ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

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 · two levels

8 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

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