ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

The unit inserts basis vectors and the counit evaluates formal linear combinations in the free-vector-space adjunction

Example

Let $k$ be a field. In the free-vector-space adjunction $k^{(-)} ⊣ U$ , the unit sends $x$ to the basis vector $e_{x}$ , and the counit

$ε_{V} : k^{(U (V))} ⟶ V$

sends a formal finite sum $\sum_{v \in E} a_{v} e_{v}$ to the actual sum $\sum_{v \in E} a_{v} v$ in $V$ .

Facts & Assumptions

Given: A set $X$ and a $k$ -vector space $V$ .

[L1]

The free-module adjunction sends a function on $X$ to its unique linear extension from $k^{(X)}$ (The free-module functor is left adjoint to the underlying-set functor).

[F1]

Every element of $k^{(X)}$ is a unique finite sum $\sum_{x \in E} a_{x} e_{x}$ , and $x \mapsto e_{x}$ is the standard basis inclusion (The free module on a set and its standard basis).

[F2]

The unit and counit of an adjunction satisfy $(ε F) (F η) = 1_{F}$ and $(U ε) (η U) = 1_{U}$ (Adjunction by unit, counit, and the triangle identities).

Verification

technique · direct

1.1

By [L1], the unit is the standard basis inclusion $x \mapsto e_{x}$ . The counit is the unique linear extension of $1_{U (V)}$ , so [F1] gives $ε_{V} (\sum a_{v} e_{v}) = \sum a_{v} v$ .

L1F1

2.1

On a basis vector $e_{x}$ , the composite $ε_{k^{(X)}} k^{(η_{X})}$ sends $e_{x}$ to $e_{e_{x}}$ and then to $e_{x}$ . Linearity and [F1] show that it is the identity on all of $k^{(X)}$ .

step 1.1F1

2.2

On $v \in V$ , the composite $U (ε_{V}) η_{U (V)}$ sends $v$ to $e_{v}$ and then to $v$ . Hence both triangle identities in [F2] hold.

step 1.1F2

3.1

When $X = \emptyset$ , $k^{(X)}$ is the zero vector space and step 2.1 is the unique linear endomorphism of it. When $V = 0$ , the formula in step 1.1 is the zero map.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 25 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Tom Leinster, Basic Category Theory, Example 2.2.1 (standard reference, not scraped)