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.