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2 Higher Category Theory
Structure
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Chapter 6: Adjoint Functors
- Section 6.1: Adjunctions in $2$-Categories
- Section 6.2: Adjoint Functors Between $\infty $-Categories
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Section 6.3: Localization
- Subsection 6.3.1: Localizations of $\infty $-Categories
- Subsection 6.3.2: Existence of Localizations
- Subsection 6.3.3: Reflective Localizations
- Subsection 6.3.4: Stability Properties of Localizations
- Subsection 6.3.5: Fiberwise Localization
- Subsection 6.3.6: Universal Localizations
- Subsection 6.3.7: Subdivision and Localization
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Chapter 7: Limits and Colimits
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Section 7.1: Limits and Colimits
- Subsection 7.1.1: Limits and Colimits in $\infty $-Categories
- Subsection 7.1.2: Limit and Colimit Diagrams
- Subsection 7.1.3: Preservation of Limits and Colimits
- Subsection 7.1.4: Relative Initial and Final Objects
- Subsection 7.1.5: Relative Limits and Colimits
- Subsection 7.1.6: Limits and Colimits of Functors
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Section 7.2: Cofinality
- Subsection 7.2.1: Cofinal Morphisms of Simplicial Sets
- Subsection 7.2.2: Cofinality and Limits
- Subsection 7.2.3: Quillen's Theorem A for $\infty $-Categories
- Subsection 7.2.4: Filtered $\infty $-Categories
- Subsection 7.2.5: Local Characterization of Filtered $\infty $-Categories
- Subsection 7.2.6: Left Fibrations over Filtered $\infty $-Categories
- Subsection 7.2.7: Cofinal Approximation
- Subsection 7.2.8: Sifted Simplicial Sets
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Section 7.3: Kan Extensions
- Subsection 7.3.1: Kan Extensions along General Functors
- Subsection 7.3.2: Kan Extensions along Inclusions
- Subsection 7.3.3: Relative Kan Extensions
- Subsection 7.3.4: Kan Extensions along Fibrations
- Subsection 7.3.5: Existence of Kan Extensions
- Subsection 7.3.6: The Universal Property of Kan Extensions
- Subsection 7.3.7: Kan Extensions in Functor $\infty $-Categories
- Subsection 7.3.8: Transitivity of Kan Extensions
- Subsection 7.3.9: Relative Colimits for Cocartesian Fibrations
- Section 7.4: Limits and Colimits of $\infty $-Categories
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Section 7.5: Homotopy Limits and Colimits
- Subsection 7.5.1: Homotopy Limits of Kan Complexes
- Subsection 7.5.2: Homotopy Limits of $\infty $-Categories
- Subsection 7.5.3: The Homotopy Limit as a Derived Functor
- Subsection 7.5.4: Homotopy Limit Diagrams
- Subsection 7.5.5: Categorical Limit Diagrams
- Subsection 7.5.6: The Homotopy Colimit as a Derived Functor
- Subsection 7.5.7: Homotopy Colimit Diagrams
- Subsection 7.5.8: Categorical Colimit Diagrams
- Subsection 7.5.9: Application: Filtered Colimits of $\infty $-Categories
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Section 7.6: Examples of Limits and Colimits
- Subsection 7.6.1: Products and Coproducts
- Subsection 7.6.2: Powers and Tensors
- Subsection 7.6.3: Pullbacks and Pushouts
- Subsection 7.6.4: Examples of Pullback and Pushout Squares
- Subsection 7.6.5: Equalizers and Coequalizers
- Subsection 7.6.6: Sequential Limits and Colimits
- Subsection 7.6.7: Small Limits
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Section 7.1: Limits and Colimits
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Chapter 8: The Yoneda Embedding
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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: 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
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Section 8.1: Twisted Arrows and Cospans
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Chapter 9: Large $\infty $-Categories
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Section 9.1: Local Objects and Factorization Systems
- Subsection 9.1.1: Local Objects
- Subsection 9.1.2: Digression: Transfinite Composition
- Subsection 9.1.3: Weakly Local Objects
- Subsection 9.1.4: The Small Object Argument
- Subsection 9.1.5: Lifting Problems in $\infty $-Categories
- Subsection 9.1.6: Weak Factorization Systems
- Subsection 9.1.7: Orthogonality
- Subsection 9.1.8: Uniqueness of Factorizations
- Subsection 9.1.9: Factorization Systems
- Section 9.2: Truncated Objects of $\infty $-Categories
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Section 9.3: Fiberwise Cocompletions
- Subsection 9.3.1: Uniqueness of Fiberwise Cocompletions
- Subsection 9.3.2: Fiberwise Cocompletions of Cocartesian Fibrations
- Subsection 9.3.3: Fiberwise Cocompletion of Cartesian Fibrations
- Subsection 9.3.4: Existence of Fiberwise Cocompletions
- Subsection 9.3.5: Digression: Morita Equivalence
- Subsection 9.3.6: Application: Flat Inner Fibrations
- Subsection 9.3.7: Flatness and Morphism Spaces
- Subsection 9.3.8: Fiberwise Cocompletion via the Yoneda Embedding
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Section 9.1: Local Objects and Factorization Systems
- Chapter 10: Exactness and Animation