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1 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: Discrete Simplicial Sets
      • Subsection 1.1.5: Directed Graphs as Simplicial Sets
      • Subsection 1.1.6: Connected Components of Simplicial Sets
      • Subsection 1.1.7: The Singular Simplicial Set of a Topological Space
      • Subsection 1.1.8: The Geometric Realization of a Simplicial Set
      • Subsection 1.1.9: 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
      • Subsection 1.2.5: The Homotopy Category of a Simplicial Set
      • Subsection 1.2.6: Example: The Path Category of a Directed Graph
    • 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 of an $\infty $-Category
      • Subsection 1.3.6: 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
      • Subsection 1.4.7: Universality of Path Categories
  • Chapter 2: Examples of $\infty $-Categories
    • Section 2.1: Monoidal Categories
      • Subsection 2.1.1: Nonunital Monoidal Categories
      • Subsection 2.1.2: Monoidal Categories
      • Subsection 2.1.3: Examples of Monoidal Categories
      • Subsection 2.1.4: Nonunital Monoidal Functors
      • Subsection 2.1.5: Lax Monoidal Functors
      • Subsection 2.1.6: Monoidal Functors
      • Subsection 2.1.7: Enriched Category Theory
    • Section 2.2: The Theory of $2$-Categories
      • Subsection 2.2.1: $2$-Categories
      • Subsection 2.2.2: Examples of $2$-Categories
      • Subsection 2.2.3: Opposite and Conjugate $2$-Categories
      • Subsection 2.2.4: Functors of $2$-Categories
      • Subsection 2.2.5: The Category of $2$-Categories
      • Subsection 2.2.6: Isomorphisms of $2$-Categories
      • Subsection 2.2.7: Strictly Unitary $2$-Categories
    • Section 2.3: The Duskin Nerve of a $2$-Category
      • Subsection 2.3.1: The Duskin Nerve
      • Subsection 2.3.2: From $2$-Categories to $\infty $-Categories
      • Subsection 2.3.3: Thin $2$-Simplices of a Duskin Nerve
      • Subsection 2.3.4: Recovering a $2$-Category from its Duskin Nerve
      • Subsection 2.3.5: Twisted Arrows and the Nerve of $\operatorname{Corr}(\operatorname{\mathcal{C}})^{\operatorname{c}}$
      • Subsection 2.3.6: The Duskin Nerve of a Strict $2$-Category
    • Section 2.4: Simplicial Categories
      • Subsection 2.4.1: Simplicial Enrichment
      • Subsection 2.4.2: Examples of Simplicial Categories
      • Subsection 2.4.3: The Homotopy Coherent Nerve
      • Subsection 2.4.4: The Path Category of a Simplicial Set
      • Subsection 2.4.5: From Simplicial Categories to $\infty $-Categories
      • Subsection 2.4.6: The Homotopy Category of a Simplicial Category
      • Subsection 2.4.7: Example: Braid Monoids
    • Section 2.5: Differential Graded Categories
      • Subsection 2.5.1: Generalities on Chain Complexes
      • Subsection 2.5.2: Differential Graded Categories
      • Subsection 2.5.3: The Differential Graded Nerve
      • Subsection 2.5.4: The Homotopy Category of a Differential Graded Category
      • Subsection 2.5.5: Digression: The Homology of Simplicial Sets
      • Subsection 2.5.6: The Dold-Kan Correspondence
      • Subsection 2.5.7: The Shuffle Product
      • Subsection 2.5.8: The Alexander-Whitney Construction
      • Subsection 2.5.9: Comparison with the Homotopy Coherent Nerve
  • Chapter 3: Kan Complexes
    • Section 3.1: The Homotopy Theory of Kan Complexes
      • Subsection 3.1.1: Kan Fibrations
      • Subsection 3.1.2: Left and Right Fibrations
      • Subsection 3.1.3: Exponentiation of Kan Fibrations
      • Subsection 3.1.4: The $\infty $-Category of Kan Complexes
      • Subsection 3.1.5: Homotopy Equivalences and Weak Homotopy Equivalences
      • Subsection 3.1.6: Anodyne Morphisms
      • Subsection 3.1.7: Fibrant Replacement
    • Section 3.2: Homotopy Groups
      • Subsection 3.2.1: Pointed Kan Complexes
      • Subsection 3.2.2: The Homotopy Groups of a Kan Complex
      • Subsection 3.2.3: The Group Structure on $\pi _{n}(X,x)$
      • Subsection 3.2.4: The Connecting Homomorphism
      • Subsection 3.2.5: The Long Exact Sequence of a Fibration
      • Subsection 3.2.6: Whitehead's Theorem
      • Subsection 3.2.7: Closure Properties of Homotopy Equivalences
    • Section 3.3: The $\operatorname{Ex}^{\infty }$ Functor
      • Subsection 3.3.1: Subdivision of Simplices
      • Subsection 3.3.2: Digression: Braced Simplicial Sets
      • Subsection 3.3.3: The Subdivision of a Simplicial Set
      • Subsection 3.3.4: The Last Vertex Map
      • Subsection 3.3.5: Comparison of $X$ with $\operatorname{Ex}(X)$
      • Subsection 3.3.6: The $\operatorname{Ex}^{\infty }$ Functor
      • Subsection 3.3.7: Application: Characterizations of Weak Homotopy Equivalences
      • Subsection 3.3.8: Application: Extending Kan Fibrations