Kerodon

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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: Digression: Commutation of Limits and Colimits
    • Subsection 9.1.5: Filtered Colimits of Spaces
    • Subsection 9.1.6: Cofinal Approximation
    • Subsection 9.1.7: Filtered Colimits of Simplicial Sets
  • Section 9.2: Local Objects and Factorization Systems
    • Subsection 9.2.1: Local Objects
    • Subsection 9.2.2: Digression: Transfinite Composition
    • Subsection 9.2.3: Weakly Local Objects
    • Subsection 9.2.4: The Small Object Argument
    • Subsection 9.2.5: Lifting Problems in $\infty $-Categories
    • Subsection 9.2.6: Weak Factorization Systems
    • Subsection 9.2.7: Orthogonality
    • Subsection 9.2.8: Uniqueness of Factorizations
    • Subsection 9.2.9: Factorization Systems
  • Section 9.3: Truncated Objects of $\infty $-Categories
    • Subsection 9.3.1: Truncated Objects
    • Subsection 9.3.2: Example: Discrete and Subterminal Objects
    • Subsection 9.3.3: Truncated Morphisms
    • Subsection 9.3.4: Monomorphisms
  • Section 9.4: Fiberwise Cocompletions
    • Subsection 9.4.1: Uniqueness of Fiberwise Cocompletions
    • Subsection 9.4.2: Fiberwise Cocompletions of Cocartesian Fibrations
    • Subsection 9.4.3: Fiberwise Cocompletion of Cartesian Fibrations
    • Subsection 9.4.4: Existence of Fiberwise Cocompletions
    • Subsection 9.4.5: Digression: Morita Equivalence
    • Subsection 9.4.6: Application: Flat Inner Fibrations
    • Subsection 9.4.7: Flatness and Morphism Spaces
    • Subsection 9.4.8: Fiberwise Cocompletion via the Yoneda Embedding