Category Theory
Zulip Server
Archive

第一章 集合论 Chapter 1 Set Theory
1.1 ZFC 公理一览 Overview of ZFC's Axioms
1.2 序结构与序数 Order Structure and Ordinals
1.3 超穷递归及其应用 Transfinite Induction and its Applications
1.4 基数 Cardinals
1.5 Grothendieck 宇宙 Grothendieck Universe
习题 Exercises
第二章 范畴论基础 Foundations of Category Theory
2.1 范畴与态射 Categories and Morphisms
2.2 函子与自然变换 Functors and Natural Transformations
2.3 函子范畴 The Categories of Functors
2.4 泛性质 Universal Properties
2.5 可表函子 Representable Functors
2.6 伴随函子 Adjoint Functors
2.7 极限 Limits
2.8 完备性 Completeness
习题 Exercises
第三章 幺半范畴 Monoidal Categories
3.1 基本定义 Basic Definitions
3.2 严格性与融贯定理 Strictness and the Coherence Theorem
3.3 辫结构 Braiding Structures
3.4 充实范畴 Enriched Categories
3.5 2-范畴一瞥 Overview of 2-Categories
习题 Exercises
第四章 群论 Group Theory
4.1 半群, 幺半群与群 Semigroups, Monoids, and Groups
4.2 同态和商群 Homomorphisms and Quotient Groups
4.3 直积, 半直积与群扩张 Direct Products, Semidirect Products, and Group Extensions
4.4 群作用和计数原理 Group Actions and Orbit-Counting Principles
4.5 Sylow 定理 Sylow's Theorems
4.6 群的合成列 Composition Series of Groups
4.7 可解群与幂零群 Solvable Groups and Nilpotent Groups
4.8 自由群 Free Groups
4.9 对称群 Symmetric Groups
4.10 群的极限和完备化 Limit of Groups and Completion
4.11 范畴中的群 Groups internal to Categories
习题 Exercises
第五章 环论初步 Introductory Ring Theory
5.1 基本概念 Basic Notions
5.2 几类特殊的环 Some Classes of Special Rings
5.3 交换环初探 Commutative Rings
5.4 间奏: Möbius 反演 Excursion: Möbius Inversion
5.5 环的极限与完备化 Limit of Rings and Completion
5.6 从幺半群环到多项式环 From Monoid Rings to Polynomial Rings
5.7 唯一分解性 Unique-Decomposability
5.8 对称多项式入门 Introduction to Symmetric Polynomials
习题 Exercises
第六章 模论 Theory of Modules
6.1 基本概念 Basic Notions
6.2 模的基本操作 Basic Operations on Modules
6.3 自由模 Free Modules
6.4 向量空间 Vector Spaces
6.5 模的张量积 Tensor Products of Modules
6.6 环变换 Change of Ring
6.7 主理想环上的有限生成模 F.G. Modules over PID
6.8 正合列入门 Introduction to Exact Sequences
6.9 投射模, 内射模, 平坦模 Projective Modules, Injective Modules, Flat Modules
6.10 链条件和模的合成列 Chain Conditions and Composition Series of Modules
6.11 半单模 Semisimple Modules
6.12 不可分解模 Irreducible Modules
习题 Exercises
第七章 代数初步 Introduction to Algebras
7.1 交换环上的代数 Algebras over Polynomial Rings
7.2 整性, 有限性和 Frobenius 定理 Integrality, Finiteness, and Frobenius theorem
7.3 代数的张量积 Tensor Product of Algebras
7.4 分次代数 Graded Algebras
7.5 张量代数 Tensor Algebras
7.6 对称代数和外代数 Symmetric Algebras and Exterior Algebras
7.7 牛刀小试: Grassmann 簇 A Small Duel: Grassmannian Varieties
7.8 行列式, 迹, 判别式 Determinants, Traces, Discriminants
习题 Exercises
第八章 域扩张 Field Extensions
8.1 扩张的几种类型 Several Types of Extensions
8.2 代数闭包 Algebraic Closures
8.3 分裂域和正规扩张 Splitting Fields and Normal Extensions
8.4 可分性 Separability
8.5 本原元素定理 Primitive Element Theorem
8.6 域扩张中的范数与迹 Norms and Traces in Field Extensions
8.7 纯不可分扩张 Purely Inseparable Extensions
8.8 超越扩张 Transcendental Extensions
8.9 张量积的应用 Applications of Tensor Products
习题 Exercises
第九章 Galois 理论 Galois Theory
9.1 有限 Galois 对应 Finite Galois Connections
9.2 无穷 Galois 对应 Infinite Galois Connections
9.3 有限域 Finite FIelds
9.4 分圆域 Cyclotomic Fields
9.5 正规基定理 Normal Basis Theorem
9.6 Kummer 理论 Kummer Theorem
9.7 根式解判准 Decision of Radical Solutions
9.8 尺规作图问题 The Problem of Euclidean Constructions
习题 Exercises
第十章 域的赋值 Valuations of Fields
10.1 滤子 Filters
10.2 Krull 赋值与完备化 Krull Valuation and Completion
10.3 域上的赋值 Valuations over Fields
10.4 绝对值, 局部域和整体域 Absolute Values, Local Fields, and Global Fields
10.5 个案研究: 单位闭圆盘 Case Study: Unit Closed Disc
10.6 一般扩域的赋值 Valuation of General Extension Fields
10.7 代数扩域的赋值 Valuation of Algebraic Extension Fields
10.8 完备域中求根 Root-Finding in Complete Fields
10.9 Witt 向量 Witt Vectors

第一章 范畴论拾遗 Chapter 1: Récoltes of Category Theory
1.1 子商 Subquotients
1.2 像, 余像和严格态射 Image, Coimage, and Strict Morphisms
1.3 加性范畴: 核, 余核 Additive Categories: Kernal, Cokernal
1.4 推广: 交换环上的线性范畴 Generalization: Linear Categories over Communtive Rings
1.5 由函子观极限 Limit, From Functors’ Perspective
1.6 滤过归纳极限 Filtered Inductive Limits
1.7 Kan 延拓 Kan Extensions
1.8 以极限构造 Kan 延拓 Kan Extensions Constructed by Limits
1.9 Gabriel–Zisman 局部化 Gabriel-Zisman Localization
1.10 沿局部化作 Kan 延拓 Kan Extension along Localization
1.11 伴随函子定理 Adjoint Functor Theorem
习题 Exercises
第二章 Abel 范畴 Abelian Categories
2.1 Abel 范畴的定义 Definition of Abelian Categories
2.2 初识复形 First View of Complexes
2.3 若干图表引理 Some Diagrammatic Lemmas
2.4 格论一瞥 Overview of Lattice Theory
2.5 直和分解 Direct Sum Decomposition
2.6 子对象和同构定理 Subobjects and Isomorphism Theorems
2.7 单性和半单性 Simplicity and Semisimplicity
2.8 正合函子, 内射对象和投射对象 Exact Functors, Injective Objects, and Project Objects
2.9 Serre 子范畴和 K0 群 Serre Subcategories and K_0-Groups
2.10 Grothendieck 范畴 Grothendieck Categories
习题 Exercise
第三章 复形 Complexes
3.1 加性范畴上的复形 Complexes in Additive Categories
3.2 Hom 复形与同伦 Hom Complexes and Homotopies
3.3 映射锥 Mapping Cones
3.4 相反范畴上的复形 Complexes in Opposite Categories
3.5 双复形 Bicomplexes
3.6 Abel 范畴上的复形 Complexes in Abelian Categories
3.7 映射锥和长正合列 Mapping Cylinders and Long Exact Sequences
3.8 练习: Hochschild 同调与上同调 Playground: Hochschild Homology and Cohomology
3.9 截断函子 Truncation Functors
3.10 双复形的上同调 Cohomology of Bicomplexes
3.11 解消 Resolutions
3.12 经典导出函子Classical Derived Functors
3.13 实例: lim1 Practical Example: \lim^1
3.14 实例: Ext 和 Tor Practical Examples: Ext and tor
3.15 K-内射和 K-投射复形 K-Injective and K-Projective Complexes
习题 Exercise
第四章 三角范畴与导出范畴 Triangulated Categories and Derived Categories
4.1 三角范畴的定义 Definition of Triangulated Category
4.2 基本性质 Basic Properties
4.3 三角范畴的局部化 Localization of Triangulated Categories
4.4 导出范畴 Derived Categories
4.5 态射和扩张 Morphisms and Extensions
4.6 三角函子与局部化 Triangulated Functors and Localizations
4.7 导出函子通论 General Theory of Derived Functors
4.8 有界导出函子 Bounded Derived Functors
4.9 实例: RHom Practical Example: RHom
4.10 实例: R lim 作为同伦极限 Practical Example: R\lim as Homotopy Limits
4.11 无界导出函子 Unbounded Derived Functors
4.12 实例: K-平坦复形和 L⊗ K-Flat Complexes and ⊗^L
习题 Exercises
第五章 谱序列 Spectral Sequences
5.1 滤过与分次结构 Filtration Structures and Grading Structures
5.2 谱序列的一般定义 General Definition of Spectral Sequences
5.3 正合偶 Exact Couples
5.4 滤过微分对象的谱序列 Spectral Sequences of Filtered Differential Objects
5.5 滤过复形的谱序列 Spectral Sequences of Filtered Complexes
5.6 双复形的谱序列及其应用 Spectral Sequences of Double Complexes
5.7 谈谈乘法结构 Discussions on Multiplicative Structures
习题 Exercises
第六章 群的同调与上同调 Group Homology and Cohomology
6.1 G-模及其解消 G-Modules and their Resolutions
6.2 群的同调与上同调 Homology and Cohomology of Groups
6.3 低次上同调: 叉同态和群扩张 Low Degree Cohomology Groups: Crossed Homomorphisms and Group Extensions
6.4 诱导模 Induced Modules
6.5 群的变换 Change of Groups
6.6 重访群扩张 Revisit on Group Extensions
6.7 实例: 循环群与自由群 Practical Examples: Cyclic Groups and Free Groups
6.8 有限指数子群 Finite Exponential Subgroups
6.9 Lyndon–Hochschild–Serre 谱序列 Lyndon–Hochschild–Serre Spectral Sequence
6.10 杯积运算 Cup Product Operation
6.11 Tate 上同调 Tate Cohomology
6.12 pro-有限群的上同调 Cohomology of Pro-Finite Groups
6.13 非交换上同调 Nonabelian Cohomologies
习题 Exercises
第七章 单子论 The Theory of Monads
7.1 幺半范畴上的代数 Algebras over Monoidal Categories
7.2 实例: 微分分次结构 Practical Example: Differential-Graded Structures
7.3 闭幺半范畴 Closed Monoidal Categories
7.4 案例研究: dg-范畴的闭结构 Case Study: Closed Structures of dg-Categories
7.5 从余代数到 Hopf 代数 From Coalgebras to Hopf Algebras
7.6 Beck 单子性定理 The Beck Monadicity Theorem
7.7 森田理论 Morita Theory
7.8 识别模范畴 Identification of Module Categories
7.9 应用: 模的下降 Application: Descents of Modules
7.10 应用: Galois 下降 Application: Galois Descents
7.11 Galois 下降和 H1 的关联 Connection between Galois Descents and and H^1
习题 Exercises
第八章 单纯形方法 Simplicial Methods
8.1 单纯形对象 Simplicial Objects
8.2 单纯形集 Simplicial Sets
8.3 实例: 范畴的脉 Practical Example: Nerves of Categories
8.4 几何实现函子 The Geometric Realization Functor
8.5 Dold–Kan 对应 Dold-Kan Correspondence
8.6 同调计算 Computing Homology
8.7 杠构造 The Bar Construction
8.8 双单纯形对象 Bisimplicial Objects
8.9 闭结构 Close Structures
8.10 重访映射锥 Revisit of Mapping Cones
习题 Exercises
第九章 对偶性 Dualities
9.1 幺半范畴中的对偶性 Duality in Monoidal Categories
9.2 对偶性: 迹和维数 Duality: Trace and Dimension
9.3 对偶性的实例 Practice Examples of Dualities
9.4 自同态余代数 The Coalgebra of Endomorphisms
9.5 重构定理 Reconstruction Theorems
9.6 Hopf 代数的重构 The Reconstruction of Hopf Algebras
9.7 淡中范畴 Tannaka Categories
9.8 有限群的淡中–Krein 定理 Tannaka-Krein Theorem of Finite Groups
习题 Exercises
附录 A 关于 Abel 范畴的延伸内容 Appendix A: Expansive Contents on Abelian Categories
A.1 米田嵌入的稠密性 Density of Yoneda Embedding
A.2 紧对象和可展示范畴 Compact Objects and Presentable Categories
A.3 Gabriel–Popescu 定理 Gabriel–Popescu Theorem
A.4 局部有限 Abel 范畴 Locally Finite Abelian Categories
习题 Exercises
附录 B 简介 ind-对象和 pro-对象 Simple Introduction to Ind-Objects and Pro-Objects
B.1 楔子: pro-有限群 Preface: Profinite Groups
B.2 关于 ind-对象与 pro-对象 On Ind-Objects and Pro-Objects
B.3 范畴的 Ind 化 Ind-Completion of Categories
B.4 函子的 Ind 化与延拓 Ind-Completion of Functors and Extensions
B.5 Abel 范畴的 Ind 化 Ind-Completion of Abelian Categories
B.6 Freyd–Mitchell 嵌入定理 Freyd–Mitchell Embedding Theorem
习题 Exercises