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.

Left roof representing a localized morphism

Definition

Depends on

Dependency tree · 2 levels

Sources