Kerodon

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8 The Yoneda Embedding

Structure

  • 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: Example: Extensions as Adjoints
    • 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: Idempotent Endomorphisms
    • Subsection 8.5.7: Homotopy Idempotent Endomorphisms
    • Subsection 8.5.8: Partial Idempotents
    • Subsection 8.5.9: 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