In mathematics, the constant sheaf on a topological space formula_1 associated to a set formula_2 is a sheaf of sets on formula_1 whose stalks are all equal to formula_2. It is denoted by formula_5 or formula_6. The constant presheaf with value formula_2 is the presheaf that assigns to each open subset of formula_1 the value formula_2, and all of whose restriction maps are the identity map formula_10. The constant sheaf associated to formula_2 is the sheafification of the constant presheaf associated to formula_2. This sheaf identifies with the sheaf of locally constant formula_2-valued functions on formula_1.
In certain cases, the set formula_2 may be replaced with an object formula_2 in some category formula_17 (e.g. when formula_17 is the category of abelian groups, or commutative rings).
Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
Basics.
Let formula_1 be a topological space, and formula_2 a set. The sections of the constant sheaf formula_5 over an open set formula_22 may be interpreted as the continuous functions formula_23, where formula_2 is given the discrete topology. If formula_22 is connected, then these locally constant functions are constant. If formula_26 is the unique map to the one-point space and formula_2 is considered as a sheaf on formula_28, then the inverse image formula_29 is the constant sheaf formula_5 on formula_1. The sheaf space of formula_5 is the projection map formula_2 (where formula_34 is given the discrete topology).
A detailed example.
Let formula_1 be the topological space consisting of two points formula_36 and formula_37 with the discrete topology. formula_1 has four open sets: formula_39. The five non-trivial inclusions of the open sets of formula_1 are shown in the chart.
A presheaf on formula_1 chooses a set for each of the four open sets of formula_1 and a restriction map for each of the inclusions (with identity map for formula_43). The constant presheaf with value formula_44, denoted formula_45, is the presheaf where all four sets are formula_44, the integers, and all restriction maps are the identity. formula_45 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, formula_48, and vacuously, any two sections in formula_49 are equal when restricted to any set in the empty family formula_50. The local identity axiom would therefore imply that any two sections in formula_49 are equal, which is false.
To modify this into a presheaf formula_52 that satisfies the local identity axiom, let formula_53, a one-element set, and give formula_52 the value formula_44 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 formula_53 is forced by the local identity axiom.
Now formula_52 is a separated presheaf (satisfies local identity), but unlike formula_45 it fails the gluing axiom. Indeed, formula_59 is disconnected, covered by non-intersecting open sets formula_60 and formula_61. Choose distinct sections formula_62 in formula_63 over formula_60 and formula_61 respectively. Because formula_66 and formula_67 restrict to the same element 0 over formula_68, the gluing axiom would guarantee the existence of a unique section formula_69 on formula_70 that restricts to formula_66 on formula_60 and formula_67 on formula_61; but the restriction maps are the identity, giving formula_75, which is false. Intuitively, formula_70 is too small to carry information about both connected components formula_60 and formula_61. 
Modifying further to satisfy the gluing axiom, let formula_79,the formula_80-valued functions on formula_59, and define the restriction maps of formula_82 to be natural restriction of functions to formula_60 and formula_61, with the zero map restricting to formula_85. Then formula_82 is a sheaf, called the constant sheaf on formula_1 with value formula_44. Since all restriction maps are ring homomorphisms, formula_82 is a sheaf of commutative rings.
References.
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