ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Density computed for a presheaf on a two-object discrete category

Example

Let $C$ be the discrete category on two objects $0$ and $1$ . Define a presheaf $P : C^{o p} \to S e t$ by

$P (0) = {x}, P (1) = {y, z} .$

Then the category of elements of $P$ has three objects and no non-identity morphisms, and the density theorem identifies $P$ as the coproduct

$P ≅ y (0) ⊔ y (1) ⊔ y (1)$

in $[C^{o p}, S e t]$ .

Facts & Assumptions

Given: The discrete two-object category $C$ and the presheaf $P$ above.

[F1]

The category of elements has objects $(c, u)$ with $u \in P (c)$ ; because $C$ is discrete, it has no non-identity arrows between distinct such objects (The category of elements of a covariant functor or a presheaf).

[F2]

The Yoneda embedding sends $0$ and $1$ to the representables $y (0)$ and $y (1)$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, Sets and functions form the large locally small category $S e t$ ).

[L1]

The density theorem expresses $P$ as the colimit of the diagram indexed by $\int P$ whose values are the representables at those objects (Density theorem for a small category).

Verification

technique · direct

1.1

The category of elements of $P$ has the three objects $(0, x)$ , $(1, y)$ , and $(1, z)$ , and no non-identity morphisms, by [F1].

2.1

Therefore the density diagram of [L1] is the discrete three-object diagram with values $y (0)$ , $y (1)$ , and $y (1)$ , by [F2]. Its colimit is the coproduct $y (0) ⊔ y (1) ⊔ y (1)$ .

F2L1step 1.1

3.1

Applying [L1] to this presheaf gives exactly that coproduct as $P$ .

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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.