Constant sheaf

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the constant sheaf on a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 associated to a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf of sets</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 whose <a href="/facts/Stalk_(sheaf)/qAD2etdW">stalks</a> are all equal to 
 
 
 
 A
 
 
 {\displaystyle A}
 
. It is denoted by 
 
 
 
 
 
 A
 _
 
 
 
 
 {\displaystyle {\underline {A}}}
 
 or 
 
 
 
 
 A
 
 X
 
 
 
 
 {\displaystyle A_{X}}
 
. The constant presheaf with value 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is the <a href="/facts/Presheaf/MM3hcRw4">presheaf</a> that assigns to each <a href="/facts/Open_set/QfTCIqMu">open subset</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 the value 
 
 
 
 A
 
 
 {\displaystyle A}
 
, and all of whose restriction maps are the identity map 
 
 
 
 A
 →
 A
 
 
 {\displaystyle A\to A}
 
. The constant sheaf associated to 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is the <a href="/facts/Sheafification/V3MnSb2f">sheafification</a> of the constant presheaf associated to 
 
 
 
 A
 
 
 {\displaystyle A}
 
. This sheaf identifies with the sheaf of locally constant 
 
 
 
 A
 
 
 {\displaystyle A}
 
-valued functions on 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
In certain cases, the set 
 
 
 
 A
 
 
 {\displaystyle A}
 
 may be replaced with an <a href="/facts/Object_(category_theory)/7WzxUThf">object</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in some <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\textbf {C}}}
 
 (e.g. when 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\textbf {C}}}
 
 is the <a href="/facts/Category_of_abelian_groups/jF3sXRFn">category of abelian groups</a>, or <a href="/facts/Category_of_commutative_rings/pIwY32bc">commutative rings</a>).
Constant sheaves of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> appear in particular as coefficients in <a href="/facts/Sheaf_cohomology/TrcpmD6k">sheaf cohomology</a>.

Constant sheaf open-in-new

Constant sheaf