TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ 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.

An object is projective exactly when Hom out of it is exact

Statement

Let $P$ and $I$ be objects of an abelian category.

$P$ is projective if and only if the functor $A (P, -)$ is exact.
$I$ is injective if and only if the functor $A (-, I)$ is exact.

Facts & Assumptions

Given: An abelian category and objects $P$ and $I$ in it.

[L1]

Hom is left exact in each variable (Hom is left exact in each variable).

[L2]

Projective objects are exactly those for which Hom out of them sends every short exact sequence to a short exact sequence (Projective object, Projective object characterisations).

[L3]

Injective objects are exactly those for which Hom into them sends every short exact sequence to a short exact sequence (Injective object, Injective object characterisations).

[L4]

An exact functor is one that is both left exact and right exact (Exact functor between abelian categories).

Proof

technique · direct

1.1

By [L1], the functor $A (P, -)$ is always left exact. Therefore, by [L4], it is exact exactly when it is also right exact on every short exact sequence. But [L2] says that extra right-end surjectivity is exactly the projective lifting property.

L1L2L4

2.1

The same argument for the contravariant Hom functor uses [L1], [L3], and [L4]: $A (-, I)$ is always left exact, and exactness is equivalent to the additional surjectivity that characterizes injectivity.

L1L3L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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 et al., Tensor Categories, Section 1.6 (standard reference, not scraped)