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

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. 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. 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.

7.6 Examples of Limits and Colimits. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §7.1, we introduced the notion of limit and colimit for an arbitrary morphism of simplicial sets $\sigma : K \rightarrow \operatorname{\mathcal{C}}$.Our goal in this section is to make the general theory more explicit for some special classes of diagrams which arise frequently in …

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$ Bibliography A. Ad á mek, J. and Rosick ý, J., On sifted colimits and generalized varieties.; Adámek, Jiří and Rosický, Jiří, Locally presentable and accessible categories. Arone, Greg and Ching, Michael, Operads and chain rules for the calculus of functors. Arone, Greg and Mahowald, Mark, The G oodwillie tower of the identity …

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 $\sigma :$

(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 …