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2 Higher Category Theory

Structure

  • Chapter 6: Adjoint Functors
    • Section 6.1: Adjunctions in $2$-Categories
      • Subsection 6.1.1: Adjunctions
      • Subsection 6.1.2: Adjuncts
      • Subsection 6.1.3: Uniqueness of Adjoints
      • Subsection 6.1.4: Adjoints of Isomorphisms
      • Subsection 6.1.5: Composition of Adjunctions
      • Subsection 6.1.6: Duality in Monoidal Categories
    • Section 6.2: Adjoint Functors Between $\infty $-Categories
      • Subsection 6.2.1: Adjunctions of $\infty $-Categories
      • Subsection 6.2.2: Reflective Subcategories
      • Subsection 6.2.3: Local Objects
      • Subsection 6.2.4: Stability Properties of Reflective Subcategories
      • Subsection 6.2.5: Correspondences
      • Subsection 6.2.6: Local Existence Criterion
      • Subsection 6.2.7: Digression: $\infty $-Categories with Short Morphisms
    • Section 6.3: Localization
      • Subsection 6.3.1: Localizations of $\infty $-Categories
      • Subsection 6.3.2: Existence of Localizations
      • Subsection 6.3.3: Reflective Localizations
      • Subsection 6.3.4: Fiberwise Localization
      • Subsection 6.3.5: Universal Localizations
      • Subsection 6.3.6: Subdivision and Localization
  • Chapter 7: Limits and Colimits
    • Section 7.1: Limits and Colimits
      • Subsection 7.1.1: Limits and Colimits in $\infty $-Categories
      • Subsection 7.1.2: Constant Diagrams
      • Subsection 7.1.3: Limit and Colimit Diagrams
      • Subsection 7.1.4: Preservation of Limits and Colimits
      • Subsection 7.1.5: Relative Initial and Final Objects
      • Subsection 7.1.6: Relative Limits and Colimits
      • Subsection 7.1.7: Limits and Colimits of Functors
    • Section 7.2: Cofinality
      • Subsection 7.2.1: Cofinal Morphisms of Simplicial Sets
      • Subsection 7.2.2: Cofinality and Limits
      • Subsection 7.2.3: Quillen's Theorem A for $\infty $-Categories
    • Section 7.3: Kan Extensions
      • Subsection 7.3.1: Kan Extensions along General Functors
      • Subsection 7.3.2: Kan Extensions along Inclusions
      • Subsection 7.3.3: Relative Kan Extensions
      • Subsection 7.3.4: Kan Extensions along Fibrations
      • Subsection 7.3.5: Existence of Kan Extensions
      • Subsection 7.3.6: The Universal Property of Kan Extensions
      • Subsection 7.3.7: Kan Extensions in Functor $\infty $-Categories
      • Subsection 7.3.8: Transitivity of Kan Extensions
      • Subsection 7.3.9: Relative Colimits for Cocartesian Fibrations
    • Section 7.4: Limits and Colimits of Spaces
      • Subsection 7.4.1: Limits of Spaces
      • Subsection 7.4.2: Digression: Functoriality of Covariant Transport
      • Subsection 7.4.3: Colimits of Spaces
      • Subsection 7.4.4: Limits of $\infty $-Categories
      • Subsection 7.4.5: Colimits of $\infty $-Categories
      • Subsection 7.4.6: Proof of the Refraction Criterion
    • Section 7.5: Homotopy Limits and Colimits
      • Subsection 7.5.1: Homotopy Limits of Kan Complexes
      • Subsection 7.5.2: Homotopy Limits of $\infty $-Categories
      • Subsection 7.5.3: The Homotopy Limit as a Derived Functor
      • Subsection 7.5.4: Homotopy Limit Diagrams
      • Subsection 7.5.5: Categorical Limit Diagrams
      • Subsection 7.5.6: The Homotopy Colimit as a Derived Functor
      • Subsection 7.5.7: Homotopy Colimit Diagrams
      • Subsection 7.5.8: Categorical Colimit Diagrams
    • Section 7.6: Examples of Limits and Colimits
      • Subsection 7.6.1: Products and Coproducts
      • Subsection 7.6.2: Pullback and Pushout Squares
      • Subsection 7.6.3: Examples of Pullback and Pushout Squares
      • Subsection 7.6.4: Equalizers and Coequalizers
      • Subsection 7.6.5: Sequential Limits and Colimits
      • Subsection 7.6.6: Small Limits
    • Section 7.7: Universality of Colimits
      • Subsection 7.7.1: Cartesian Natural Transformations
      • Subsection 7.7.2: Descent Diagrams
      • Subsection 7.7.3: Cartesian Closed $\infty $-Categories
      • Subsection 7.7.4: Disjoint Coproducts
      • Subsection 7.7.5: The Mather Cube Theorem Revisted
      • Subsection 7.7.6: Descent in the $\infty $-Category of Spaces
  • Chapter 8: The Yoneda Embedding
    • Section 8.1: Twisted Arrows and Cospans
      • Subsection 8.1.1: The Twisted Arrow Construction
      • Subsection 8.1.2: Homotopy Transport for Twisted Arrows
      • Subsection 8.1.3: The Cospan Construction
      • Subsection 8.1.4: Cospans in $\infty $-Categories
      • Subsection 8.1.5: Thin $2$-Simplices of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$
      • Subsection 8.1.6: Restricted Cospans
      • Subsection 8.1.7: Comparing $\operatorname{\mathcal{C}}$ with $\operatorname{Cospan}(\operatorname{\mathcal{C}})$
      • Subsection 8.1.8: Morphisms in the Duskin Nerve
      • Subsection 8.1.9: Cospan Fibrations
      • Subsection 8.1.10: Beck-Chevalley Fibrations
    • Section 8.2: Couplings of $\infty $-Categories
      • Subsection 8.2.1: Representable Couplings
      • Subsection 8.2.2: Morphisms of Couplings
      • Subsection 8.2.3: Representations of Couplings
      • Subsection 8.2.4: Presentations of Representable Couplings
      • Subsection 8.2.5: Adjunctions as Couplings
      • Subsection 8.2.6: Balanced Couplings
    • Section 8.3: The Yoneda Embedding
      • Subsection 8.3.1: Yoneda's Lemma
      • Subsection 8.3.2: Profunctors of $\infty $-Categories
      • Subsection 8.3.3: Hom-Functors for $\infty $-Categories
      • Subsection 8.3.4: Representable Profunctors
      • Subsection 8.3.5: Recognition of Hom-Functors
      • Subsection 8.3.6: Strict Models for Hom-Functors
    • Section 8.4: Cocompletion
      • Subsection 8.4.1: Dense Functors
      • Subsection 8.4.2: Density of Yoneda Embeddings
      • Subsection 8.4.3: Cocompletion via the Yoneda Embedding
      • Subsection 8.4.4: Cocontinuous Extensions
      • Subsection 8.4.5: Adjoining Colimits to $\infty $-Categories
      • Subsection 8.4.6: Recognition of Cocompletions
      • Subsection 8.4.7: Slices of Cocompletions
    • Section 8.5: Retracts and Idempotents
      • Subsection 8.5.1: Retracts in $\infty $-Categories
      • Subsection 8.5.2: Idempotents in Ordinary Categories
      • Subsection 8.5.3: Idempotents in $\infty $-Categories
      • Subsection 8.5.4: Idempotent Completeness
      • Subsection 8.5.5: Idempotent Completion
      • Subsection 8.5.6: Morita Equivalence
      • Subsection 8.5.7: Idempotent Endomorphisms
      • Subsection 8.5.8: Homotopy Idempotent Endomorphisms
      • Subsection 8.5.9: Partial Idempotents
      • Subsection 8.5.10: The Thompson Groupoid
    • Section 8.6: Conjugate and Dual Fibrations
      • Subsection 8.6.1: Conjugate Fibrations
      • Subsection 8.6.2: Existence of Conjugate Fibrations
      • Subsection 8.6.3: Uniqueness of Conjugate Fibrations
      • Subsection 8.6.4: Dual Fibrations
      • Subsection 8.6.5: Existence of Dual Fibrations
      • Subsection 8.6.6: Comparison of Dual and Conjugate Fibrations
      • Subsection 8.6.7: The Opposition Functor
      • Subsection 8.6.8: Contravariant Transport Representations
    • Section 8.7: Fiberwise Cocompletions
      • Subsection 8.7.1: Uniqueness of Fiberwise Cocompletions
      • Subsection 8.7.2: Fiberwise Cocompletions of Cocartesian Fibrations
      • Subsection 8.7.3: Functoriality of Cocompletion
      • Subsection 8.7.4: Fiberwise Cocompletion of Cartesian Fibrations
      • Subsection 8.7.5: Existence of Fiberwise Cocompletions
      • Subsection 8.7.6: Application: Flat Inner Fibrations
      • Subsection 8.7.7: Flatness and Morphism Spaces
      • Subsection 8.7.8: Fiberwise Cocompletion via the Yoneda Embedding
  • Chapter 9: Large $\infty $-Categories
    • Section 9.1: Filtered $\infty $-Categories
      • Subsection 9.1.1: Filtered $\infty $-Categories
      • Subsection 9.1.2: Local Characterization of Filtered $\infty $-Categories
      • Subsection 9.1.3: Fibrations over Filtered $\infty $-Categories
      • Subsection 9.1.4: Digression: Commutation of Limits and Colimits
      • Subsection 9.1.5: Filtered Colimits of Spaces
      • Subsection 9.1.6: Filtered Colimits of Simplicial Sets
      • Subsection 9.1.7: Approximation by Simplicial Subsets
      • Subsection 9.1.8: Approximation by Partially Ordered Sets
      • Subsection 9.1.9: Finitary Functors
      • Subsection 9.1.10: Filtered Colimits of $\infty $-Categories
    • Section 9.2: Compact Objects and Ind-Completion
      • Subsection 9.2.1: Ind-Completion
      • Subsection 9.2.2: Compact Objects
      • Subsection 9.2.3: Recognition of Ind-Completions
      • Subsection 9.2.4: Flat Functors
      • Subsection 9.2.5: Exact Functors
      • Subsection 9.2.6: Compactly Generated $\infty $-Categories
      • Subsection 9.2.7: Finiteness Conditions on Spaces
      • Subsection 9.2.8: Finiteness Conditions on $\infty $-Categories
      • Subsection 9.2.9: Transitivity of $\operatorname{Ind}$-Completion
    • Section 9.3: Local Objects and Factorization Systems
      • Subsection 9.3.1: Digression: Transfinite Composition
      • Subsection 9.3.2: Weakly Local Objects
      • Subsection 9.3.3: The Small Object Argument
      • Subsection 9.3.4: Lifting Problems in $\infty $-Categories
      • Subsection 9.3.5: Weak Factorization Systems
      • Subsection 9.3.6: Orthogonality
      • Subsection 9.3.7: Uniqueness of Factorizations
      • Subsection 9.3.8: Factorization Systems
    • Section 9.4: Truncated Objects of $\infty $-Categories
      • Subsection 9.4.1: Truncated Objects
      • Subsection 9.4.2: Example: Discrete and Subterminal Objects
      • Subsection 9.4.3: Truncated Morphisms
      • Subsection 9.4.4: Monomorphisms
  • 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 Nerve 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