Item 3 (02J3) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 12: Presheaves and the Yoneda Lemma
Section 12.1: Presheaves
Subsection 12.1.6: Properties of Categories of Presheaves
Item 3 (cite)

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12.1.6 Properties of Categories of Presheaves

Proposition 12.1.6.1.1 ▶ Properties of Categories of Presheaves

Let $\mathcal{C}$ be a category.

1.
Functoriality. The assignment $\mathcal{C}\mapsto \mathsf{PSh}(\mathcal{C})$ defines a functor

\[ \mathsf{PSh}\colon \mathsf{Cats}\to \mathsf{Cats} \]

up to some set-theoretic considerations.¹
2.
Interaction With Slice Categories. Let $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$. We have an equivalence of categories

\[ \mathsf{PSh}(\mathcal{C}_{/X})\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{PSh}(\mathcal{C})_{/h_{X}}. \]

3.
Interaction With Categories of Elements. Let $\mathcal{F}\in \operatorname {\mathrm{Obj}}(\mathsf{PSh}(\mathcal{C}))$. We have an equivalence of categories

\[ \mathsf{PSh}(\textstyle \int _{\mathcal{C}}\mathcal{F})\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{PSh}(\mathcal{C})_{/\mathcal{F}}. \]

¹For instance:
- •
  The $\mathsf{Cats}$ in the source of $\mathsf{PSh}$ could be small categories, and then the $\mathsf{Cats}$ in the right would be locally small categories.
- •
  The $\mathsf{Cats}$ in the source of $\mathsf{PSh}$ could be locally small categories, and then the $\mathsf{Cats}$ on the right would be large categories.
In general, one can systematise and formalise this using Grothendieck universes.

Proof of Proposition 12.1.6.1.1.

Item 1: Functoriality

Omitted.

Item 2: Interaction With Slice Categories

Omitted.

Item 3: Interaction With Categories of Elements

Omitted.

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