Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

3 Kan Complexes

Structure

  • Section 3.1: The Homotopy Theory of Kan Complexes
    • Subsection 3.1.1: Kan Fibrations
    • Subsection 3.1.2: Left and Right Fibrations
    • Subsection 3.1.3: Exponentiation of Kan Fibrations
    • Subsection 3.1.4: The $\infty $-Category of Kan Complexes
    • Subsection 3.1.5: Homotopy Equivalences and Weak Homotopy Equivalences
    • Subsection 3.1.6: Anodyne Morphisms
    • 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: The Connecting Homomorphism
    • Subsection 3.2.5: The Long Exact Sequence of a Fibration
    • Subsection 3.2.6: Whitehead's Theorem
    • Subsection 3.2.7: Closure Properties of Homotopy Equivalences
  • Section 3.3: The $\operatorname{Ex}^{\infty }$ Functor
    • Subsection 3.3.1: Subdivision of Simplices
    • Subsection 3.3.2: Digression: Braced Simplicial Sets
    • 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