Kerodon is an online textbook on categorical homotopy theory and related mathematics. It currently consists of a handful of chapters, but should grow (slowly) over time. It is modeled on the Stacks project , and is maintained by Jacob Lurie .

an online resource for homotopy-coherent mathematics. Forage Part 1: Foundations. Chapter 1: The Language of $\infty $-Categories; Chapter 2: Examples of $\infty $-Categories; Chapter 3: Kan Complexes; Chapter 4: The Homotopy Theory of $\infty $-Categories; Chapter 5: Fibrations of $\infty $-Categories. Part 2: Higher Category Theory. Chapter 6: Adjoint Functors ...

An online resource for homotopy-coherent mathematics. $\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$ 1 Foundations Structure. Chapter 1: The Language of $\infty $-Categories . Section 1.1: Simplicial Sets . Subsection 1.1.1: Face Operators ; Subsection 1.1.2: Degeneracy Operators

4.3.3 Joins of Simplicial Sets. Our next goal is to extend the join operation of Definition 4.3.2.1 to the setting of $\infty $-categories (and more general simplicial sets). We begin with a slightly more general discussion. Let $\operatorname{Lin}$ denote the category whose objects are finite linearly ordered sets and whose morphisms are nondecreasing functions.

1 The Language of $\infty $-Categories. A principal goal of algebraic topology is to understand topological spaces by means of algebraic and combinatorial invariants. Let us consider some elementary examples.

Tags explained The tag system. Each tag refers to a unique item (section, lemma, theorem, etc.) in order for this project to be referenceable. These tags don't change even if the item moves within the text.

(see Remark 1.1.1.7).In §1.1.1, we prove a partial converse: from a collection of sets $\{ S_{n} \} $ and face operators $\{ d^{n}_{i}: S_{n} \rightarrow S_{n-1} \} $ which satisfy (), we can uniquely reconstruct the data of a semisimplicial set: that is, a (contravariant) set-valued functor on the subcategory $\operatorname{{\bf \Delta }}_{\operatorname{inj}} \subset \operatorname{{\bf ...

An online resource for homotopy-coherent mathematics. We refer the reader to [] for a more detailed discussion (including an extension to the setting of topological groups).. Proof of Proposition 1.3.5.2. Suppose first that $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex; we wish to show that $\operatorname{\mathcal{C}}$ is a groupoid.

Kerodon $\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$ about; changes; recent comments; bibliography. next. statistics. comments (0) Part 1: Foundations Chapter 1: The Language of $\infty $-Categories Section 1.2: From Topological Spaces to Simplicial Sets ...

An online resource for homotopy-coherent mathematics. where $\operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{C}})$ denotes the full ...