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

1.1

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.

2.1

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.

F1algebra∎

Depends on

Derived category of an abelian category

Used by

ex-inverting-a-quasi-isomorphism-by-a-reversed-roof.md Example
fs-localization-identifies-a-quasi-isomorphism-with-an-identity-morphism.md False statement
A complex is zero in the derived category exactly when it is acyclic Proposition

Dependency tree · two levels

5 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

13.11.1–13.11.6 (standard reference, not scraped)