DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Inverse image presheaf and inverse image sheaf

Definition

Let $f : X \to Y$ be a continuous map, and let $G$ be a presheaf on $Y$ . The inverse-image presheaf

f_{p} G

on $X$ is defined by

(f_{p} G) (U) : = \underset{f (U) \subseteq V}{\underset{\to}{l i m}} G (V),

where $V$ runs over the open neighbourhoods of $f (U)$ in $Y$ , ordered by reverse inclusion. If $U^{'} \subseteq U$ , then every neighbourhood of $f (U)$ is a neighbourhood of $f (U^{'})$ , so there is a natural restriction map $(f_{p} G) (U) \to (f_{p} G) (U^{'})$ .

The inverse image sheaf of a sheaf $G$ on $Y$ is the sheafification of this presheaf:

f^{- 1} G : = a (f_{p} G) .

When $U$ is open in $X$ and $V$ is an open set of $Y$ with $f (U) \subseteq V$ , there is a canonical map

G (V) ⟶ (f_{p} G) (U)

into the colimit class represented by $V$ .

Inverse image presheaf and inverse image sheaf

Definition

Depends on

Dependency tree · 2 levels

Sources