DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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A presheaf on a topological space

Definition

Let $X$ be a topological space. A presheaf of sets on $X$ is a contravariant functor $F : Open (X) ⟶ S e t$ (Covariant functor, identity functor, composite functor, and contravariant functor, Sets and functions form the large locally small category $S e t$ ).

Equivalently, a presheaf of sets on $X$ consists of:

a set $F (U)$ for every open set $U \subseteq X$ ;
for every inclusion $V \subseteq U$ , a restriction map $ρ_{V}^{U} : F (U) ⟶ F (V),$

such that $ρ_{U}^{U} = {id}_{F (U)}$ and, whenever $W \subseteq V \subseteq U$ , $ρ_{W}^{U} = ρ_{W}^{V} \circ ρ_{V}^{U} .$

Because the arrows of $Open (X)$ are the inclusions $V \to U$ with $V \subseteq U$ , contravariance is exactly the rule that sections over a larger open set restrict to sections over a smaller one.

Depends on

Used by

Bounded continuous functions need not form a sheaf Counterexample
The constant presheaf need not be a sheaf on a disconnected open set Counterexample
A sheaf on a topological space Definition
Direct image of a sheaf along a continuous map Definition
Inverse image presheaf and inverse image sheaf Definition
Morphisms of presheaves Definition
Presheaves and sheaves of groups, rings, and modules Definition
Sections, restrictions, and global sections of a presheaf Definition
Separated presheaves Definition
Sheaf total quotient rings Definition
Support of a section of a sheaf Definition
Tensor product of abelian sheaves and its total complex Definition
The plus construction for a presheaf Definition
The stalk of a presheaf at a point Definition
The sheaf of all functions to an abelian group is flasque Example
Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
Constant sheaves on irreducible spaces are flasque and acyclic Lemma
Extension by zero and the closed complement: a short exact sequence Lemma
Filtered colimits of sheaves and sections over compact opens Lemma
Finite simplicial constructible sheaves have an actual derived incidence model Lemma
Locally constant functions form a sheaf with constant stalks Lemma
Morphisms from the constant sheaf are global sections Lemma
The constant sheaf is the sheaf of locally constant functions Lemma

Dependency tree · two levels

10 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, Sheaves on Spaces, Definition 3.1 (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, Class 3 (standard reference, not scraped)