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

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Tensor product in a multitensor category is biexact

Statement

In a multitensor category, $X \otimes -$ and $- \otimes X$ are exact for every object $X$ . Hence the tensor product is biexact.

Facts & Assumptions

Given: A multitensor category and an object $X$ .

[F1]

Every object has left and right duals (Tensor and multitensor categories).

[L1]

A dual supplies the relevant adjunctions of tensoring functors (Duality yields adjunctions of tensoring functors).

[L2]

Right adjoints preserve limits and left adjoints preserve colimits (Right adjoints preserve every limit that exists, Left adjoints preserve every colimit that exists).

[L3]

An additive functor between abelian categories is exact iff it preserves kernels and cokernels (An additive functor is exact exactly when it preserves kernels and cokernels).

Proof

technique · direct

1.1F1L1given

By [F1] choose left and right duals of $X$ . By [L1], each of $X \otimes -$ and $- \otimes X$ is both a left and a right adjoint (using the appropriate dual).

1.2L2F1

Thus each tensoring functor preserves kernels and cokernels by [L2]. It is additive because the tensor product is $k$ -bilinear.

2.1step 1.2L3∎

By [L3] both functors are exact. Since $X$ was arbitrary, tensoring is exact in either variable.

Depends on

Used by

Dependency tree · two levels

17 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, Proposition 4.2.1 (standard reference, not scraped)