$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
8 The Yoneda Embedding
Structure
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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
- Section 8.3: The Yoneda Embedding
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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
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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
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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: Dual Fibrations
- Subsection 8.6.4: Existence of Dual Fibrations
- Subsection 8.6.5: Cocartesian Duality via Cospans
- Subsection 8.6.6: Comparison of Dual and Conjugate Fibrations
- Subsection 8.6.7: The Opposition Functor