Constant sheaf

 ( 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 $\backslash underline\{A\}$ or $A\_X$. The constant presheaf with value $A$ is the presheaf that assigns to each open set of $X$ the value $A$, and all of whose restriction maps are the identity map $A\backslash to\; A$. 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 $\backslash textbf\{C\}$ (e.g. when $\backslash textbf\{C\}$ is the category of abelian groups, or commutative rings).

Constant sheaves of appear in particular as coefficients in sheaf cohomology.

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Basics Let $X$ be a topological space, and $A$ a set. The sections of the constant sheaf $\backslash underline\{A\}$ over an open set $U$ may be interpreted as the continuous functions $U\backslash to\; A$, where $A$ is given the discrete topology. If $U$ is connected space, then these locally constant functions are constant. If $f:X\backslash to\backslash \{\backslash text\{pt\}\backslash \}$ is the unique map to the one-point space and $A$ is considered as a sheaf on $\backslash \{\backslash text\{pt\}\backslash \}$, then the inverse image $f^\{-1\}A$ is the constant sheaf $\backslash underline\{A\}$ on $X$. The sheaf space of $\backslash underline\{A\}$ is the projection map $A$ (where $X\backslash times\; A\backslash to\; X$ is given the discrete topology).

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A detailed example Let $X$ be the topological space consisting of two points $p$ and $q$ with the discrete topology. $X$ has four open sets: $\backslash varnothing,\; \backslash \{p\backslash \},\; \backslash \{q\backslash \},\; \backslash \{p,q\backslash \}$. 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 inclusion map (with identity map for $U\backslash subset\; U$). The constant presheaf with value $\backslash textbf\{Z\}$, denoted $F$, is the presheaf where all four sets are $\backslash textbf\{Z\}$, 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, $\backslash varnothing\; =\; \backslash bigcup\backslash nolimits\_\{U\backslash in\backslash \{\backslash \}\}\; U$, and vacuously, any two sections in $F(\backslash varnothing)$ are equal when restricted to any set in the empty family $\backslash \{\backslash \}$. The local identity axiom would therefore imply that any two sections in $F(\backslash varnothing)$ are equal, which is false.

To modify this into a presheaf $G$ that satisfies the local identity axiom, let $G(\backslash varnothing)=0$, a one-element set, and give $G$ the value $\backslash textbf\{Z\}$ 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 $G(\backslash varnothing)=0$ is forced by the local identity axiom.

Now $G$ is a separated presheaf (satisfies local identity), but unlike $F$ it fails the gluing axiom. Indeed, $\backslash \{p,q\backslash \}$ is Connected space, covered by non-intersecting open sets $\backslash \{p\backslash \}$ and $\backslash \{q\backslash \}$. Choose distinct sections $m\backslash neq\; n$ in $\backslash mathbf\; Z$ over $\backslash \{p\backslash \}$ and $\backslash \{q\backslash \}$ respectively. Because $m$ and $n$ restrict to the same element 0 over $\backslash varnothing$, the gluing axiom would guarantee the existence of a unique section $s$ on $G(\backslash \{p,q\backslash \})$ that restricts to $m$ on $\backslash \{p\backslash \}$ and $n$ on $\backslash \{q\backslash \}$; but the restriction maps are the identity, giving $m\; =\; s\; =\; n$, which is false. Intuitively, $G(\backslash \{p,q\backslash \})$ is too small to carry information about both connected components $\backslash \{p\backslash \}$ and $\backslash \{q\backslash \}$.

Modifying further to satisfy the gluing axiom, let

$H(\backslash \{p,q\backslash \})\; =\; \backslash mathrm\{Fun\}(\backslash \{p,q\backslash \},\backslash mathbf\{Z\})\backslash cong\; \backslash Z\backslash times\backslash Z$,

the $\backslash mathbf\; Z$-valued functions on $\backslash \{p,q\backslash \}$, and define the restriction maps of $H$ to be natural restriction of functions to $\backslash \{p\backslash \}$ and $\backslash \{q\backslash \}$, with the zero map restricting to $\backslash varnothing$. Then $H$ is a sheaf, called the constant sheaf on $X$ with value $\backslash textbf\{Z\}$. Since all restriction maps are ring homomorphisms, $H$ is a sheaf of commutative rings.

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See also

• Locally constant sheaf

• Section II.1 of
• Section 2.4.6 of

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