LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01‡ 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 sufficiently long generator-extension iteration is injective

Statement

Assume the Axiom of Choice. Let $A$ be a locally small Grothendieck category with generator $U$ , and let $(M_{α})$ be the transfinite iteration of the one-step generator extension functor starting at an object $M$ . Let $κ$ bound the cardinalities of the sets of subobjects of all subobjects $N \subseteq U$ . If $λ$ is a limit ordinal with $cf (λ) > κ$ , then $M_{λ}$ is injective.

Facts & Assumptions

Given: The Axiom of Choice, the transfinite tower $(M_{α})$ in a locally small Grothendieck category with generator $U$ , the bound $κ$ from [L2], and a limit ordinal $λ$ with $cf (λ) > κ$ .

[L1]

Extension from subobjects of the fixed generator detects injectivity (Extension from subobjects of a generator detects injectivity).

[L2]

If $cf (λ) > κ$ , every map from a subobject of the generator to $M_{λ}$ factors through an earlier stage, and all transition maps are monic (Transfinite iteration of the generator extension preserves monomorphisms and factorizes small-source maps).

[L3]

Every map from a subobject of the generator into one stage extends across the generator at the next stage (The one-step generator map is a functorial monomorphism).

Proof

technique · direct

1.1

Let $f : N \to M_{λ}$ with $N \subseteq U$ . By [L2], write $f = j_{α, λ} g$ for some $α < λ$ and $g : N \to M_{α}$ . Since $λ$ is a limit ordinal, $α + 1 < λ$ . By [L3], there is $h : U \to M_{α + 1}$ whose restriction to $N$ is $η_{M_{α}} g$ . Compatibility of the transition maps gives $j_{α + 1, λ} h ∣_{N} = j_{α + 1, λ} η_{M_{α}} g = j_{α, λ} g = f,$ so $j_{α + 1, λ} h$ extends $f$ across $N \subseteq U$ .

L2L3givenconstruct

2.1

Thus every map from every subobject of $U$ extends to $U$ . By [L1], this makes $M_{λ}$ injective.

L1step 1.1∎

Depends on

Used by

Grothendieck abelian categories have functorial injective embeddings Theorem

Dependency tree · two levels

12 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

The Stacks Project, Section 19.11: Injectives in Grothendieck categories (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)