Constant sheaf

In mathematics, the constant sheaf on a topological space

is the presheaf that assigns to each open subset of

is the sheafification of the constant presheaf associated to

is the category of abelian groups, or commutative rings).

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

The sections of the constant sheaf

may be interpreted as the continuous functions

is the unique map to the one-point space and

be the topological space consisting of two points

The five non-trivial inclusions of the open sets of

, the integers, and all restriction maps are the identity.

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.

are equal when restricted to any set in the empty family

The local identity axiom would therefore imply that any two sections in

that satisfies the local identity axiom, let

, a one-element set, and give

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.

is forced by the local identity axiom.

is a separated presheaf (satisfies local identity), but unlike

it fails the gluing axiom.

is disconnected, covered by non-intersecting open sets

Choose distinct sections

, the gluing axiom would guarantee the existence of a unique section

; but the restriction maps are the identity, giving

is too small to carry information about both connected components

Modifying further to satisfy the gluing axiom, let

to be natural restriction of functions to

Since all restriction maps are ring homomorphisms,

is a sheaf of commutative rings.