TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Density theorem for a small category

Statement

Let $C$ be small and let $P : C^{o p} \to S e t$ be a presheaf. Let

$D_{P} : \int P ⟶ [C^{o p}, S e t]$

be the diagram sending an object $(c, x)$ of the category of elements of $P$ to the representable presheaf $y (c)$ (The category of elements of a covariant functor or a presheaf, The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).

Then $P$ is a colimit of $D_{P}$ . Equivalently, for every presheaf $Q$ , cocones from $D_{P}$ to $Q$ are in bijection with natural transformations $P \Rightarrow Q$ .

Facts & Assumptions

Given: A small category $C$ , a presheaf $P : C^{o p} \to S e t$ , and the Yoneda embedding $y$ .

[F1]

The category of elements $\int P$ has objects $(c, x)$ with $x \in P (c)$ , and a morphism $(c, x) \to (d, y)$ is an arrow $f : c \to d$ with $x = P (f) (y)$ (The category of elements of a covariant functor or a presheaf).

[F2]

For small $C$ , the Yoneda embedding is the functor $y : C \to [C^{o p}, S e t]$ sending $c$ to $C (-, c)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small, Small, locally small, and large categories).

[L1]

For a presheaf $Q$ , natural transformations $y (c) = C (-, c) \Rightarrow Q$ are in bijection with elements of $Q (c)$ , naturally in both variables (For a presheaf $P$ , $Nat (C (-, a), P) ≅ P (a)$ naturally in $a$ and $P$ ).

Proof

technique · direct

1.1F1F2L1

For each object $(c, x)$ of $\int P$ , [L1] gives a natural transformation $λ_{(c, x)} : y (c) \Rightarrow P$ corresponding to the element $x \in P (c)$ . If $f : (c, x) \to (d, y)$ in $\int P$ , then $x = P (f) (y)$ by [F1], so naturality in [L1] gives $λ_{(c, x)} = λ_{(d, y)} \circ y (f)$ . Therefore the family $λ_{(c, x)}$ is a cocone from $D_{P}$ to $P$ .

1.2F1L1

Let $Q$ be any presheaf. By [L1], giving a cocone from $D_{P}$ to $Q$ is exactly giving, for each object $(c, x)$ of $\int P$ , an element $q_{c, x} \in Q (c)$ such that whenever $f : (c, x) \to (d, y)$ in $\int P$ , one has $q_{c, x} = Q (f) (q_{d, y})$ . But this is precisely the naturality condition for the assignment $τ_{c} (x) : = q_{c, x}$ to define a natural transformation $τ : P \Rightarrow Q$ . Thus cocones from $D_{P}$ to $Q$ are exactly natural transformations $P \Rightarrow Q$ .

2.1step 1.1step 1.2∎

Under the bijection of step 1.2, the canonical cocone of step 1.1 corresponds to the identity transformation $1_{P}$ . Hence that cocone is universal, and $P$ is the colimit of $D_{P}$ .

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Theorem 6.5.9 (standard reference, not scraped)
T. Leinster, Basic Category Theory, Theorem 6.2.17 (standard reference, not scraped)