TheoremStatement: Literature-sourcedProof: 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.

Injective object characterisations

Statement

For an object $I$ of an abelian category, the following are equivalent:

$I$ is injective.
For every short exact sequence $0 \to K \to E \to M \to 0,$ the induced sequence $0 \to A (M, I) \to A (E, I) \to A (K, I) \to 0$ is exact.
Every monomorphism $I ↣ E$ splits.

Facts & Assumptions

Given: An object $I$ in an abelian category.

[L1]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

[L2]

Projective objects are characterized by exactness of Hom and by splitting of epimorphisms onto them (Projective object characterisations).

[L3]

Injectivity is the dual lifting property (Injective object).

Proof

technique · direct

1.1

By [L1], the opposite category $A^{o p}$ is abelian. In that opposite category, the object $I$ is projective exactly when it is injective in $A$ , because monomorphisms and epimorphisms are exchanged.

L1L3

2.1

Apply the projective characterization [L2] to $I$ inside $A^{o p}$ . The exactness statement there becomes exactness of $A (-, I)$ on short exact sequences in $A$ , and splitting of an epimorphism onto $I$ in the opposite category is splitting of a monomorphism out of $I$ in $A$ .

L1L2step 1.1

3.1

Therefore conditions 1, 2, and 3 are equivalent in $A$ .

L3step 2.1∎

Depends on

Used by

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

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