ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The colimit of an increasing chain of sets is its union

Example

For inclusions $X_{0} \subseteq X_{1} \subseteq \dots$ , the colimit in $S e t$ is $X = ⋃_{n \geq 0} X_{n}$ with the inclusion maps.

Facts & Assumptions

Given: The increasing chain of sets.

[F1]

A colimit cocone admits a unique map to every other cocone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[L1]

A Set-colimit is a quotient of the tagged union by the identifications induced by diagram arrows (Set has all small colimits, realized as a quotient of a set-indexed disjoint union).

Verification

technique · define the induced map by a containing stage

1.1

The inclusions $i_{n} : X_{n} \to X$ form a cocone. Let $f_{n} : X_{n} \to Y$ be any cocone, so $f_{m} ∣_{X_{n}} = f_{n}$ whenever $n \leq m$ .

given

1.2

For $x \in X$ , choose any $n$ with $x \in X_{n}$ and set $f (x) = f_{n} (x)$ . If $x \in X_{n} \cap X_{m}$ , then at the later stage $r = \max {n, m}$ the cocone equations give $f_{n} (x) = f_{r} (x) = f_{m} (x)$ . Thus $f$ is well-defined.

given

2.1

The equations $f i_{n} = f_{n}$ hold by definition and determine $f$ on every element of the union, so the factor is unique. By [F1], $X$ is the colimit.

F1step 1.2

3.1

In [L1], tagged copies of one element appearing at different stages are identified at a common later stage. Hence its quotient is canonically the same ordinary union, without any assumption that the stages are disjoint.

L1step 1.2∎

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: 27 results over 13 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

T. Leinster, Basic Category Theory, Example 5.2.8 (standard reference, not scraped)