False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Left adjoints preserve limits

Statement

Every left adjoint preserves all limits that exist.

Facts & Assumptions

Given: The free-group functor $F : S e t \to G r p$ .

[F1]

The functor $F$ is left adjoint to the underlying-set functor $U : G r p \to S e t$ (The free-group functor is left adjoint to the underlying-set functor).

[F2]

The reduced words on $X ⊔ X^{- 1}$ form the free group on $X$ , with each $x \in X$ represented by a one-letter word (Reduced words form the free group on an alphabet).

[F3]

A terminal object admits exactly one morphism from every object, hence exactly one endomorphism (Initial object, terminal object, and zero object).

Refutation

technique · direct

1.1F2algebra

A singleton $1 = {x}$ is terminal in $S e t$ . By [F2], $F (1)$ contains the distinct empty word and one-letter word $x$ , so its identity homomorphism differs from its trivial endomorphism.

2.1step 1.1F1F3∎

Therefore $F (1)$ is not terminal by [F3], although $F$ is a left adjoint by [F1]. The terminal-object limit is not preserved, so the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Tom Leinster, Basic Category Theory, Example 2.1.3 (standard reference, not scraped)