algebraic topology, for example as in the course Algebraic Topology I, will be useful, although not strictly necessary.
For the first few lectures we will rely on this book (in progress) as a source for the basics of simplicial sets.
Course schedule
Sept 7: Simplicial sets, basic definitions and first examples. Geometric realization and its CW structure. All the material in this lecture is explained in more detail in Sections 2.1-2.3 of the book. Ieke's notes.
Sept 14: Simplicial homotopy. Products of simplicial sets and shuffles. Geometric realization preserves products (and homotopies). Classifying spaces of groups and categories. Gijs' informal notes.
Sept 21: Kan complexes, Kan fibrations, and anodyne morphisms. Pushout products of monomorphisms with anodynes. The geometric realization of a Kan fibration is a Serre fibration. Ieke's notes.
Sept 28: More on minimal fibrations. The realization of the singular complex of a space X is weakly equivalent to X. Ieke's notes.
Oct 5: Quillen's Theorem A. Simplicial spaces and bisimplicial sets. Gijs' notes.
Oct 12: The diagonal of a bisimplicial set. Quillen's Theorem B. Gijs' notes.
Oct 19: Bott periodicity as an application of Theorem B. Ieke's notes.
Some extra notes on fiber bundles.
Oct 26: The group completion theorem. First notions of algebraic K-theory. Gijs' notes.
Nov 2: The Q-construction. Ieke's notes.
Nov 9: The localization theorem. Ieke's notes.
Nov 16: Resolution and dévissage. Gijs' notes.
Nov 23: Proof of dévissage. Group completion vs the Q-construction. Gijs' notes.
Nov 30: Thomason's theorem on homotopy colimits. Ieke's notes.
Dec 7: Spaces with G-action. Simplicial sets over BG. Elmendorf's theorem. Gijs' notes.