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

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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.
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The zig-zag identities in finite-dimensional vector spaces

Example

For a finite-dimensional vector space $V$ with basis $(v_{i})$ and dual basis $(v_{i}^{*})$ , the two zig-zag identities become ordinary Kronecker-delta computations.

Facts & Assumptions

Given: A finite-dimensional vector space $V$ with basis $(v_{i})$ and dual basis $(v_{i}^{*})$ .

[L1]

The standard duality data on $V$ and $V^{*}$ exist (Finite-dimensional vector spaces are rigid).

Verification

technique · direct

1.1

By Finite-dimensional vector spaces are rigid, the categorical dual data are $V^{*}$ , the evaluation pairing, and the coevaluation $1 \mapsto \sum_{i} v_{i} \otimes v_{i}^{*}$ .

givenL1

2.1

On a basis vector $v_{j}$ , the first zig-zag gives $\sum_{i} v_{i}^{*} (v_{j}) v_{i} = \sum_{i} δ_{i j} v_{i} = v_{j}$ . On a dual basis vector $v_{j}^{*}$ , the second zig-zag gives $\sum_{i} v_{j}^{*} (v_{i}) v_{i}^{*} = \sum_{i} δ_{j i} v_{i}^{*} = v_{j}^{*}$ .

step 1.1algebra

3.1

Hence the abstract zig-zag identities of The zig-zag identities reduce here to the familiar basis-and-dual-basis calculation.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Michael Muger, Tensor Categories: A Selective Guided Tour, Section 1.5 (standard reference, not scraped)