TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Projective object characterisations

Statement

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

$P$ is projective.
For every short exact sequence $0 \to K \to E \to M \to 0,$ the induced sequence $0 \to A (P, K) \to A (P, E) \to A (P, M) \to 0$ is exact.
Every epimorphism $E ↠ P$ splits.

Facts & Assumptions

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

[L1]

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

[L2]

In an abelian category, the pullback of an epimorphism is an epimorphism (The pullback of an epimorphism is an epimorphism).

[F1]

For every short exact sequence, the functor $A (P, -)$ is left exact; projectivity is exactly the extra surjectivity at the right-hand end.

Proof

technique · direct

1.1

Assume $P$ is projective. Then [L1] gives a lift of every map $P \to M$ across every epimorphism $E ↠ M$ , so the last map in condition 2 is surjective. Together with the left exactness in [F1], this proves condition 2.

L1F1assume-hyp

1.2

Condition 2 clearly implies condition 1, because surjectivity of $A (P, E) \to A (P, M)$ for every short exact sequence is exactly the lifting property [L1].

L1assume-hyp

1.3

If $P$ is projective and $q : E ↠ P$ is epic, apply [L1] to $1_{P} : P \to P$ . A lift $s : P \to E$ with $q s = 1_{P}$ is a section, so $q$ splits.

L1assume-hyp

1.4

Assume condition 3. Given an epimorphism $q : E ↠ M$ and a map $f : P \to M$ , form the pullback of $q$ along $f$ . By [L2], its projection to $P$ is epic, so condition 3 makes it split. Composing such a section with the other pullback leg gives a lift of $f$ across $q$ . Thus $P$ is projective.

L2assume-hypconstruct

2.1

Steps 1.1 and 1.2 prove $1 \Leftrightarrow 2$ , and steps 1.3 and 1.4 prove $1 \Leftrightarrow 3$ . Hence all three conditions are equivalent.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

The abelian category of finite abelian groups has no nonzero projective object Counterexample
A coproduct of projectives is projective and a product of injectives is injective Theorem
A direct summand of a projective is projective Theorem
A projective generator detects isomorphisms Theorem
Injective object characterisations Theorem
Module categories have enough projectives Theorem

Dependency tree · two levels

7 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)