TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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).

F1L1given

1.2

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

L2F1

2.1

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

step 1.2L3∎

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)