LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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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.
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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

1.1L1L2L3algebra

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 $M$ .

2.1L1step 1.1construct

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 $M$ .

3.1step 1.1step 2.1∎

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

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)