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

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A tensor functor is just a strong monoidal functor

Statement

False claim. A tensor functor is just a strong monoidal functor.

Facts & Assumptions

Given: The two conventions.

[F1]

Strong monoidal means that the tensor and unit comparison maps are isomorphisms (Lax, strong, and strict monoidal functors).

[F2]

A tensor functor is also $k$ -linear, exact, and faithful (Tensor functors between tensor categories).

Refutation

technique · direct

1.1F1construct

Over $k = C$ , send a finite-dimensional vector space $V$ to its conjugate vector space $\overline{V}$ and a linear map to the same underlying additive map. The canonical maps $\overline{V} \otimes \overline{W} ≅ \overline{V \otimes W}$ and $C ≅ \overline{C}$ make this a strong monoidal endofunctor of ${V e c t}_{C}^{f d}$ .

2.1step 1.1F2∎

It is not $C$ -linear on hom-spaces: for a nonreal scalar $λ$ , the images of $λ f$ and of $λ F (f)$ differ. Hence it fails the $k$ -linearity required by [F2], so a strong monoidal functor need not be a tensor functor under this convention.

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, Definition 4.2.5 (standard reference, not scraped)