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.

Tensoring with a dualizable object preserves projectives

Statement

If $P$ is projective and $X$ is dualizable in a multitensor category, then $P \otimes X$ and $X \otimes P$ are projective.

Facts & Assumptions

Given: A projective object $P$ , a left dual $X^{\lor}$ of $X$ , and a right dual $^{\lor} X$ of $X$ .

[F1]

Tensoring with $X^{\lor}$ or $^{\lor} X$ on either side is exact (Tensor product in a multitensor category is biexact).

[F2]

Projectivity is the lifting property against epimorphisms (Projective object).

[L1]

Tensor--dual adjunction identifies the relevant Hom functors (Duality yields adjunctions of tensoring functors).

Proof

technique · direct

1.1

The left dual gives $Hom (P \otimes X, -) ≅ Hom (P, - \otimes X^{\lor}),$ while the mirrored adjunction for the right dual gives $Hom (X \otimes P, -) ≅ Hom (P,^{\lor} X \otimes -) .$

L1given

2.1

An epimorphism remains an epimorphism after applying either exact tensor functor on the right sides of step 1.1. Then [F2] lifts every map out of $P$ , and transport through the corresponding adjunction proves the lifting property for both stated tensor products.

step 1.1F1F2∎

Depends on

Used by

The unit is projective exactly when the tensor category is semisimple Corollary

Dependency tree · two levels

10 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 4.2.12 (standard reference, not scraped)