False statementConstruction: 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.

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 · two levels

18 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

E. Riehl, Category Theory in Context, Section 3.6 (standard reference, not scraped)