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

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An exact k-linear strong monoidal functor to a nonzero multitensor category is faithful

Statement

Every exact $k$ -linear strong monoidal functor from a tensor category to a multitensor category whose unit is nonzero is faithful.

Facts & Assumptions

Given: An exact $k$ -linear strong monoidal functor $F : C \to D$ , where $C$ is a tensor category, $D$ is a multitensor category, and $1_{D} \neq 0$ .

[F2]

For nonzero $X$ , coevaluation $1 \to X \otimes X^{\lor}$ is monic (Evaluation is epic and coevaluation monic for nonzero objects).

Proof

technique · direct

1.1F2given

If $X \neq 0$ , [F2] and exactness imply that $F (1) \to F (X \otimes X^{\lor})$ is monic. Strong monoidality identifies its source with the nonzero unit of $D$ , so its target is nonzero. Since $F (X \otimes X^{\lor}) ≅ F (X) \otimes F (X^{\lor})$ , this forces $F (X) \neq 0$ .

2.1step 1.1given∎

If $F (f) = 0$ , exactness gives $F (im f) = 0$ . Step 1.1 therefore implies $im f = 0$ , hence $f = 0$ in the abelian source category. Thus $F$ is faithful.

Depends on

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

Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Remark 4.3.10 (standard reference, not scraped)