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: Example: Monoids as Simplicial Sets
    • Subsection 1.2.3: Recovering a Category from its Nerve
    • Subsection 1.2.4: Characterization of Nerves
    • Subsection 1.2.5: The Nerve of a Groupoid
    • Subsection 1.2.6: The Homotopy Category of a Simplicial Set
    • Subsection 1.2.7: 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: 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: Connectivity and Contractibility
    • Subsection 3.2.5: The Connecting Homomorphism
    • Subsection 3.2.6: The Long Exact Sequence of a Fibration
    • Subsection 3.2.7: Connectivity of Morphisms
    • Subsection 3.2.8: Whitehead's Theorem for Kan Complexes
    • Subsection 3.2.9: 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: 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 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
• 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
  • Section 5.4: Size Conditions on $\infty $-Categories
    • Subsection 5.4.1: Ordinals and Well-Orderings
    • Subsection 5.4.2: Cardinals and Cardinality
    • Subsection 5.4.3: Smallness of Sets
    • Subsection 5.4.4: Small Simplicial Sets
    • Subsection 5.4.5: Essential Smallness
    • Subsection 5.4.6: Minimal $\infty $-Categories
    • Subsection 5.4.7: Small Kan Complexes
    • Subsection 5.4.8: Local Smallness
    • Subsection 5.4.9: Smallness of Fibrations
  • Section 5.5: $(\infty ,2)$-Categories
    • Subsection 5.5.1: Definitions
    • Subsection 5.5.2: Interior Fibrations
    • Subsection 5.5.3: Slices of $(\infty ,2)$-Categories
    • Subsection 5.5.4: The Local Thinness Criterion
    • Subsection 5.5.5: The Pith of an $(\infty ,2)$-Category
    • Subsection 5.5.6: The Four-out-of-Five Property
    • Subsection 5.5.7: Functors of $(\infty ,2)$-Categories
    • Subsection 5.5.8: Strict $(\infty ,2)$-Categories
    • Subsection 5.5.9: Comparison of Homotopy Transport Representations
  • Section 5.6: The $\infty $-Categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$
    • Subsection 5.6.1: The $\infty $-Category of Spaces
    • Subsection 5.6.2: Digression: Slicing and the Homotopy Coherent Nerve
    • Subsection 5.6.3: The $\infty $-Category of Pointed Spaces
    • Subsection 5.6.4: The $\infty $-Category of $\infty $-Categories
    • Subsection 5.6.5: The $(\infty ,2)$-Category of $\infty $-Categories
    • Subsection 5.6.6: $\infty $-Categories with a Distinguished Object
  • Section 5.7: Classification of Cocartesian Fibrations
    • Subsection 5.7.1: Elements of Category-Valued Functors
    • Subsection 5.7.2: Elements of $\operatorname{\mathcal{QC}}$-Valued Functors
    • Subsection 5.7.3: Comparison with the Category of Elements
    • Subsection 5.7.4: Comparison with the Weighted Nerve
    • Subsection 5.7.5: The Universality Theorem
    • Subsection 5.7.6: Application: Corepresentable Functors
    • Subsection 5.7.7: Application: Extending Cocartesian Fibrations
    • Subsection 5.7.8: Transport Witnesses
    • Subsection 5.7.9: Proof of the Universality Theorem
• Chapter 10: Exactness and Animation
  • Section 10.1: Simplicial Objects of $\infty $-Categories
    • Subsection 10.1.1: Geometric Realization
    • Subsection 10.1.2: Semisimplicial Objects
    • Subsection 10.1.3: Split Simplicial Objects
    • Subsection 10.1.4: The Čech Nerve
  • Section 10.2: Regular $\infty $-Categories
    • Subsection 10.2.1: Sieves
    • Subsection 10.2.2: Subterminal Objects
    • Subsection 10.2.3: Monomorphisms
    • Subsection 10.2.4: Quotient Morphisms
    • Subsection 10.2.5: Images
    • Subsection 10.2.6: Universal Quotient Morphisms
    • Subsection 10.2.7: 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: The Universal Property of Twisted Arrows
    • Subsection 11.9.2: Adjunctions as Profunctors
    • Subsection 11.9.3: Towers of Kan Fibrations
    • Subsection 11.9.4: Classification of Fibrations
    • Subsection 11.9.5: Proof of the Universality Theorem
    • Subsection 11.9.6: 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