DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Left roof representing a localized morphism

Definition

For a multiplicative system as in Multiplicative system in a category, a left roof from $X$ to $Y$ is the typed pair $(s, f)$ with $s : U \to X$ in $S$ and $f : U \to Y$ . Its intended localized value is $Q (f) Q (s)^{- 1}$ . Thus the common vertex is the source of both arrows. This is a syntactic presentation; existence of the localized category is a subsequent theorem.

Depends on

Multiplicative system in a category

Used by

Common refinement equivalence of roofs Definition

Dependency tree · two levels

3 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

10.3.1–10.3.14, pp. 379–384 (standard reference, not scraped)