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.

Every small colimit can be constructed as a coequalizer between coproducts over the arrows and objects of the index category

Statement

Let $D : J \to C$ be small. If the coproducts

$R = ∐_{u : j \to k} D (j), S = ∐_{j} D (j)$

and the coequalizer of $d, c : R ⇉ S$ , where on the $u$ -summand $d ι_{u} = ι_{j}$ and $c ι_{u} = ι_{k} D (u)$ , exist, then that coequalizer is a colimit of $D$ .

Facts & Assumptions

Given: The displayed coproducts and coequalizer.

[L1]

The product-equalizer construction yields every small limit when its constituent limits exist (Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category).

[L2]

Formal duality exchanges limits with colimits, products with coproducts, and equalizers with coequalizers (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

[F1]

Coproducts give unique maps out of families of summands (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

[F2]

A coequalizer gives a unique factor for each arrow that equalizes its pair (Equalizers and coequalizers as limits and colimits of a parallel pair).

Proof

technique · duality

1.1

Apply [L1] to $D^{o p} : J^{o p} \to C^{o p}$ . By [L2], its object-indexed product becomes $S$ , its arrow-indexed product becomes $R$ , and the two coordinate maps become $d$ and $c$ with exactly the displayed summand equations.

L1L2F1

2.1

The equalizer universal property in $C^{o p}$ becomes the coequalizer property [F2] in $C$ . The limiting cone equations become $q ι_{j} = q ι_{k} D (u)$ , and existence and uniqueness of mediating arrows both reverse to the colimit clauses.

L2F2step 1.1

3.1

Thus the coequalizer is a colimit. When $J$ is empty, both coproducts are initial and the construction returns the initial-object colimit, exactly dual to the boundary case in [L1].

L1L2step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 9 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, Theorem 3.5.11 (standard reference, not scraped)