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I Higher Category Theory

Structure

  • Chapter 1: The Language of $\infty $-Categories
    • Section 1.1: Simplicial Sets
      • Subsection 1.1.1: Simplicial and Cosimplicial Objects
      • Subsection 1.1.2: Simplices and Horns
      • Subsection 1.1.3: The Skeletal Filtration
      • Subsection 1.1.4: Directed Graphs as Simplicial Sets
      • Subsection 1.1.5: The Singular Simplicial Set of a Topological Space
      • Subsection 1.1.6: The Geometric Realization of Simplicial Set
      • Subsection 1.1.7: Kan Complexes
    • Section 1.2: The Nerve of a Category
      • Subsection 1.2.1: Construction of the Nerve
      • Subsection 1.2.2: Recovering a Category from its Nerve
      • Subsection 1.2.3: Characterization of Nerves
      • Subsection 1.2.4: The Nerve of a Groupoid
    • Section 1.3: $\infty $-Categories
      • Subsection 1.3.1: Objects and Morphisms
      • Subsection 1.3.2: The Opposite of an $\infty $-Category
      • Subsection 1.3.3: Homotopies of Morphisms
      • Subsection 1.3.4: Composition of Morphisms
      • Subsection 1.3.5: The Homotopy Category
      • Subsection 1.3.6: The Universal Property of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$
      • Subsection 1.3.7: Equivalences
    • Section 1.4: Functors of $\infty $-Categories
      • Subsection 1.4.1: Examples of Functors
      • Subsection 1.4.2: Commutative Diagrams
      • Subsection 1.4.3: The $\infty $-Category of Functors
      • Subsection 1.4.4: Digression: Lifting Properties
      • Subsection 1.4.5: Trivial Kan Fibrations
      • Subsection 1.4.6: Uniqueness of Composition