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. $\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$ 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

An online resource for homotopy-coherent mathematics ... Kerodon $\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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.

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.

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.

2.4.3 The Homotopy Coherent Nerve. Let $\operatorname{Top}$ denote the category of topological spaces and let $\operatorname{N}_{\bullet }(\operatorname{Top})$ denote its nerve (Construction 1.3.1.1).Then $\operatorname{N}_{\bullet }(\operatorname{Top})$ is a simplicial set whose $2$-simplices can be identified with diagrams of topological spaces $\sigma :$

An online resource for homotopy-coherent mathematics. 4.7 Size Conditions on $\infty $-Categories. Recall that a small category $\operatorname{\mathcal{C}}$ consists of the following data:

(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 ...

see Theorem 2.3.4.1.In other words, the formation of Duskin nerves induces a fully faithful embedding from the category $\operatorname{2Cat}_{\operatorname{ULax}}$ of Definition 2.2.5.5 to the category of simplicial sets.. By virtue of Theorem 2.3.4.1, it is mostly harmless to abuse terminology by identifying a $2$-category $\operatorname{\mathcal{C}}$ with the simplicial set $\operatorname{N ...