CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16‡ 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.

GAFT recovers the published free-group adjunction, and the comma-initial criterion the abelianisation adjunction

Statement

The general adjoint functor theorem applies to the underlying-set functor $U : G r p \to S e t$ using the canonical solution sets of Normal-subgroup quotients of a fixed free group give a canonical solution set for the underlying-set functor on groups. With the published free groups supplied as the initial comma objects, its functor form gives the published free-group adjunction.

Likewise, the published abelianisation arrows supply singleton solution sets and initial comma objects for the inclusion $A b ↪ G r p$ , and assembling that supplied family recovers the published abelianisation adjunction. That second assembly uses only A left adjoint exists exactly when chosen initial objects are supplied in every comma category and not the GAFT functor form, whose hypotheses of completeness, local smallness and continuity are not established here for the abelian inclusion.

Facts & Assumptions

Given: The published universal-arrow data named in the Statement. The completeness, local smallness and continuity that the objectwise theorem needs are not assumed here; they are established in step 1.1 from [L4], [L5] and [L6].

[L1]

Canonical normal-subgroup quotients give a solution set for $U : G r p \to S e t$ (Normal-subgroup quotients of a fixed free group give a canonical solution set for the underlying-set functor on groups).

[L2]

Objectwise GAFT produces initial comma objects, and supplied initial comma objects assemble into a left adjoint (General adjoint functor theorem, objectwise initial-object form, General adjoint functor theorem, data-supplied functor form).

[L3]

Chosen free groups define a left adjoint to $U$ , and abelianisation defines a left adjoint to $A b ↪ G r p$ with its quotient arrows as units (The free-group functor is left adjoint to the underlying-set functor, Abelianisation is left adjoint to the inclusion of abelian groups).

[L4]

The category $G r p$ of groups and group homomorphisms has all small limits and all small colimits (Grp is complete and cocomplete).

[L5]

Groups and group homomorphisms form the large locally small category $G r p$ (Groups and group homomorphisms form the large locally small category $G r p$ ).

[L6]

If $F$ is left adjoint to $G$ and a diagram $D$ has a limit $(L, λ)$ , then $(G L, G λ)$ is a limit of $G D$ ; thus $G$ preserves every limit that exists (Right adjoints preserve every limit that exists).

[L7]

A left adjoint to $G : D \to C$ is supplied exactly by choosing, for every $c \in C$ , an initial object $(F c, η_{c})$ of the comma category $(c ↓ G)$ ; these choices determine $F$ on morphisms and the adjunction uniquely (A left adjoint exists exactly when chosen initial objects are supplied in every comma category).

Proof

technique · direct

1.1

The objectwise theorem in [L2] needs $G r p$ complete and locally small and $U$ continuous. Fact [L4] gives all small limits in $G r p$ , so it is complete, and [L5] gives local smallness. By [L3] the chosen free-group functor is left adjoint to $U$ , so [L6] makes $U$ preserve every limit that exists, in particular every small limit; hence $U$ is continuous.

L2L3L4L5L6

2.1

For the underlying-set functor, step 1.1 discharges those hypotheses and [L1] supplies the solution sets required by objectwise GAFT, while the chosen free-group universal arrows in [L3] supply the initial comma objects. The functor form of [L2] therefore assembles exactly the free-group adjunction.

step 1.1L1L2L3

3.1

For the abelian inclusion, each unit in [L3] is itself a singleton solution set and a supplied initial object of the corresponding comma category. Assembling that supplied family into a left adjoint is exactly [L7], which asks only for an initial object in every comma category and carries no completeness or continuity hypothesis; so the abelianisation adjunction is recovered from the supplied family by [L7]. The functor form of [L2] is not invoked for this branch, since its Statement does require $A$ complete and locally small and $U$ continuous, and those hypotheses are not established here for $A b ↪ G r p$ .

step 2.1L3L7∎

Depends on

Used by

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Sources

E. Riehl, Category Theory in Context, examples 4.7.4 and 4.7.6 (standard reference, not scraped)