TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

The free-group functor is left adjoint to the underlying-set functor

Statement

Choosing a free group $(F (X), i_{X})$ on every set $X$ defines a functor $F : S e t \to G r p$ , and

$F ⊣ U : G r p \to S e t,$

where $U$ is the underlying-set functor. The adjunction bijection sends a group homomorphism $φ : F (X) \to G$ to the function $U (φ) i_{X} : X \to U (G)$ .

Facts & Assumptions

Given: A chosen free group $(F (X), i_{X})$ for every set $X$ .

[F1]

A free group on $X$ has the property that every function $u : X \to G$ extends uniquely to a homomorphism $\hat{u} : F (X) \to G$ with $\hat{u} i_{X} = u$ (Free group on a set of generators).

[F2]

Two free groups on the same set are uniquely isomorphic by an isomorphism preserving the generator maps (Free groups on the same set are uniquely isomorphic compatibly with their generators).

[F3]

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

[L1]

Chosen objectwise universal arrows assemble uniquely into a left adjoint (Chosen objectwise universal arrows assemble uniquely into a left adjoint).

Proof

technique · direct

1.1F1construct

For a function $a : X \to Y$ , apply [F1] to $i_{Y} a : X \to F (Y)$ and define $F (a)$ as its unique extending homomorphism.

1.2F1F3

For each $G$ , restriction along $i_{X}$ and extension by [F1] are inverse maps between $G r p (F (X), G)$ and $S e t (X, U (G))$ .

2.1step 1.1F1

Uniqueness in [F1] gives $F (1_{X}) = 1_{F (X)}$ and $F (b a) = F (b) F (a)$ , since both sides agree with the relevant generator map. Thus $F$ is a functor and $i : 1_{S e t} \Rightarrow U F$ is natural.

3.1step 1.1step 2.1step 1.2L1

Equivalently, $(F (X), i_{X})$ is a universal arrow from $X$ to $U$ , so [L1] gives $F ⊣ U$ and the asserted natural bijection.

4.1F1F2∎

For $X = \emptyset$ , [F1] says $F (X)$ is initial in $G r p$ , so the same construction applies. By [F2], replacing any chosen free-group model changes $F (X)$ only by the unique generator-preserving isomorphism.

Depends on

Used by

Dependency tree · two levels

14 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

Emily Riehl, Category Theory in Context, 2nd ed., Example 4.1.10 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Example 2.1.3 (standard reference, not scraped)