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1 Foundations

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: Isomorphisms
    • 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
      • Subsection 2.2.8: The Homotopy Category of a $2$-Category
    • 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: Anodyne Morphisms
      • Subsection 3.1.3: Exponentiation of Kan Fibrations
      • Subsection 3.1.4: Covering Maps
      • Subsection 3.1.5: The Homotopy Category of Kan Complexes
      • Subsection 3.1.6: Homotopy Equivalences and Weak Homotopy Equivalences
      • 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: Contractibility
      • Subsection 3.2.7: Whitehead's Theorem for Kan Complexes
      • Subsection 3.2.8: Closure Properties of Homotopy Equivalences
    • Section 3.3: The $\operatorname{Ex}^{\infty }$ Functor
      • Subsection 3.3.1: Digression: Braced Simplicial Sets
      • Subsection 3.3.2: The Subdivision of a Simplex
      • 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
      • Subsection 3.3.9: Towers of Kan Fibrations
    • Section 3.4: Homotopy Pullback and Homotopy Pushout Squares
      • Subsection 3.4.1: Homotopy Pullback Squares
      • Subsection 3.4.2: Homotopy Pushout Squares
      • Subsection 3.4.3: Mather's Second Cube Theorem
      • Subsection 3.4.4: Mather's First Cube Theorem
      • Subsection 3.4.5: Digression: Weak Homotopy Equivalences of Semisimplicial Sets
      • Subsection 3.4.6: Excision
      • Subsection 3.4.7: The Seifert van-Kampen Theorem
    • Section 3.5: Comparison with Topological Spaces
      • Subsection 3.5.1: Digression: Finite Simplicial Sets
      • Subsection 3.5.2: Exactness of Geometric Realization
      • Subsection 3.5.3: Weak Homotopy Equivalences in Topology
      • Subsection 3.5.4: The Unit Map $u: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$
      • Subsection 3.5.5: Comparison of Homotopy Categories
      • Subsection 3.5.6: Serre Fibrations
  • Chapter 4: The Homotopy Theory of $\infty $-Categories
    • Section 4.1: Inner Fibrations
      • Subsection 4.1.1: Inner Fibrations of Simplicial Sets
      • Subsection 4.1.2: Subcategories of $\infty $-Categories
      • Subsection 4.1.3: Inner Anodyne Morphisms
      • Subsection 4.1.4: Exponentiation for Inner Fibrations
      • Subsection 4.1.5: Inner Covering Maps
    • Section 4.2: Left and Right Fibrations
      • Subsection 4.2.1: Left and Right Fibrations of Simplicial Sets
      • Subsection 4.2.2: Fibrations in Groupoids
      • Subsection 4.2.3: Left and Right Covering Maps
      • Subsection 4.2.4: Left Anodyne and Right Anodyne Morphisms
      • Subsection 4.2.5: Exponentiation for Left and Right Fibrations
      • Subsection 4.2.6: The Homotopy Extension Lifting Property
    • Section 4.3: The Slice and Join Constructions
      • Subsection 4.3.1: Slices of Categories
      • Subsection 4.3.2: Joins of Categories
      • Subsection 4.3.3: Joins of Simplicial Sets
      • Subsection 4.3.4: Joins of Topological Spaces
      • Subsection 4.3.5: Slices of Simplicial Sets
      • Subsection 4.3.6: Slices of $\infty $-Categories
      • Subsection 4.3.7: Slices of Left and Right Fibrations
    • Section 4.4: Isomorphisms and Isofibrations
      • Subsection 4.4.1: Isofibrations of $\infty $-Categories
      • Subsection 4.4.2: Isomorphisms and Lifting Properties
      • Subsection 4.4.3: The Core of an $\infty $-Category
      • Subsection 4.4.4: Natural Isomorphisms
      • Subsection 4.4.5: Exponentiation for Isofibrations
    • Section 4.5: Equivalence
      • Subsection 4.5.1: Equivalences of $\infty $-Categories
      • Subsection 4.5.2: Categorical Equivalences of Simplicial Sets
      • Subsection 4.5.3: Categorical Pushout Diagrams
      • Subsection 4.5.4: Detecting Equivalences of $\infty $-Categories
      • Subsection 4.5.5: Application: Universal Property of the Join
      • Subsection 4.5.6: Lifting Property of Isofibrations
      • Subsection 4.5.7: Isofibrations of Simplicial Sets
    • Section 4.6: Morphism Spaces
      • Subsection 4.6.1: Morphism Spaces
      • Subsection 4.6.2: Fully Faithful and Essentially Surjective Functors
      • Subsection 4.6.3: Digression: Categorical Mapping Cylinders
      • Subsection 4.6.4: Oriented Fiber Products
      • Subsection 4.6.5: Pinched Morphism Spaces
      • Subsection 4.6.6: Morphism Spaces in the Homotopy Coherent Nerve
      • Subsection 4.6.7: Composition of Morphisms
  • Chapter 5: Fibrations of $\infty $-Categories
    • Section 5.1: Cartesian Fibrations
      • Subsection 5.1.1: Cartesian Edges of Simplicial Sets
      • Subsection 5.1.2: Cartesian Morphisms of $\infty $-Categories
      • Subsection 5.1.3: Locally Cartesian Edges
      • Subsection 5.1.4: Cartesian Fibrations
      • Subsection 5.1.5: Equivalences of Fibered $\infty $-Categories
      • Subsection 5.1.6: Equivalence of Inner Fibrations
      • Subsection 5.1.7: Example: Path Fibrations
    • Section 5.2: Covariant Transport
      • Subsection 5.2.1: Exponentiation for Cartesian Fibrations
      • Subsection 5.2.2: Covariant Transport Functors
      • Subsection 5.2.3: Transitivity of Covariant Transport
      • Subsection 5.2.4: Digression: The Relative Join
      • Subsection 5.2.5: Fibrations over the $1$-Simplex
      • Subsection 5.2.6: Fibrations over the $n$-Simplex
      • Subsection 5.2.7: Parametrized Covariant Transport
    • Section 5.3: $(\infty ,2)$-Categories
      • Subsection 5.3.1: Definitions
      • Subsection 5.3.2: Interior Fibrations
      • Subsection 5.3.3: Slices of $(\infty ,2)$-Categories
      • Subsection 5.3.4: The Local Thinness Criterion
      • Subsection 5.3.5: The Pith of an $(\infty ,2)$-Category
      • Subsection 5.3.6: The Four-out-of-Five Property
      • Subsection 5.3.7: Functors of $(\infty ,2)$-Categories
      • Subsection 5.3.8: Strict $(\infty ,2)$-Categories
      • Subsection 5.3.9: Comparison of Homotopy Transport Representations
    • Section 5.4: The $\infty $-Categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$
      • Subsection 5.4.1: The $\infty $-Category of Spaces
      • Subsection 5.4.2: Digression: Slicing and the Homotopy Coherent Nerve
      • Subsection 5.4.3: The $\infty $-Category of Pointed Spaces
      • Subsection 5.4.4: The $\infty $-Category of $\infty $-Categories
      • Subsection 5.4.5: The $(\infty ,2)$-Category of $\infty $-Categories
      • Subsection 5.4.6: $\infty $-Categories with a Distinguished Object
    • Section 5.5: The Category of Elements
      • Subsection 5.5.1: Elements of Set-Valued Functors
      • Subsection 5.5.2: Elements of Category-Valued Functors
      • Subsection 5.5.3: The Weighted Nerve
      • Subsection 5.5.4: Elements of $\operatorname{\mathcal{QC}}$-Valued Functors
      • Subsection 5.5.5: Comparison with the Category of Elements
      • Subsection 5.5.6: Comparison with the Weighted Nerve
    • Section 5.6: Classification of Cocartesian Fibrations
      • Subsection 5.6.1: Covering Space Theory
      • Subsection 5.6.2: The Covariant Transport Representation
      • Subsection 5.6.3: Application: Fibrations of Ordinary Categories
      • Subsection 5.6.4: Application: Extending Cocartesian Fibrations
      • Subsection 5.6.5: Transport Witnesses
      • Subsection 5.6.6: Uniqueness of the Transport Representation