TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Images commute with tensor products in a multitensor category

Statement

For morphisms $f : X \to X^{'}$ and $g : Y \to Y^{'}$ in a multitensor category, the canonical map $im (f) \otimes im (g) \to im (f \otimes g)$ is an isomorphism.

Facts & Assumptions

Given: Morphisms $f : X \to X^{'}$ and $g : Y \to Y^{'}$ .

[F1]

Tensoring in either variable is exact (Tensor product in a multitensor category is biexact).

[F2]

An image is the kernel of a cokernel (Image and coimage in a category with kernels and cokernels).

Proof

technique · direct

1.1

Factor $f$ and $g$ as an epimorphism followed by a monomorphism through their images, as specified by [F2].

F2given

2.1

Exactness in [F1] preserves those epimorphisms and monomorphisms after tensoring, first in one variable and then in the other. Hence $f \otimes g$ factors as an epimorphism onto $im (f) \otimes im (g)$ followed by a monomorphism.

step 1.1F1

3.1

In an abelian category this epi--mono factorization identifies its middle object with the image. Therefore the displayed canonical map is an isomorphism.

step 2.1F2∎

Depends on

Used by

The unit object of a multitensor category is semisimple Theorem

Dependency tree · two levels

8 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 4.2.8 (standard reference, not scraped)