Alphabeta MathUniversal Properties, Representables and the Yoneda Lemma
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✓ 13 results · all verified · 0 also independently AI-judged
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2 · Logical flowchart
3 · Items
Definition
The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category
Theorem
The assignments C(a,-) and C(-,a) are functors to Set
Theorem
The hom-assignment C(-,-):CᵒᵖtimesCtoSet is a bifunctor
Definition
Generalized elements and their shapes
Definition
Presheaves, covariantly and contravariantly representable functors, and representations
Proposition
Initial and terminal objects are exactly the representations of the constant singleton functor
Theorem
Initial and terminal objects are unique up to a unique isomorphism
Lemma
Evaluation at the identity gives Nat(C(a,-),F)≅ F(a) and proves that the natural-transformation collection is a set
Theorem
The Yoneda bijection Nat(C(a,-),F)≅ F(a) is natural in both a and F
Corollary
For a presheaf P, Nat(C(-,a),P)≅ P(a) naturally in a and P
Remark
Local smallness does not make every natural-transformation collection a set, but the Yoneda construction proves sethood in the representable-source case
Definition
The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding
Theorem
The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective
Corollary
Objects a and b are isomorphic exactly when C(-,a) and C(-,b) are naturally isomorphic
Definition
Universal elements of covariant functors and presheaves
Theorem
A representation is equivalently a universal element with a unique factorisation property
Theorem
Representing objects are unique up to a unique isomorphism compatible with their universal elements
Definition
The category of elements of a covariant functor or a presheaf
Theorem
Universal elements are initial in a covariant category of elements and terminal in a presheaf category of elements
Definition
Universal arrows from an object to a functor and from a functor to an object
Theorem
Universal arrows to a functor are initial in comma categories, and universal arrows from a functor are terminal
Sources
- Emily Riehl, Category Theory in Context, Chapter 2, Section 2.1
- Tom Leinster, Basic Category Theory, Chapter 4, Section 4.1
- Tom Leinster, Basic Category Theory, Definitions 4.1.1 and 4.1.16
- Tom Leinster, Basic Category Theory, Definition 4.1.22 and Remarks 4.1.23
- Tom Leinster, Basic Category Theory, Definition 4.1.25
- Emily Riehl, Category Theory in Context, Definition 2.1.4
- Tom Leinster, Basic Category Theory, Definitions 4.1.3 and 4.1.17
- Emily Riehl, Category Theory in Context, Definition 2.1.3
- Emily Riehl, Category Theory in Context, Chapters 1 and 2
- Tom Leinster, Basic Category Theory, Chapter 4
- Emily Riehl, Category Theory in Context, Theorem 2.2.4
- Tom Leinster, Basic Category Theory, Theorem 4.2.1
- Emily Riehl, Category Theory in Context, Exercise 2.2.i
- Emily Riehl, Category Theory in Context, Remark 2.2.7
- Emily Riehl, Category Theory in Context, Corollary 2.2.8
- Tom Leinster, Basic Category Theory, Definition 4.1.21
- Tom Leinster, Basic Category Theory, Corollary 4.3.7
- Emily Riehl, Category Theory in Context, Proposition 2.3.1
- Tom Leinster, Basic Category Theory, Corollary 4.3.10
- Emily Riehl, Category Theory in Context, Definition 2.3.3
- Tom Leinster, Basic Category Theory, Corollaries 4.3.2 and 4.3.3
- Emily Riehl, Category Theory in Context, Corollary 2.3.2
- Justin Campbell, Harvard Math 55b tutorial notes, Corollary 1.2.1
- Emily Riehl, Category Theory in Context, Definitions 2.4.1 and 2.4.2
- Tom Leinster, Basic Category Theory, Section 6.2
- Emily Riehl, Category Theory in Context, Proposition 2.4.8
- Emily Riehl, Category Theory in Context, Sections 4.2 and 4.7
- Emily Riehl, Category Theory in Context, Proposition 2.4.8 and Theorem 4.2.7(v)--(vi)