TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ 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.

A coproduct of projectives is projective and a product of injectives is injective

Statement

Assume an abelian category satisfies AB3 and AB3*.

Every finite coproduct of projective objects is projective, and every finite product of injective objects is injective.
For an arbitrary small family, the same conclusion holds provided one may choose one lift or one extension for each index in each lifting problem; in particular it holds under the Axiom of Choice.

Facts & Assumptions

Given: An abelian category satisfying AB3 and AB3*, and small families $(P_{i})$ of projective objects and $(I_{i})$ of injective objects.

[L1]

AB3 and AB3* supply the required coproducts and products (The axioms AB3 and AB3*).

[L2]

Projective objects are characterized by the lifting property against epimorphisms, and injective objects dually by the extension property against monomorphisms (Projective object characterisations, Injective object characterisations).

Proof

technique · direct

1.1

By [L1], let $P = ∐_{i} P_{i}$ . Given an epimorphism $q : E ↠ M$ and a morphism $f : P \to M$ , write $f_{i} = f ι_{i}$ on the coproduct summands. Because each $P_{i}$ is projective, [L2] gives a lift ${\tilde{f}}_{i} : P_{i} \to E$ of $f_{i}$ . For a finite family these lifts are chosen explicitly; for an arbitrary small family they are exactly the stated choice-dependent data. The coproduct universal property then assembles the ${\tilde{f}}_{i}$ into a lift $\tilde{f} : P \to E$ , so $P$ is projective.

L1L2chooseconstruct

1.2

The injective claim is dual. By [L1], let $I = \prod_{i} I_{i}$ . Given a monomorphism $m : M ↣ E$ and a morphism $f : M \to I$ , write $f_{i} = π_{i} f$ . Each injective object $I_{i}$ admits an extension ${\tilde{f}}_{i} : E \to I_{i}$ by [L2]. The product universal property assembles them into $\tilde{f} : E \to I$ extending $f$ . So $I$ is injective.

L1L2chooseconstruct

2.1

Steps 1.1 and 1.2 prove the finite case without extra choice and the arbitrary small-family case with the stated choice boundary.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.6 (standard reference, not scraped)