Reporter: Andrey Lazarev, Lancaster University

Report location: Tencent meeting

https://meeting.tencent.com/s/X079S8uLsWTA

Conference ID: 401 7495 7545

School contact: Sheng Yunhe shengyh@jlu.edu.cn

Model categories in algebra and topology: a minicourse

Abstract: this course will describe a modern approach to homotopy theory based on model categories, invented just over 50 years ago by the British mathematician Daniel Quillen. Model categories are an abstraction of the homotopy category of topological spaces but have applications extending far beyond algebraic topology , namely in algebraic geometry, homological algebra, representation theory, deformation theory and other fields. We will explain how model categories give a unified approach to classical homotopy theory and homological algebra.

Lecture 1: basic notions of category theory

Categories and functors, equivalence of categories. Adjoint and representable functors. Natural transformations, limits and colimits. Examples.

Lecture 2: basic notions of homological algebra

Chain complexes and their homology, chain homotopy, quasi-isomorphisms. Tensor products of complexes and complexes of homomorphisms. Projective and injective modules.

Lecture 3: basic notions of homotopy theory

Homotopy of continuous maps, homotopy equivalences of topological spaces. Cylinders and path spaces. Homotopy groups and weak homotopy equivalences.

Lecture 4: model categories I

Axioms of model categories, left and right homotopies. Fibrant and cofibrant objects.

Lecture 5: model categories II

The construction of the homotopy category of a model category. Derived functors. Localization of categories.

Lecture 6: derived category of a ring

Construction of the unbounded derived category of a ring. Small object argument. Projective and injective resolutions. Functors Tor and Ext.

Lecture 7: homotopy category of spaces

Construction of the model category of topological spaces and its homotopy category. CW complexes.

Lecture 8: future directions

Further examples of model categories, Quillen adjunctions and Quillen equivalences. Constructing new model categories from old. Infinity-categories.

Brief introduction of the speaker:

Andrey Lazarev, professor at Lancaster University, UK, engaged in the research of algebraic topology and homotopy theory, editor-in-chief of Bull. Lond. Math. Soc., in Adv. Math., Proc. Lond. Math. Soc., J. Noncommut . Published many high-level papers in magazines such as Geom.