False statementConstruction: 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.

FALSE: colimits in Grp are computed by taking the Set-colimit of the underlying diagram

Statement refuted

The underlying set of every colimit in $G r p$ is the colimit of the underlying Set-diagram.

Facts & Assumptions

Given: The empty diagram in $G r p$ .

[L1]

$G r p$ has all small colimits (Grp is complete and cocomplete).

[L2]

An empty-diagram colimit is an initial object (Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects).

[F1]

Groups and homomorphisms form $G r p$ , while sets and functions form $S e t$ (Groups and group homomorphisms form the large locally small category $G r p$ , Sets and functions form the large locally small category $S e t$ ).

Refutation

technique · empty diagram

1.1

By [L1] and [L2], the empty-diagram colimit in $G r p$ is its initial object, the trivial group. Its underlying set is a singleton.

L1L2F1

1.2

The underlying diagram is still empty. Its Set-colimit is the initial set $\emptyset$ by [L2], not a singleton.

L2F1

2.1

Thus the underlying-set functor does not preserve even the empty colimit, and the universal statement is false.

step 1.1step 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: 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, Section 3.6 (standard reference, not scraped)