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

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.

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

1.1F2given

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

2.1step 1.1F1

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.

3.1step 2.1F2∎

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

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)