TheoremStatement: Literature-sourcedProof: 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 free-module functor is left adjoint to the underlying-set functor

Statement

Fix a unital ring $R$ . The assignment $X \mapsto R^{(X)}$ extends to a functor from sets to left $R$ -modules, and it is left adjoint to the underlying-set functor

$R^{(-)} ⊣ U : R - M o d \to S e t .$

The natural bijection sends an $R$ -linear map $T : R^{(X)} \to M$ to the function $x \mapsto T (e_{x})$ .

Facts & Assumptions

Given: A unital ring $R$ , a set $X$ , and a left $R$ -module $M$ .

[F1]

Every function $u : X \to M$ extends uniquely to an $R$ -linear map $\overset{ˉ}{u} : R^{(X)} \to M$ with $\overset{ˉ}{u} (e_{x}) = u (x)$ (Universal property of the free module on a set).

[F2]

Left $R$ -modules and module homomorphisms form the locally small category $R - M o d$ (Left modules over a fixed ring and module homomorphisms form the large locally small category $R - M o d$ ).

[L1]

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

Proof

technique · direct

1.1

For a function $a : X \to Y$ , define $R^{(a)} : R^{(X)} \to R^{(Y)}$ as the unique linear map sending $e_{x}$ to $e_{a (x)}$ .

F1construct

1.2

Restricting a linear map to the standard basis and extending a function by [F1] are inverse operations, naturally in $X$ and $M$ .

F1F2

2.1

Uniqueness in [F1] gives $R^{(1_{X})} = 1$ and $R^{(b a)} = R^{(b)} R^{(a)}$ , so this is a functor and the basis inclusions are natural.

step 1.1F1

3.1

Thus the standard-basis map is a universal arrow from $X$ to $U$ , and [L1] gives the asserted adjunction.

step 1.1step 2.1step 1.2L1

4.1

When $X = \emptyset$ , $R^{(X)}$ is the zero module and [F1] gives the unique map from it to every $R$ -module, so no separate nonempty-basis hypothesis is required.

F1∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 13 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

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)