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.

The one-step generator map is a functorial monomorphism

Statement

In a locally small Grothendieck category, for the one-step generator extension functor $M \mapsto M (M)$ , the canonical map $η_{M} : M \to M (M)$ is a monomorphism, natural in $M$ . In addition, every indexed map $N \to M$ from a subobject $N \subseteq U$ extends to a map $U \to M (M)$ one stage later.

Facts & Assumptions

Given: An object $M$ in a locally small Grothendieck category with fixed generator $U$ .

[L1]

In a locally small Grothendieck category, the one-step generator extension is defined by a pushout over the set of subobjects of $U$ and maps into $M$ (The one-step generator extension functor).

[L2]

Pushouts of monomorphisms are monomorphisms in an abelian category (The pushout of a monomorphism is a monomorphism).

[L3]

AB5 implies AB4, so every small coproduct of monomorphisms in a Grothendieck category is monic (AB5 implies AB4).

Proof

technique · direct

1.1

In the defining pushout square of [L1], the left vertical map is the coproduct of the subobject inclusions $N ↪ U$ , hence is monic by [L3]. Therefore the induced map $η_{M} : M \to M (M)$ is monic by [L2]. For every $g : M \to M^{'}$ , the morphism $M (g)$ supplied by [L1] is the map of pushouts induced by $g$ , so $M (g) η_{M} = η_{M^{'}} g .$ Thus the monomorphisms $η_{M}$ are natural in $M$ .

L1L2L3algebra

2.1

For each $(N, φ) \in S_{M}$ , let $h_{N, φ} : U \to M (M)$ be the lower pushout map restricted to the corresponding summand. Commutativity of the defining square gives $h_{N, φ} i_{N} = η_{M} φ .$ Hence every indexed map $φ : N \to M$ extends across $N ↪ U$ after one application of $M$ .

L1step 1.1construct

3.1

Hence $η_{M}$ is a functorial monomorphism and every indexed generator-subobject map extends after one application of $M$ .

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

11 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)