DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Direct image of a sheaf along a continuous map

Definition

Let $f : X \to Y$ be a continuous map, and let $F$ be a presheaf on $X$ . The direct image presheaf $f_{*} F$ on $Y$ is defined on an open set $V \subseteq Y$ by

(f_{*} F) (V) : = F (f^{- 1} (V)) .

If $V^{'} \subseteq V$ , the restriction map

(f_{*} F) (V) ⟶ (f_{*} F) (V^{'})

is the restriction map of $F$ for the inclusion $f^{- 1} (V^{'}) \subseteq f^{- 1} (V)$ .

Thus direct image is just precomposition of the presheaf with the inverse-image map on open sets.

Depends on

A presheaf on a topological space

Used by

Dependency tree · two levels

4 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

The Stacks Project, Section 6.21: Continuous maps and sheaves (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Sections 2.2.H and 2.7 (standard reference, not scraped)