TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

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

1.1L1L3

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.

2.1L1L2step 1.1

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 $A$ .

3.1L3step 2.1∎

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

Depends on

Used by

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)