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9 Large $\infty $-Categories

Structure

  • 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: Cofinal Functors of Filtered $\infty $-Categories
    • 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: Filtered Colimits of $\infty $-Categories
  • Section 9.2: Finitary Functors and Compact Objects
    • Subsection 9.2.1: Filtered Cocompleteness
    • Subsection 9.2.2: Finitary Functors
    • Subsection 9.2.3: Example: Sequential Cocompleteness
    • Subsection 9.2.4: Fiber Products of Filtered $\infty $-Categories
    • Subsection 9.2.5: Compact Objects
    • Subsection 9.2.6: Finiteness Conditions on Kan Complexes
    • Subsection 9.2.7: Finiteness Conditions on $\infty $-Categories
    • Subsection 9.2.8: Closure Properties of Compact Objects
    • Subsection 9.2.9: Compactness in $\infty $-Categories of Sections
  • Section 9.3: Ind-Completions of $\infty $-Categories
    • Subsection 9.3.1: Ind-Completion
    • Subsection 9.3.2: Recognition of Ind-Completions
    • Subsection 9.3.3: Functoriality of Ind-Completion
    • Subsection 9.3.4: Flat Functors
    • Subsection 9.3.5: Exact Functors
    • Subsection 9.3.6: Transitivity of Ind-Completion
    • Subsection 9.3.7: Final Objects of Ind-Completions
  • Section 9.4: Compact Generation and Accessibility
    • Subsection 9.4.1: Compactly Generated $\infty $-Categories
    • Subsection 9.4.2: Compact Functors
    • Subsection 9.4.3: Compact Generation of Oriented Fiber Products
    • Subsection 9.4.4: Compact Generation of Homotopy Fiber Products
    • Subsection 9.4.5: Compactly Generated Fibrations
    • Subsection 9.4.6: Accessible $\infty $-Categories
    • Subsection 9.4.7: Accessible Functors
    • Subsection 9.4.8: Stability Properties of Accessible $\infty $-Categories
  • Section 9.5: Presentable $\infty $-Categories
    • Subsection 9.5.1: Presentable $\infty $-Categories
    • Subsection 9.5.2: Cocontinuous Functors
    • Subsection 9.5.3: Stability Properties of Presentable $\infty $-Categories
    • Subsection 9.5.4: Limits in Presentable $\infty $-Categories
    • Subsection 9.5.5: Representable and Corepresentable Functors
    • Subsection 9.5.6: The Adjoint Functor Theorem
  • Section 9.6: Local Objects and Factorization Systems
    • Subsection 9.6.1: Digression: Transfinite Composition
    • Subsection 9.6.2: Weakly Local Objects
    • Subsection 9.6.3: The Small Object Argument
    • Subsection 9.6.4: Lifting Problems in $\infty $-Categories
    • Subsection 9.6.5: Weak Factorization Systems
    • Subsection 9.6.6: Orthogonality
    • Subsection 9.6.7: Uniqueness of Factorizations
    • Subsection 9.6.8: Factorization Systems
  • Section 9.7: Truncated Objects of $\infty $-Categories
    • Subsection 9.7.1: Truncated Objects
    • Subsection 9.7.2: Example: Discrete and Subterminal Objects
    • Subsection 9.7.3: Truncated Morphisms
    • Subsection 9.7.4: Monomorphisms