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

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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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Injective comparison maps exist

Statement

Assume the Axiom of Dependent Choice.

Let $u : A \to B$ be a morphism, and let $I^{∙}$ and $J^{∙}$ be injective resolutions of $A$ and $B$ . Then there exists a coaugmentation-preserving cochain map $I^{∙} \to J^{∙}$ extending $u$ .

Facts & Assumptions

Given: A morphism $u : A \to B$ and injective resolutions $I^{∙}$ of $A$ and $J^{∙}$ of $B$ .

[L1]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

[L2]

An injective resolution is the cochain datum to be dualized (Injective resolutions in an abelian category).

[L3]

Projective comparison maps exist (Projective comparison maps exist).

Proof

technique · direct

1.1L1L2L3construct

By [L1], pass to the opposite abelian category. Reversing arrows turns the given injective resolutions from [L2] into projective resolutions there, and $u : A \to B$ becomes a morphism in the opposite direction. Apply [L3] in the opposite category to obtain the required comparison map.

2.1step 1.1∎

Translating that chain map back to the original category reverses arrows again and yields a cochain map $I^{∙} \to J^{∙}$ extending $u$ .

Injective comparison maps exist

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources