8 Categories
This chapter contains some elementary material about categories, functors, and natural transformations. Notably, we discuss and explore:
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Categories (Section 8.1).
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The quadruple adjunction $\pi _{0}\dashv {\webleft (-\webright )_{\mathsf{disc}}}\dashv \text{Obj}\dashv {\webleft (-\webright )_{\mathsf{indisc}}}$ between the category of categories and the category of sets (Section 8.2).
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Groupoids, categories in which all morphisms admit inverses (Section 8.3).
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Functors (Section 8.4).
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The conditions one may impose on functors in decreasing order of importance:
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Section 8.5 introduces the foundationally important conditions one may impose on functors, such as faithfulness, conservativity, essential surjectivity, etc.
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Section 8.6 introduces more conditions one may impose on functors that are still important but less omni-present than those of Section 8.5, such as being dominant, being a monomorphism, being pseudomonic, etc.
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Section 8.7 introduces some rather rare or uncommon conditions one may impose on functors that are nevertheless still useful to explicit record in this chapter.
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Natural transformations (Section 8.8).
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The various categorical and 2-categorical structures formed by categories, functors, and natural transformations (Section 8.9).
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Section 8.1: Categories
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Subsection 8.1.1: Foundations
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Definition 8.1.1.1.1: Categories
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Notation 8.1.1.1.2: Further Notation for Morphisms in Categories
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Definition 8.1.1.1.3: Size Conditions on Categories
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Subsection 8.1.2: Examples of Categories
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Subsection 8.1.3: Posetal Categories
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Definition 8.1.3.1.1: Posetal Categories
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Proposition 8.1.3.1.2: Properties of Posetal Categories
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Subsection 8.1.4: Subcategories
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Subsection 8.1.5: Skeletons of Categories
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Definition 8.1.5.1.1: Skeletons of Categories
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Definition 8.1.5.1.2: Skeletal Categories
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Proposition 8.1.5.1.3: Properties of Skeletons of Categories
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Subsection 8.1.6: Precomposition and Postcomposition
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Definition 8.1.6.1.1: Precomposition and Postcomposition Functions
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Proposition 8.1.6.1.2: Properties of Pre/Postcomposition
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Section 8.2: The Quadruple Adjunction With Sets
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Subsection 8.2.1: Statement
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Proposition 8.2.1.1.1: The Quadruple Adjunction Between $\mathsf{Sets}$ and $\mathsf{Cats}$
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Subsection 8.2.2: Connected Components and Connected Categories
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Subsubsection 8.2.2.1: Connected Components of Categories
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Definition 8.2.2.1.1: Connected Components of Categories
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Subsubsection 8.2.2.2: Sets of Connected Components of Categories
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Definition 8.2.2.2.1: Sets of Connected Components of Categories
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Proposition 8.2.2.2.2: Properties of Sets of Connected Components
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Subsubsection 8.2.2.3: Connected Categories
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Subsection 8.2.3: Discrete Categories
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Definition 8.2.3.1.1: Discrete Categories
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Proposition 8.2.3.1.2: Properties of Discrete Categories on Sets
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Subsection 8.2.4: Indiscrete Categories
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Definition 8.2.4.1.1: Indiscrete Categories
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Proposition 8.2.4.1.2: Properties of Indiscrete Categories on Sets
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Section 8.3: Groupoids
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Subsection 8.3.1: Foundations
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Definition 8.3.1.1.1: Isomorphisms
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Notation 8.3.1.1.2: The Set of Isomorphisms Between Two Objects in a Category
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Definition 8.3.1.1.3: Groupoids
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Subsection 8.3.2: The Groupoid Completion of a Category
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Definition 8.3.2.1.1: The Groupoid Completion of a Category
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Construction 8.3.2.1.2: Construction of the Groupoid Completion of a Category
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Proposition 8.3.2.1.3: Properties of Groupoid Completion
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Subsection 8.3.3: The Core of a Category
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Definition 8.3.3.1.1: The Core of a Category
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Notation 8.3.3.1.2: Alternative Notation for the Core of a Category
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Construction 8.3.3.1.3: Construction of the Core of a Category
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Proposition 8.3.3.1.4: Properties of the Core of a Category
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Section 8.4: Functors
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Subsection 8.4.1: Foundations
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Definition 8.4.1.1.1: Functors
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Notation 8.4.1.1.2: Subscript and Superscript Notation for Functors
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Notation 8.4.1.1.3: Additional Notation for Functors
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Example 8.4.1.1.4: Identity Functors
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Definition 8.4.1.1.5: Composition of Functors
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Proposition 8.4.1.1.6: Elementary Properties of Functors
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Subsection 8.4.2: Contravariant Functors
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Subsection 8.4.3: Forgetful Functors
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Subsection 8.4.4: The Natural Transformation Associated to a Functor
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Definition 8.4.4.1.1: The Natural Transformation Associated to a Functor
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Proposition 8.4.4.1.2: Properties of Natural Transformations Associated to Functors
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Section 8.5: Conditions on Functors
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Subsection 8.5.1: Faithful Functors
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Definition 8.5.1.1.1: Faithful Functors
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Proposition 8.5.1.1.2: Properties of Faithful Functors
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Subsection 8.5.2: Full Functors
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Definition 8.5.2.1.1: Full Functors
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Proposition 8.5.2.1.2: Properties of Full Functors
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Question 8.5.2.1.3: Better Characterisations of Functors With Full Precomposition
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Subsection 8.5.3: Fully Faithful Functors
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Definition 8.5.3.1.1: Fully Faithful Functors
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Proposition 8.5.3.1.2: Properties of Fully Faithful Functors
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Subsection 8.5.4: Conservative Functors
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Definition 8.5.4.1.1: Conservative Functors
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Proposition 8.5.4.1.2: Properties of Conservative Functors
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Question 8.5.4.1.3: Characterisations of Functors With Conservative Pre/Postcomposition
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Subsection 8.5.5: Essentially Injective Functors
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Definition 8.5.5.1.1: Essentially Injective Functors
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Question 8.5.5.1.2: Characterisations of Functors With Essentially Injective Pre/Postcomposition
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Subsection 8.5.6: Essentially Surjective Functors
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Definition 8.5.6.1.1: Essentially Surjective Functors
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Question 8.5.6.1.2: Characterisations of Functors With Essentially Surjective Pre/Postcomposition
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Subsection 8.5.7: Equivalences of Categories
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Definition 8.5.7.1.1: Equivalences of Categories
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Proposition 8.5.7.1.2: Properties of Equivalences of Categories
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Subsection 8.5.8: Isomorphisms of Categories
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Definition 8.5.8.1.1: Isomorphisms of Categories
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Example 8.5.8.1.2: Equivalent But Non-Isomorphic Categories
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Proposition 8.5.8.1.3: Properties of Isomorphisms of Categories
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Section 8.6: More Conditions on Functors
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Subsection 8.6.1: Dominant Functors
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Definition 8.6.1.1.1: Dominant Functors
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Proposition 8.6.1.1.2: Properties of Dominant Functors
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Question 8.6.1.1.3: Characterisations of Functors With Dominant Pre/Postcomposition
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Subsection 8.6.2: Monomorphisms of Categories
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Definition 8.6.2.1.1: Monomorphisms of Categories
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Proposition 8.6.2.1.2: Properties of Monomorphisms of Categories
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Question 8.6.2.1.3: Characterisations of Functors With Monic Pre/Postcomposition
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Subsection 8.6.3: Epimorphisms of Categories
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Definition 8.6.3.1.1: Epimorphisms of Categories
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Proposition 8.6.3.1.2: Properties of Epimorphisms of Categories
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Question 8.6.3.1.3: Characterisations of Functors With Epic Pre/Postcomposition
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Subsection 8.6.4: Pseudomonic Functors
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Definition 8.6.4.1.1: Pseudomonic Functors
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Proposition 8.6.4.1.2: Properties of Pseudomonic Functors
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Subsection 8.6.5: Pseudoepic Functors
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Definition 8.6.5.1.1: Pseudoepic Functors
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Proposition 8.6.5.1.2: Properties of Pseudoepic Functors
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Question 8.6.5.1.3: Characterisations of Pseudoepic Functors
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Question 8.6.5.1.4: Must a Pseudomonic and Pseudoepic Functor Be an Equivalence of Categories
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Question 8.6.5.1.5: Characterisations of Functors With Pseudoepic Pre/Postcomposition
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Section 8.7: Even More Conditions on Functors
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Subsection 8.7.1: Injective on Objects Functors
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Definition 8.7.1.1.1: Injective on Objects Functors
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Proposition 8.7.1.1.2: Properties of Injective on Objects Functors
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Subsection 8.7.2: Surjective on Objects Functors
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Definition 8.7.2.1.1: Surjective on Objects Functors
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Subsection 8.7.3: Bijective on Objects Functors
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Definition 8.7.3.1.1: Bijective on Objects Functors
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Subsection 8.7.4: Functors Representably Faithful on Cores
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Subsection 8.7.5: Functors Representably Full on Cores
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Subsection 8.7.6: Functors Representably Fully Faithful on Cores
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Subsection 8.7.7: Functors Corepresentably Faithful on Cores
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Subsection 8.7.8: Functors Corepresentably Full on Cores
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Subsection 8.7.9: Functors Corepresentably Fully Faithful on Cores
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Section 8.8: Natural Transformations
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Subsection 8.8.1: Transformations
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Definition 8.8.1.1.1: Transformations
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Notation 8.8.1.1.2: The Set of Transformations Between Two Functors
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Subsection 8.8.2: Natural Transformations
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Definition 8.8.2.1.1: Natural Transformations
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Remark 8.8.2.1.2: Picturing Natural Transformations in Diagrams
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Notation 8.8.2.1.3: The Set of Natural Transformations Between Two Functors
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Example 8.8.2.1.4: Identity Natural Transformations
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Definition 8.8.2.1.5: Equality of Natural Transformations
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Subsection 8.8.3: Vertical Composition of Natural Transformations
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Definition 8.8.3.1.1: Vertical Composition of Natural Transformations
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Proposition 8.8.3.1.2: Properties of Vertical Composition of Natural Transformations
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Subsection 8.8.4: Horizontal Composition of Natural Transformations
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Definition 8.8.4.1.1: Horizontal Composition of Natural Transformations
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Definition 8.8.4.1.2: Whiskering of Functors With Natural Transformations
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Proposition 8.8.4.1.3: Properties of Horizontal Composition of Natural Transformations
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Subsection 8.8.5: Properties of Natural Transformations
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Proposition 8.8.5.1.1: Natural Transformations as Categorical Homotopies
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Subsection 8.8.6: Natural Isomorphisms
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Definition 8.8.6.1.1: Natural Isomorphisms
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Proposition 8.8.6.1.2: Properties of Natural Isomorphisms
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Section 8.9: Categories of Categories
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Subsection 8.9.1: Functor Categories
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Definition 8.9.1.1.1: Functor Categories
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Proposition 8.9.1.1.2: Properties of Functor Categories
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Subsection 8.9.2: The Category of Categories and Functors
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Definition 8.9.2.1.1: The Category of Categories and Functors
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Proposition 8.9.2.1.2: Properties of the Category $\mathsf{Cats}$
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Subsection 8.9.3: The $2$-Category of Categories, Functors, and Natural Transformations
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Definition 8.9.3.1.1: The $2$-Category of Categories
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Proposition 8.9.3.1.2: Properties of the 2-Category $\mathsf{Cats}_{\mathsf{2}}$
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Subsection 8.9.4: The Category of Groupoids
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Definition 8.9.4.1.1: The Category of Small Groupoids
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Subsection 8.9.5: The $2$-Category of Groupoids
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Definition 8.9.5.1.1: The $2$-Category of Small Groupoids