TheoremStatement: Literature-sourcedProof: 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.

Set has all small colimits, realized as a quotient of a set-indexed disjoint union

Statement

Every small diagram $D : J \to S e t$ has a colimit. It is the quotient of the tagged union $S = {(j, x) : x \in D (j)}$ by the least equivalence relation containing

$(j, x) \sim (k, D (u) (x)) (u : j \to k) .$

Facts & Assumptions

Given: A small diagram $D : J \to S e t$ .

[F1]

Smallness makes the object and morphism collections sets, and cocompleteness means existence of all small colimits (Finite, small, and large limits and colimits; complete and cocomplete categories).

[F2]

Sets and functions form $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[F3]

An equivalence relation is reflexive, symmetric, and transitive (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

[L1]

A function on a set factors uniquely through its quotient precisely when it is constant on equivalence classes (Let $\sim$ be an equivalence relation on $A$ with quotient map $π : A \to A / \sim$ , and let $f : A \to B$ . There is a function $g : A / \sim \to B$ with $g \circ π = f$ if and only if $a \sim a^{'}$ implies $f (a) = f (a^{'})$ ; and such a $g$ is then unique).

[F4]

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

Proof

technique · construction

1.1

By [F1], $S$ is a set. Intersecting all equivalence relations on $S$ that contain the displayed pairs gives the least such relation $\sim$ ; let $Q = S / \sim$ .

F1F3

2.1

Define $ρ_{j} : D (j) \to Q$ by $ρ_{j} (x) = [j, x]$ . Each generating relation gives $ρ_{k} D (u) = ρ_{j}$ , so $ρ$ is a cocone.

step 1.1

2.2

For a cocone $ξ_{j} : D (j) \to X$ , define $h : S \to X$ by $h (j, x) = ξ_{j} (x)$ . The cocone equations make $h$ equal on every generating pair, hence on the equivalence relation they generate.

givenstep 1.1

2.3

If $J$ is empty, then $S = Q = \emptyset$ ; the empty set has one function to every set, so the same construction is the initial-set colimit.

F2step 1.1

3.1

By [L1], there is a unique $\overset{ˉ}{h} : Q \to X$ with $\overset{ˉ}{h} [j, x] = ξ_{j} (x)$ , equivalently $\overset{ˉ}{h} ρ_{j} = ξ_{j}$ for every $j$ . Any map with these equations has the same composite with the quotient map and therefore equals $\overset{ˉ}{h}$ .

L1step 2.2

4.1

By [F4], the cocone is colimiting. Since $D$ was arbitrary, $S e t$ is cocomplete.

F1F4step 2.1step 3.1step 2.3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 46 results over 15 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

E. Riehl, Category Theory in Context, Proposition 3.6.1 (standard reference, not scraped)