ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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 internal hom of abelian groups

Example

In $A b$ , viewed as $Z - M o d$ , the internal hom from $B$ to $C$ is the abelian group ${Hom}_{Z} (B, C)$ .

Facts & Assumptions

Given: Abelian groups $A, B, C$ .

[L1]

In a right-closed monoidal category, an internal hom is determined by its evaluation-transposition bijection (The internal hom and its evaluation morphism).

[L2]

The Hom-tensor adjunction holds for modules over a commutative ring, hence for abelian groups (Hom-tensor adjunction: ${Hom}_{R} (M \otimes_{R} N, P) ≅ {Hom}_{R} (M, {Hom}_{R} (N, P))$ ).

Verification

technique · direct

1.1givenalgebra

The evaluation morphism is the group homomorphism $ev : {Hom}_{Z} (B, C) \otimes_{Z} B \to C$ , $ϕ \otimes b \mapsto ϕ (b)$ . It is bilinear, so it is well defined.

2.1step 1.1L1L2

By Hom-tensor adjunction: ${Hom}_{R} (M \otimes_{R} N, P) ≅ {Hom}_{R} (M, {Hom}_{R} (N, P))$ , for every abelian group $A$ there is a natural isomorphism $Hom (A \otimes B, C) ≅ Hom (A, Hom (B, C))$ . This is exactly the transposition property from The internal hom and its evaluation morphism.

3.1step 2.1L1∎

Therefore ${Hom}_{Z} (B, C)$ is the internal hom in $A b$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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 4.4.15 (standard reference, not scraped)
Hom-tensor adjunction for modules (standard reference, not scraped)