Vladimir Hinich · Research paper · Graduate · English
PDF
Public full text; checked against the title and contents.已核对题名和目录，可公开阅读全文。
Revisits simplicial localization in infinity-categorical language and studies the passage from Quillen adjunctions to localized categories.以无穷范畴语言重新讨论单纯局部化，并研究Quillen伴随如何传到局部化后的范畴。
Preparation: Model categories; simplicial categories; adjunctions.模型范畴、单纯范畴与伴随。
Use: Compare simplicial localization with the induced adjunction from a Quillen pair.比较单纯局部化与Quillen对所诱导的伴随。
An undergraduate text on primes, congruences, arithmetic functions, quadratic residues and continued fractions, with later introductions to analytic topics.
Wissam Raji · Open textbook · Introductory · English
Online text
Licensed CC BY HTML book on LibreTexts; full chapter/section reading pages opened without login.
An undergraduate text on primes, congruences, arithmetic functions, quadratic residues and continued fractions, with later introductions to analytic topics.
Preparation: Elementary algebra and proof-writing; later chapters need analysis and abstract algebra.
Use: A first structured number-theory reading route, followed by selected advanced chapters.
Version: LibreTexts open HTML edition; edition date not stated · Checked 2026-10-13
Develops divisibility, prime numbers, congruences, arithmetic functions, primality tests and public-key cryptography through short theorem-based chapters.
W. Edwin Clark; revised by Jim Hefferon · Textbook · Introductory · English
PDF
Reviser's university-hosted archive provides the complete 82-page PDF; its notice permits noncommercial redistribution.
Develops divisibility, prime numbers, congruences, arithmetic functions, primality tests and public-key cryptography through short theorem-based chapters.
Preparation: School algebra; basic sets and functions. Calculus I is suggested in the preface, chiefly for mathematical maturity.
Use: Build proof-writing skill while learning elementary arithmetic theory.
Version: Hefferon revision December 2003; PDF compiled January 5, 2004 · Checked 2026-10-13
Connects classical topics with computation, cryptography and elliptic curves. Broader than a minimal congruences course; later chapters benefit from abstract algebra.
William Stein · Textbook · Introductory · English (legacy metadata)
PDF
Connects classical topics with computation, cryptography and elliptic curves. Broader than a minimal congruences course; later chapters benefit from abstract algebra.
Preparation: Proofs; basic algebra, rings and fields.
Oleg Viro, Oleg Ivanov, Nikita Netsvetaev and Viatcheslav Kharlamov
Introductory · PDF
A problem-led development of general topology, followed by fundamental groups and covering spaces. This record uses the full Russian book, not the shortened English file without proofs.
Oleg Viro, Oleg Ivanov, Nikita Netsvetaev and Viatcheslav Kharlamov · Problem textbook · Introductory · Russian
PDF
The author’s university book page links the complete Russian PDF.
A problem-led development of general topology, followed by fundamental groups and covering spaces. This record uses the full Russian book, not the shortened English file without proofs.
Preparation: Set theory; proof writing; elementary group theory for later sections.
Use: Work through examples and proof exercises before a homology course.
Version: Third Russian edition, 2010 · Checked 2026-10-13
A substantial treatment of higher categories for mathematically mature readers. A later step after ordinary category theory, not a shortcut around its foundations.
Emily Riehl and Dominic Verity · Monograph · Advanced · English (legacy metadata)
PDF
A substantial treatment of higher categories for mathematically mature readers. A later step after ordinary category theory, not a shortcut around its foundations.
Thermodynamic and probabilistic entropy ideas organize elliptic and parabolic estimates, conservation laws, Hamilton–Jacobi equations and stochastic examples.
Lawrence C. Evans · Lecture notes · Graduate · English
PDF
The complete 213-page text opened at math.berkeley.edu.
Thermodynamic and probabilistic entropy ideas organize elliptic and parabolic estimates, conservation laws, Hamilton–Jacobi equations and stochastic examples.
Preparation: Graduate real analysis and a first PDE course.
Use: Study how entropy structures yield estimates and select physically meaningful solutions.
Version: Author-hosted notes; date not stated · Checked 2026-10-13
Learning notes organized around analogies with algebraic topology and complex geometry. Some listed sections remain unwritten, so this is a selective companion rather than a complete first course.
Online lecture text; chapters can be read separately.在线讲义，可按章节阅读。
Original title: 给几何人的平展上同调
Learning notes organized around analogies with algebraic topology and complex geometry. Some listed sections remain unwritten, so this is a selective companion rather than a complete first course.围绕代数拓扑、复几何中的类比组织的学习笔记。部分列出的章节尚未写成，适合作为补充阅读，不宜当作完整的第一门课程。
Preparation: Schemes, sheaves and homological algebra.概形、层和同调代数。
Use: Compare specific results with Milne or SGA; keep a systematic reference available for omitted arguments.按主题与Milne或SGA对照阅读，省略的论证需要查系统教材。
Version: 2023 learning notes; some sections unwritten2023年学习笔记；部分章节尚未写成 · Checked 2026-11-03
Alexander Grothendieck and Michèle Raynaud · Research monograph · Graduate · French
PDF
Public full text; checked against the title and contents.已核对题名和目录，可公开阅读全文。
Original title: Revêtements étales et groupe fondamental
Foundations of the algebraic fundamental group through finite etale coverings and descent; this is the corrected electronic re-edition.通过有限平展覆盖与下降建立代数基本群理论；链接为修订后的电子重印版。
P. D. Magnus, Tim Button, Aaron Thomas-Bolduc and Richard Zach, with contributors
Introductory · Free to read
Introduces formal languages, truth tables and natural deduction. Exercises and solutions support a first logic course; metatheory is developed further in the next text.
P. D. Magnus, Tim Button, Aaron Thomas-Bolduc and Richard Zach, with contributors · Textbook · Introductory · English (legacy metadata)
Online text
Introduces formal languages, truth tables and natural deduction. Exercises and solutions support a first logic course; metatheory is developed further in the next text.
Lecture notes emphasizing simplicial homology and its technical foundations, followed by singular and cellular methods and the universal coefficient theorem. The recorded version does not develop the cohomology ring.
从前有一只呜喵 (notes credited in the introduction) · Course notes · Intermediate · Chinese
Online text
Online lecture text; chapters can be read separately.在线讲义，可按章节阅读。
Original title: 代数拓扑基础
Lecture notes emphasizing simplicial homology and its technical foundations, followed by singular and cellular methods and the universal coefficient theorem. The recorded version does not develop the cohomology ring.重点讲单纯同调及其技术细节，随后讨论奇异同调、胞腔方法和万有系数定理。这个记录版本没有继续展开上同调环。
Preparation: Point-set topology; groups, quotient groups and linear algebra.点集拓扑、群与商群、线性代数。
Use: Work through simplicial chains and relative homology before using the exact-sequence arguments.先计算单纯链与相对同调，再使用后面的正合序列论证。