ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-04

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 dual of a finite-dimensional vector space as a categorical dual

Example

Let $V$ be a finite-dimensional vector space over a field $k$ . Its algebraic dual $V^{*}$ , together with evaluation $f \otimes v \mapsto f (v)$ and coevaluation $1 \mapsto \sum_{i} v_{i} \otimes v_{i}^{*}$ for a basis $(v_{i})$ and dual basis $(v_{i}^{*})$ , is a categorical dual of $V$ .

Facts & Assumptions

Given: A finite-dimensional vector space $V$ .

[L1]

Finite-dimensional vector spaces are rigid with dual object $V^{*}$ (Finite-dimensional vector spaces are rigid).

Verification

technique · direct

1.1

Theorem Finite-dimensional vector spaces are rigid proves that these maps make $V^{*}$ both a left dual and a right dual of $V$ in ${V e c t}_{k}$ .

givenL1

2.1

So the familiar linear-algebra dual object is exactly a categorical dual in the monoidal sense.

step 1.1∎

Depends on

Finite-dimensional vector spaces are rigid

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Example 2.10.12 (standard reference, not scraped)