12 Presheaves and the Yoneda Lemma

This chapter contains some material about presheaves and the Yoneda lemma.

This chapter is under revision. TODO:

1. 1.

   Subsection properties of categories of copresheaves

2. 2.

   Adjointness of tensor product of functors

3. 3.

   Limit of category of elements (instead of colimit)

4. 4.

   Category of elements where objects are natural transformations $\mathcal{F}\Rightarrow h_{X}$ instead of the other way around. Is this related to Isbell duality?

5. 5.

   Motivate the proof of the Yoneda lemma as in Martin’s comment here:

6. 6.

   Add discussion of universal properties

7. 7.

   Add $h_{g\circ f}=h_{g}\circ h_{f}$ to properties of representable natural transformations

• Section 12.1: Presheaves
  • Subsection 12.1.1: Foundations
    • Definition 12.1.1.1.1: Presheaves on a Category
    • Example 12.1.1.1.2: Presheaves on One-Object Categories
    • Definition 12.1.1.1.3: Morphisms of Presheaves
    • Definition 12.1.1.1.4: The Category of Presheaves on a Category
    • Remark 12.1.1.1.5: Unwinding Definition 12.1.1.1.4
  • Subsection 12.1.2: Representable Presheaves
    • Definition 12.1.2.1.1: Representable Presheaves
    • Example 12.1.2.1.2: Representable Presheaves on One-Object Categories
    • Proposition 12.1.2.1.3: Uniqueness of Representing Objects Up to Isomorphism
  • Subsection 12.1.3: Representable Natural Transformations
    • Definition 12.1.3.1.1: Representable Natural Transformations
  • Subsection 12.1.4: The Yoneda Embedding
    • Definition 12.1.4.1.1: The Yoneda Embedding
    • Remark 12.1.4.1.2: On the Usage of ${\text{よ}}$ to Denote the Yoneda Embedding
    • Proposition 12.1.4.1.3: Properties of the Yoneda Embedding
  • Subsection 12.1.5: The Yoneda Lemma
    • Theorem 12.1.5.1.1: The Yoneda Lemma
  • Subsection 12.1.6: Properties of Categories of Presheaves
    • Proposition 12.1.6.1.1: Properties of Categories of Presheaves
• Section 12.2: Copresheaves
  • Subsection 12.2.1: Foundations
    • Definition 12.2.1.1.1: Copresheaves on a Category
    • Example 12.2.1.1.2: Copresheaves on One-Object Categories
    • Definition 12.2.1.1.3: Morphisms of Copresheaves
    • Definition 12.2.1.1.4: The Category of Copresheaves on a Category
    • Remark 12.2.1.1.5: Unwinding Definition 12.2.1.1.4
  • Subsection 12.2.2: Corepresentable Copresheaves
    • Definition 12.2.2.1.1: Corepresentable Copresheaves
    • Example 12.2.2.1.2: Corepresentable Copresheaves on One-Object Categories
    • Proposition 12.2.2.1.3: Uniqueness of Corepresenting Objects Up to Isomorphism
  • Subsection 12.2.3: Corepresentable Natural Transformations
    • Definition 12.2.3.1.1: Corepresentable Natural Transformations
  • Subsection 12.2.4: The Contravariant Yoneda Embedding
    • Definition 12.2.4.1.1: The Contravariant Yoneda Embedding
    • Proposition 12.2.4.1.2: Properties of the Contravariant Yoneda Embedding
  • Subsection 12.2.5: The Contravariant Yoneda Lemma
    • Theorem 12.2.5.1.1: The Contravariant Yoneda Lemma
• Section 12.3: Restricted Yoneda Embeddings and Yoneda Extensions
  • Subsection 12.3.1: Foundations
    • Definition 12.3.1.1.1: The Restricted Yoneda Embedding Associated to a Functor
    • Remark 12.3.1.1.2: Unwinding Definition 12.3.1.1.1
    • Example 12.3.1.1.3: Examples of Restricted Yoneda Embeddings
    • Proposition 12.3.1.1.4: Properties of the Restricted Yoneda Embedding
  • Subsection 12.3.2: The Yoneda Extension Functor
    • Definition 12.3.2.1.1: The Yoneda Extension Functor
    • Example 12.3.2.1.2: Examples of Yoneda Extensions
    • Proposition 12.3.2.1.3: Properties of Yoneda Extensions
• Section 12.4: Functor Tensor Products
  • Subsection 12.4.1: The Tensor Product of Presheaves With Copresheaves
    • Definition 12.4.1.1.1: The Tensor Product of Presheaves With Copresheaves
    • Remark 12.4.1.1.2: Unwinding Definition 12.4.1.1.1
    • Example 12.4.1.1.3: The Tensor Product of Presheaves With Copresheaves on One Object Categories
    • Proposition 12.4.1.1.4: Properties of Tensor Products of Presheaves With Copresheaves
  • Subsection 12.4.2: The Tensor of a Presheaf With a Functor
    • Definition 12.4.2.1.1: The Tensor of a Presheaf With a Functor
    • Remark 12.4.2.1.2: Unwinding Definition 12.4.2.1.1
    • Proposition 12.4.2.1.3: Properties of Tensors of Presheaves With Functors
  • Subsection 12.4.3: The Tensor of a Copresheaf With a Functor
    • Definition 12.4.3.1.1: The Tensor of a Copresheaf With a Functor
    • Remark 12.4.3.1.2: Unwinding Definition 12.4.3.1.1
    • Proposition 12.4.3.1.3: Properties of Tensors of Copresheaves With Functors