HADR Higher Algebra Done Right

Higher Algebra is a book by Professor Jacob Lurie. Its main purpose is to establish theories of several kinds of algebra, such as commutative algebra and associative algebra, in ∞-categories (especially stable ∞-categories). Higher Algebra 是 Jacob Lurie 教授的著作, 主要目的是建立 ∞-范畴 (特别是稳定 ∞-范畴) 中交换代数和结合代数等几种代数的理论.

Higher Algebra Done Right is a group founded in 2023 by several students at Qiuzhen College from the classes of 2020 through 2023. At that time we began learning the concepts of higher category theory, dreaming of an essential understanding of Higher Algebra stripped of its model-theoretic shell. Three years later, in the summer of 2026, we finally felt that we had enough experience to read every chapter of Higher Algebra and to understand its model-independent essence. So it is time to hold this seminar and turn that experience into a shared consensus. Higher Algebra Done Right 是求真书院的几位 0 至 3 字班同学于 2023 年建立的群组. 那时我们开始学习高阶范畴的概念, 梦想对 Higher Algebra 获得一个剥去模型外壳的本质的理解. 三年后, 2026 年的夏天, 我们终于认为自己具备足够的阅历读懂 Higher Algebra 的每个章节, 以及其 “模型无关” (model-independent) 的本质. 因此是时候举办讨论班, 将这些阅历固化下来成为共识了.

Organization会务组织

The seminar is planned for the Fall 2026 semester. To finish the plan within one semester, there will be one talk each week. The Fall 2026 semester begins on September 14. 讨论班计划于 2026 年秋季学期举办. 为了在一学期之内完成计划, 每周举办一场. 2026 年秋季学期从 9 月 14 日开始.

Time时间

Mondays, 22:00-23:00 (Beijing Time)每周一北京时间 22:00-23:00

Venue地点

None (online only)暂无 (纯线上)

By default I (Wang Jinyi) will be the speaker, but before each talk anyone is welcome to volunteer to lead it. Please provide an abstract, ideally notes (Chinese, TeX/Markdown), following the conventions below. You do not need to follow the exposition of HA; covering the corresponding topics is enough. 主讲人默认为我 (王进一), 每一讲之前都欢迎任何人报名认领本次的主讲. 请提供摘要, 最好能提供讲稿 (中文, tex/markdown), 参照下文的重要约定. 不需要按照 HA 的讲法, 覆盖对应话题即可.

Online Participation在线参会

All talks will be streamed live via Tencent Meeting. 讨论班的每次报告均会通过腾讯会议直播并录制.

Meeting ID: 会议号： 598-747-9669 Copy 复制

Prerequisites前置知识

Participants should have some mathematical maturity and be familiar with the following concepts: 讨论班需要参与者具有一定数学成熟度, 并了解以下概念:

• Basic concepts of ordinary category theory 传统范畴论的基本概念
  • limits, colimits, adjoint functors;极限, 余极限, 伴随函子;
  • associative algebras, commutative algebras;结合代数, 交换代数;
  • fibered categories, straightening–unstraightening;纤维范畴, 直化–反直化;
  • presentable categories, the adjoint functor theorem;可表现范畴, 伴随函子定理;
• Basic concepts of ordinary homotopy theory 传统同伦论基本概念
  • homotopy, homotopy equivalence, homotopy limits/colimits;同伦, 同伦等价, 同伦极限/余极限;
  • spectra, spectral sequences.谱, 谱序列.

The following background would be helpful: 具有如下背景则更好:

• homotopy type theory (HoTT) — identity types, the univalence axiom;同伦类型论 (HoTT) - 等式类型, 泛等公理;
• topos theory.意象理论.

Conventions重要约定

We hope all participants accept anima as a primitive mathematical concept, in the same way that sets are primitive in traditional mathematics. 希望所有参与者接受以生象 (anima) 为原生概念的数学, 如同传统数学中的集合一般.

By default, we will call an (∞,1)-category a category, an (∞,n)-category an n-category, and a stable ∞-category a stable category. Traditional “categories” will be called (1,1)-categories and automatically regarded as (∞,1)-categories; in particular, we will not use the notion of a “nerve”. 我们将默认称 (∞,1)-范畴为范畴, (∞,n)-范畴为 n-范畴, 稳定 ∞-范畴为稳定范畴. 传统的 “范畴” 将称为 (1,1)-范畴并自动视为 (∞,1)-范畴, 即我们不会使用 “脉” (nerve) 的概念.

Category-theoretic concepts will mean their ∞-categorical versions by default. For example, we will not say “homotopy” limits or “derived” tensor products; we will simply say limits and tensor products. 范畴论概念将默认为 ∞-范畴论的版本, 例如我们不会说 “同伦” 极限, “导出” 张量积, 而是直接说极限, 张量积.

Terminology翻译

Model independence模型无关

We hope all participants use model-independent language as much as possible. In particular, please avoid the following notions: 希望所有参与者尽量使用模型无关的语言, 即尽量避免使用如下概念:

• the simplicial set model of ∞-categories and Kan complexes (although simplicial anima and Segal anima are reasonable); ∞-范畴的单纯集模型, Kan 复形 (但单纯生象与 Segal 生象的概念是合理的);
• the topological category model of ∞-categories, including concrete topological spaces (instead, use anima presented by topological spaces); ∞-范畴的拓扑范畴模型, 包括具体的拓扑空间 (转而使用拓扑空间表现的生象);
• the notion of a model category, including fibrant replacement; 模型范畴的概念, 包括 fibrant replacement;
• the notion of a fibered category (instead, use functors to Cat); 纤维范畴的概念 (转而使用到 Cat 的函子);
• the chain complex model of stable categories (unless the purpose of the talk is to introduce chain complexes) and differential graded categories (dg categories). 稳定范畴的链复形模型 (除非报告的目的就是介绍链复形), 微分分次范畴 (dg category).

We will ignore issues of “large” and “small” categories. For example, the category of categories Cat may itself be viewed as a category; we trust that a suitable hierarchy of type-theoretic universes can handle these issues. 我们将忽略范畴的 “大” 与 “小” 的问题, 例如范畴的范畴 Cat 本身也可以视为一个范畴; 相信适当的类型论宇宙层级能够处理这些问题.

Since the mathematical foundations for ∞-categories are temporarily incomplete, we will not be able to pursue complete rigor. 由于 ∞-范畴相关数学基础的暂时缺失, 我们将无法追求完全的严谨性.

Use of AI toolsAI 工具的使用

This seminar encourages the use of AI tools. Following the Leiden Declaration, participants are asked to disclose the tools they used and their purposes, and to take responsibility for the accuracy of the final content. 本讨论班鼓励 AI 工具的使用, 并希望参与者根据《莱顿宣言》, 透明地披露使用的工具以及目的, 并对最终内容的准确性负责.

The text portions of this draft seminar plan did not use any AI tools. 这份讨论班计划拟稿的文字部分未使用任何 AI 工具.