ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26‡ 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.

A Kan extension computing the free-group functor

Example

Let $1$ be the one-object category, so $[1^{o p}, S e t] ≅ S e t$ . Let $y : 1 \to S e t$ be the Yoneda embedding. Choose a free group $F ({*})$ on the one-element set ${*}$ , and let $F : 1 \to G r p$ send the unique object to that group.

Then the left Kan extension ${Lan}_{y} F : S e t \to G r p$ is the free-group functor.

Facts & Assumptions

Given: The one-object category $1$ , the Yoneda embedding $y$ , and the functor whose value is a chosen free group on ${*}$ .

[L1]

The free-cocompletion theorem makes ${Lan}_{y} F$ left adjoint to $G r p (F -, -)$ (The presheaf category on a small category is the free cocompletion).

[L2]

Choosing a free group on every set gives a free-group functor left adjoint to the underlying-set functor; at ${*}$ its adjunction bijection is $G r p (F ({*}), G) ≅ U (G)$ naturally in $G$ (The free-group functor is left adjoint to the underlying-set functor).

[L3]

Two left adjoints to the same functor are uniquely naturally isomorphic compatibly with their adjunction data (Adjoints are unique up to a unique natural isomorphism compatible with the adjunction data).

[F1]

The presheaf category on $1$ is $S e t$ (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding, Sets and functions form the large locally small category $S e t$ ).

[F2]

Groups and group homomorphisms form the locally small category $G r p$ , and $G r p$ has all small colimits (Groups and group homomorphisms form the large locally small category $G r p$ , Grp is complete and cocomplete).

Verification

technique · direct

1.1

By [F1], the functor ${Lan}_{y} F$ has domain $S e t$ , and [F2] verifies the locally small and cocomplete target hypotheses of [L1]. Hence [L1] makes it left adjoint to the functor $G \mapsto G r p (F ({*}), G)$ .

F1F2L1

2.1

By [L2], the functor $G \mapsto G r p (F ({*}), G)$ is naturally isomorphic to the underlying-set functor on groups. Therefore ${Lan}_{y} F$ is a left adjoint to the underlying-set functor.

L2step 1.1

3.1

The published free-group functor is also a left adjoint to the underlying-set functor by [L2], so [L3] identifies ${Lan}_{y} F$ with that free functor.

L2L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

35 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, 2nd ed., Proposition 6.5.11 (standard reference, not scraped)