ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

A representable presheaf on a poset is the indicator of a principal down-set

Example

Let $(P, \leq)$ be a partially ordered set, viewed as a category, and fix $p \in P$ . The representable presheaf $P (-, p)$ has the object values

$P (q, p) = {\begin{cases} {*}, & q \leq p, \\ \emptyset, & q ≰ p . \end{cases}$

Consequently its nonempty support is the principal down-set $↓ p = {q \in P : q \leq p}$ .

Facts & Assumptions

Given: A partially ordered set $(P, \leq)$ and an element $p \in P$ .

[F1]

Every partial order is a preorder, and a preorder is reflexive and transitive (Preorder and monotone map).

[L1]

A preorder determines a category with at most one morphism between any two objects, and functors between such categories are exactly the monotone maps (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps). In that category the morphism $q \to r$ is present exactly when $q \leq r$ , which is how this example reads every hom-set below.

[F2]

The presheaf represented by $p$ is the contravariant hom-functor $P (-, p)$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[F3]

There is exactly one function with empty domain, and a function is determined by its values (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain).

Verification

technique · direct

1.1

By [L1], the hom-set $P (q, p)$ has one element when $q \leq p$ and no elements when $q ≰ p$ , proving the displayed object table.

2.1

If $r \leq q$ , presheaf restriction along the unique arrow $r \to q$ maps the unique member of $P (q, p)$ to the composite $r \to q \to p$ , the unique member of $P (r, p)$ . Whenever the source is empty, there is instead the unique empty function of [F3]. These are all possible restriction maps.

step 1.1F1L1F2F3

2.2

The object value is nonempty exactly when $q \leq p$ , so its support is precisely $↓ p$ . Reflexivity gives $p \in ↓ p$ , and transitivity makes the support downward closed.

step 1.1F1

3.1

Steps 1.1 and 2.1 compute the entire representable presheaf, including every empty value and restriction map, and step 2.2 identifies its support.

step 1.1step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Tom Leinster, Basic Category Theory, Example 1.1.8(e) and Definitions 4.1.16--4.1.17 (standard reference, not scraped)