TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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 unit object of a tensor category is simple

Statement

The unit object of a tensor category is simple.

Facts & Assumptions

Given: A tensor category $C$ .

[F1]

The unit is semisimple in every multitensor category (The unit object of a multitensor category is semisimple).

[F2]

In a tensor category $End (1) ≅ k$ (Tensor and multitensor categories).

Proof

technique · direct

1.1F1given

By [F1], write $1$ as a finite direct sum of simple objects. A decomposition with at least two nonzero summands supplies a nontrivial idempotent projection in $End (1)$ .

2.1step 1.1F2∎

But [F2] identifies this endomorphism algebra with the field $k$ , whose only idempotents are $0$ and $1$ . Thus there is one nonzero summand, and $1$ is simple.

Depends on

Used by

Dependency tree · two levels

11 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, Theorem 4.3.8(i) (standard reference, not scraped)