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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
      • Subsection 1.1.3: Dimensions of Simplicial Sets
      • Subsection 1.1.4: The Skeletal Filtration
      • Subsection 1.1.5: Discrete Simplicial Sets
      • Subsection 1.1.6: Directed Graphs as Simplicial Sets
    • Section 1.2: From Topological Spaces to Simplicial Sets
      • Subsection 1.2.1: Connected Components of Simplicial Sets
      • Subsection 1.2.2: The Singular Simplicial Set of a Topological Space
      • Subsection 1.2.3: The Geometric Realization of a Simplicial Set
      • Subsection 1.2.4: Horns
      • Subsection 1.2.5: Kan Complexes
    • Section 1.3: From Categories to Simplicial Sets
      • Subsection 1.3.1: The Nerve of a Category
      • Subsection 1.3.2: Example: Monoids as Simplicial Sets
      • Subsection 1.3.3: Recovering a Category from its Nerve
      • Subsection 1.3.4: Characterization of Nerves
      • Subsection 1.3.5: The Nerve of a Groupoid
      • Subsection 1.3.6: The Homotopy Category of a Simplicial Set
      • Subsection 1.3.7: Example: The Path Category of a Directed Graph
    • Section 1.4: $\infty $-Categories
      • Subsection 1.4.1: Objects and Morphisms
      • Subsection 1.4.2: The Opposite of an $\infty $-Category
      • Subsection 1.4.3: Homotopies of Morphisms
      • Subsection 1.4.4: Composition of Morphisms
      • Subsection 1.4.5: The Homotopy Category of an $\infty $-Category
      • Subsection 1.4.6: Isomorphisms
    • Section 1.5: Functors of $\infty $-Categories
      • Subsection 1.5.1: Examples of Functors
      • Subsection 1.5.2: Commutative Diagrams
      • Subsection 1.5.3: The $\infty $-Category of Functors
      • Subsection 1.5.4: Digression: Lifting Properties
      • Subsection 1.5.5: Trivial Kan Fibrations
      • Subsection 1.5.6: Uniqueness of Composition
      • Subsection 1.5.7: 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: 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 for 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: Contractibility
      • Subsection 3.2.5: The Connecting Homomorphism
      • Subsection 3.2.6: The Long Exact Sequence of a Fibration
      • 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
    • 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: Truncations and Postnikov Towers
      • Subsection 3.5.1: Connectivity
      • Subsection 3.5.2: Connectivity as a Lifting Property
      • Subsection 3.5.3: Coskeletal Simplicial Sets
      • Subsection 3.5.4: Weakly Coskeletal Simplicial Sets
      • Subsection 3.5.5: Higher Groupoids
      • Subsection 3.5.6: Higher Fundamental Groupoids
      • Subsection 3.5.7: Truncated Kan Complexes
      • Subsection 3.5.8: The Postnikov Tower of a Kan Complex
      • Subsection 3.5.9: Truncated Morphisms
    • Section 3.6: Comparison with Topological Spaces
      • Subsection 3.6.1: Digression: Finite Simplicial Sets
      • Subsection 3.6.2: Exactness of Geometric Realization
      • Subsection 3.6.3: Weak Homotopy Equivalences in Topology
      • Subsection 3.6.4: The Unit Map $u: X \rightarrow \operatorname{Sing}_{\bullet }(|X|)$
      • Subsection 3.6.5: Comparison of Homotopy Categories
      • Subsection 3.6.6: 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 Pullback Squares
      • Subsection 4.5.3: Categorical Equivalence
      • Subsection 4.5.4: Categorical Pushout Squares
      • Subsection 4.5.5: Isofibrations of Simplicial Sets
      • Subsection 4.5.6: Isofibrant Diagrams
      • Subsection 4.5.7: Detecting Equivalences of $\infty $-Categories
      • Subsection 4.5.8: Application: Universal Property of the Join
      • Subsection 4.5.9: Relative Exponentiation
    • 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: Digression: Diagrams in Slice $\infty $-Categories
      • Subsection 4.6.7: Initial and Final Objects
      • Subsection 4.6.8: Morphism Spaces in the Homotopy Coherent Nerve
      • Subsection 4.6.9: Composition of Morphisms
    • Section 4.7: Size Conditions on $\infty $-Categories
      • Subsection 4.7.1: Ordinals and Well-Orderings
      • Subsection 4.7.2: Cardinals and Cardinality
      • Subsection 4.7.3: Small Sets
      • Subsection 4.7.4: Small Simplicial Sets
      • Subsection 4.7.5: Essential Smallness
      • Subsection 4.7.6: Minimal $\infty $-Categories
      • Subsection 4.7.7: Small Kan Complexes
      • Subsection 4.7.8: Local Smallness
      • Subsection 4.7.9: Small Fibrations
    • Section 4.8: Truncations in Higher Category Theory
      • Subsection 4.8.1: $(n,1)$-Categories
      • Subsection 4.8.2: Locally Truncated $\infty $-Categories
      • Subsection 4.8.3: Minimality Conditions
      • Subsection 4.8.4: Higher Homotopy Categories
      • Subsection 4.8.5: Full and Faithful Functors
      • Subsection 4.8.6: Essentially Categorical Functors
      • Subsection 4.8.7: Categorically Connective Functors
      • Subsection 4.8.8: Relative Higher Homotopy Categories
      • Subsection 4.8.9: Categorically Connective Morphisms of Simplicial Sets
  • 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: Locally Cartesian Fibrations
      • Subsection 5.1.6: Fiberwise Equivalence
      • Subsection 5.1.7: Equivalence of Inner 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: Example: The Relative Join
      • Subsection 5.2.4: Fibrations over the $1$-Simplex
      • Subsection 5.2.5: The Homotopy Transport Representation
      • Subsection 5.2.6: Elements of Set-Valued Functors
      • Subsection 5.2.7: Covering Space Theory
      • Subsection 5.2.8: Parametrized Covariant Transport
    • Section 5.3: Fibrations over Ordinary Categories
      • Subsection 5.3.1: The Strict Transport Representation
      • Subsection 5.3.2: Homotopy Colimits of Simplicial Sets
      • Subsection 5.3.3: The Weighted Nerve
      • Subsection 5.3.4: Scaffolds of Cocartesian Fibrations
      • Subsection 5.3.5: Application: Classification of Cocartesian Fibrations
      • Subsection 5.3.6: Application: Relative Exponentials
      • Subsection 5.3.7: Application: Path Fibrations
      • Subsection 5.3.8: Digression: Minimal Fibrations
    • Section 5.4: $(\infty ,2)$-Categories
      • Subsection 5.4.1: Definitions
      • Subsection 5.4.2: Interior Fibrations
      • Subsection 5.4.3: Slices of $(\infty ,2)$-Categories
      • Subsection 5.4.4: The Local Thinness Criterion
      • Subsection 5.4.5: The Pith of an $(\infty ,2)$-Category
      • Subsection 5.4.6: The Four-out-of-Five Property
      • Subsection 5.4.7: Functors of $(\infty ,2)$-Categories
      • Subsection 5.4.8: Strict $(\infty ,2)$-Categories
      • Subsection 5.4.9: Comparison of Homotopy Transport Representations
    • Section 5.5: The $\infty $-Categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$
      • Subsection 5.5.1: The $\infty $-Category of Spaces
      • Subsection 5.5.2: Digression: Slicing and the Homotopy Coherent Nerve
      • Subsection 5.5.3: The $\infty $-Category of Pointed Spaces
      • Subsection 5.5.4: The $\infty $-Category of $\infty $-Categories
      • Subsection 5.5.5: The $(\infty ,2)$-Category of $\infty $-Categories
      • Subsection 5.5.6: $\infty $-Categories with a Distinguished Object
    • Section 5.6: Classification of Cocartesian Fibrations
      • Subsection 5.6.1: Elements of Category-Valued Functors
      • Subsection 5.6.2: Elements of $\operatorname{\mathcal{QC}}$-Valued Functors
      • Subsection 5.6.3: Comparison with the Category of Elements
      • Subsection 5.6.4: Comparison with the Weighted Nerve
      • Subsection 5.6.5: The Universality Theorem
      • Subsection 5.6.6: Application: Corepresentable Functors
      • Subsection 5.6.7: Application: Extending Cocartesian Fibrations
      • Subsection 5.6.8: Transport Witnesses
      • Subsection 5.6.9: Proof of the Universality Theorem
  • Chapter 10: Exactness and Animation
    • Section 10.1: Sifted $\infty $-Categories
      • Subsection 10.1.1: Sifted Simplicial Sets
    • Section 10.2: Simplicial Objects of $\infty $-Categories
      • Subsection 10.2.1: Geometric Realization
      • Subsection 10.2.2: Semisimplicial Objects
      • Subsection 10.2.3: Skeletal Simplicial Objects
      • Subsection 10.2.4: Coskeletal Simplicial Objects
      • Subsection 10.2.5: The ČechNerve of a Morphism
      • Subsection 10.2.6: Split Simplicial Objects
    • Section 10.3: Regular $\infty $-Categories
      • Subsection 10.3.1: Sieves
      • Subsection 10.3.2: Quotient Morphisms
      • Subsection 10.3.3: Images
      • Subsection 10.3.4: Universal Quotient Morphisms
      • Subsection 10.3.5: Regular $\infty $-Categories
  • Chapter 11: Retired Tags
    • Section 11.1: Expanded Tags
    • Section 11.2: Discarded Diagrams
    • Section 11.3: Tags without Context
    • Section 11.4: Obsolete Constructions
    • Section 11.5: Tags Lost in Reorganization
    • Section 11.6: Reclassified Tags
    • Section 11.7: Retired Sections
    • Section 11.8: Temporarily Removed Tags
    • Section 11.9: Retired Subsections
      • Subsection 11.9.1: Obsolete Discussion of Balanced Couplings
      • Subsection 11.9.2: Connectivity of Morphisms
      • Subsection 11.9.3: The Universal Property of Twisted Arrows
      • Subsection 11.9.4: Adjunctions as Profunctors
      • Subsection 11.9.5: Towers of Kan Fibrations
      • Subsection 11.9.6: Classification of Fibrations
      • Subsection 11.9.7: Proof of the Universality Theorem
      • Subsection 11.9.8: The Category of Simplices
    • Section 11.10: Temporarily Retired Subsections
      • Subsection 11.10.1: Notational Junk
      • Subsection 11.10.2: The Transport Representation
      • Subsection 11.10.3: Classical Stuff
      • Subsection 11.10.4: Classical Grothendieck Construction for Lax Functors
      • Subsection 11.10.5: Explicit Transport for Set-Valued Functors
      • Subsection 11.10.6: Relative Homotopy Equivalences
      • Subsection 11.10.7: Covariant and Contravariant Equivalences