Constant sheaf
Object in mathematical sheaf theory
title: "Constant sheaf" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["sheaf-theory"] description: "Object in mathematical sheaf theory" topic_path: "general/sheaf-theory" source: "https://en.wikipedia.org/wiki/Constant_sheaf" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Object in mathematical sheaf theory ::
In mathematics, the constant sheaf on a topological space X associated to a set A is a sheaf of sets on X whose stalks are all equal to A. It is denoted by \underline{A} or A_X. The constant presheaf with value A is the presheaf that assigns to each open subset of X the value A, and all of whose restriction maps are the identity map A\to A. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A. This sheaf may be identified with the sheaf of locally constant A-valued functions on X.
In certain cases, the set A may be replaced with an object A in some category \textbf{C} (e.g. when \textbf{C} is the category of abelian groups, or commutative rings).
Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
Basics
Let X be a topological space, and A a set. The sections of the constant sheaf \underline{A} over an open set U may be interpreted as the continuous functions U\to A, where A is given the discrete topology. If U is connected, then these locally constant functions are constant. If f:X\to{\text{pt}} is the unique map to the one-point space and A is considered as a sheaf on {\text{pt}}, then the inverse image f^{-1}A is the constant sheaf \underline{A} on X. The sheaf space of \underline{A} is the projection map A (where X\times A\to X is given the discrete topology).
A detailed example
::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/e6/Constantpresheaf.png" caption="Constant presheaf on a two-point discrete space"] ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/b5/2_point_discrete_space.png" caption="Two-point discrete topological space"] ::
Let X be the topological space consisting of two points p and q with the discrete topology. X has four open sets: \varnothing, {p}, {q}, {p,q}. The five non-trivial inclusions of the open sets of X are shown in the chart.
A presheaf on X chooses a set for each of the four open sets of X and a restriction map for each of the inclusions (with identity map for U\subset U). The constant presheaf with value \textbf{Z}, denoted F, is the presheaf where all four sets are \textbf{Z}, the integers, and all restriction maps are the identity. F is a functor on the diagram of inclusions (a presheaf), because it is constant. It satisfies the gluing axiom, but is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets, \varnothing = \bigcup\nolimits_{U\in{}} U , and vacuously, any two sections in F(\varnothing) are equal when restricted to any set in the empty family {} . The local identity axiom would therefore imply that any two sections in F(\varnothing) are equal, which is false.
To modify this into a presheaf G that satisfies the local identity axiom, let G(\varnothing)=0, a one-element set, and give G the value \textbf{Z} on all non-empty sets. For each inclusion of open sets, let the restriction be the unique map to 0 if the smaller set is empty, or the identity map otherwise. Note that G(\varnothing)=0 is forced by the local identity axiom. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bf/Constantsheaf_intermediate_step.png" caption="Intermediate step for the constant sheaf"] ::
Now G is a separated presheaf (satisfies local identity), but unlike F it fails the gluing axiom. Indeed, {p,q} is disconnected, covered by non-intersecting open sets {p} and {q}. Choose distinct sections m\neq n in \mathbf Z over {p} and {q} respectively. Because m and n restrict to the same element 0 over \varnothing, the gluing axiom would guarantee the existence of a unique section s on G({p,q}) that restricts to m on {p} and n on {q}; but the restriction maps are the identity, giving m = s = n , which is false. Intuitively, G({p,q}) is too small to carry information about both connected components {p} and {q}.
::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/94/Constant_sheaf_with_product.png" caption="Constant sheaf on a two-point topological space"] ::
Modifying further to satisfy the gluing axiom, let H({p,q}) = \mathrm{Fun}({p,q},\mathbf{Z})\cong \Z\times\Z ,the \mathbf Z -valued functions on {p,q}, and define the restriction maps of H to be natural restriction of functions to {p} and {q}, with the zero map restricting to \varnothing . Then H is a sheaf, called the constant sheaf on X with value \textbf{Z}. Since all restriction maps are ring homomorphisms, H is a sheaf of commutative rings.
References
- Section II.1 of
- Section 2.4.6 of {{Citation | last=Tennison | first=B.R. | title=Sheaf theory | isbn=978-0-521-20784-3 | year=1975 | publisher=Cambridge University Press
References
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