PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The localization functor sends quasi isomorphisms to isomorphisms

Statement

For every quasi-isomorphism $s : X \to Y$ , $Q (s)$ is invertible in the derived category, with inverse represented by the roof $Y \overset{s}{\leftarrow} X \overset{1_{X}}{\to} X$ .

Facts & Assumptions

Given: For every quasi-isomorphism $s : X \to Y$ , $Q (s)$ is invertible in the derived category, with inverse represented by the roof $Y \overset{s}{\leftarrow} X \overset{1_{X}}{\to} X$ .

[F1]

The derived category is the roof localization of the homotopy category at quasi-isomorphisms (Derived category of an abelian category).

Proof

1.1F1

The derived category is the localization at quasi-isomorphisms, so the proposed inverse is the allowed roof $(s, 1_{X})$ . This includes $s = 1_{0}$ at the zero complex.

2.1F1algebra∎

Composing the roof with $Q (s)$ gives the identity roof of $X$ in one order and $(s, s)$ in the other. The latter refines $(1_{Y}, 1_{Y})$ via $s$ and $1_{X}$ , so both products are identities.

The localization functor sends quasi isomorphisms to isomorphisms

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources