報 告 人：Andrey Lazarev，Lancaster University
報告地點：騰訊會議
https://meeting.tencent.com/s/X079S8uLsWTA
會議 ID：401 7495 7545
校内聯系人：生雲鶴 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.
授課日期 Date of Lecture |
課程名稱（講座題目） Name (Title) of Lecture |
授課時間 Duration (Beijing Time) |
參與人數 Number of Participants |
March5, 2021 |
basic notions of category theory |
17:00-18:00 |
30 |
March8, 2021 |
basic notions of homological algebra |
17:00-18:00 |
30 |
March11, 2021 |
basic notions of homotopy theory |
17:00-18:00 |
30 |
March12, 2021 |
model categories I |
17:00-18:00 |
30 |
March16, 2021 |
model categories II |
17:00-18:00 |
30 |
March17, 2021 |
derived category of a ring |
17:00-18:00 |
30 |
March18, 2021 |
homotopy category of spaces |
17:00-18:00 |
30 |
March19, 2021 |
future directions |
17:00-18:00 |
30 |
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.
報告人簡介：
Andrey Lazarev，英國蘭卡斯特大學教授，從事代數拓撲與同倫論的研究， Bull. Lond. Math. Soc.雜志主編，在Adv. Math.、 Proc. Lond. Math. Soc.、 J. Noncommut. Geom.等雜志上發表多篇高水平論文。
