ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

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A matrix multifusion category with nonsimple unit

Example

For $n > 1$ , finite $n \times n$ matrices of finite-dimensional vector spaces, with matrix multiplication using $\oplus$ and $\otimes_{k}$ , form a multifusion category whose unit is not simple.

Facts & Assumptions

Given: A field $k$ and an integer $n > 1$ .

[F1]

${V e c t}_{k}^{f d}$ is fusion (Finite-dimensional vector spaces form a fusion category).

[F2]

A multifusion category is finite semisimple multitensor category (Fusion and multifusion categories).

Verification

technique · direct

1.1F1givenconstruct

Let objects be $n \times n$ matrices $(V_{i j})$ and set $(V \otimes W)_{i j} = ⨁_{r} V_{i r} \otimes_{k} W_{r j}$ . The matrix $I$ with $k$ on the diagonal and $0$ off it is a unit.

2.1step 1.1F1F2

Entrywise semisimplicity and finite direct sums from [F1] give the finite semisimple rigid structure required in [F2].

3.1step 1.1F2∎

But $I = E_{11} \oplus \dots \oplus E_{n n}$ is a nontrivial direct sum when $n > 1$ . Hence this is multifusion, not fusion.

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

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Example 4.1.3 (standard reference, not scraped)