Constant sheaf explained
associated to a
set
is a
sheaf of sets on
whose
stalks are all equal to
. It is denoted by
or
. The
constant presheaf with value
is the presheaf that assigns to each
open subset of
the value
, and all of whose restriction maps are the identity map
. The constant sheaf associated to
is the sheafification of the constant presheaf associated to
. This sheaf identifies with the sheaf of locally constant
-valued functions on
.
[1] In certain cases, the set
may be replaced with an
object
in some
category
(e.g. when
is the
category of abelian groups, or commutative rings).
Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.
Basics
Let
be a topological space, and
a set. The sections of the constant sheaf
over an open set
may be interpreted as the continuous functions
, where
is given the
discrete topology. If
is
connected, then these locally constant functions are constant. If
is the unique
map to the one-point space and
is considered as a sheaf on
, then the
inverse image
is the constant sheaf
on
. The sheaf space of
is the projection map
(where
is given the discrete topology).
A detailed example
Let
be the topological space consisting of two points
and
with the
discrete topology.
has four open sets:
\varnothing,\{p\},\{q\},\{p,q\}
. The five non-trivial inclusions of the open sets of
are shown in the chart.
A presheaf on
chooses a set for each of the four open sets of
and a restriction map for each of the
inclusions (with identity map for
). The
constant presheaf with value
, denoted
, is the presheaf where all four sets are
, 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. This is because the empty set is covered by the empty family of sets,
\varnothing=cup\nolimitsU\in\{\
} U , and vacuously, any two sections in
are equal when restricted to any set in the empty family
. The local identity axiom would therefore imply that any two sections in
are equal, which is false.
To modify this into a presheaf
that satisfies the local identity axiom, let
, a one-element set, and give
the value
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
is forced by the local identity axiom.
Now
is a separated presheaf (satisfies local identity), but unlike
it fails the gluing axiom. Indeed,
is
disconnected, covered by non-intersecting open sets
and
. Choose distinct sections
in
over
and
respectively. Because
and
restrict to the same element 0 over
, the gluing axiom would guarantee the existence of a unique section
on
that restricts to
on
and
on
; but the restriction maps are the identity, giving
, which is false. Intuitively,
is too small to carry information about both connected components
and
.
Modifying further to satisfy the gluing axiom, let
H(\{p,q\})=Fun(\{p,q\},Z)\cong\Z ⊗ \Z
,
the
-valued functions on
, and define the restriction maps of
to be natural restriction of functions to
and
, with the zero map restricting to
. Then
is a sheaf, called the
constant sheaf on
with value
. Since all restriction maps are ring homomorphisms,
is a sheaf of commutative rings.
See also
References
- Section II.1 of
- Section 2.4.6 of
Notes and References
- Web site: Does the extension by zero sheaf of the constant sheaf have some nice description? . 2022-07-08 . Mathematics Stack Exchange . en.