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 A or AX. 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 → A. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A.
In certain cases, the set A may be replaced with an object A in some category C (e.g. when C is the category of abelian groups, or commutative rings).
Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
... The complex ICp(X) is given by starting with the constant sheaf on the open set X−Xn−2 and repeatedly extending it to larger open sets X−Xn−k ... numbers p(k)−n are sometimes written as p(k).) By replacing the constant sheaf on X−Xn−2 with a local system, one can use Deligne's formula to define intersection cohomology with coefficients in a local system ...
... The constant presheaf with value Z, which we will denote F, is the presheaf which chooses all four sets to be Z, the integers, and all restriction maps to be the ... F is a functor, hence a presheaf, because it is constant ... F satisfies the gluing axiom, but it is not a sheaf because it fails the local identity axiom on the empty set ...
... class with fixed extremities of the path, is the identity on constant paths and such that composition of paths corresponds to compositions of morphisms ... In sheaf theory terms, a constant sheaf has locally constant functions as its sections ... Consider instead a sheaf F, such that locally on X it is a constant sheaf ...
Famous quotes containing the word constant:
“The thirsty earth soaks up the rain,
And drinks, and gapes for drink again.
The plants suck in the earth, and are
With constant drinking fresh and fair.”
—Abraham Cowley (16181667)