Editing Sheaf (mathematics) (section)

=== Further examples ===

==== Sheaf of sections of a continuous map ====
Any continuous map <math>f:Y\to X</math> of topological spaces determines a sheaf <math>\Gamma(Y/X)</math> on <math>X</math> by setting
:<math>\Gamma(Y/X)(U) = \{s: U \to Y, f \circ s = \operatorname{id}_U\}.</math>
Any such <math>s</math> is commonly called a [[section (category theory)|section]] of ''<math>f</math>'', and this example is the reason why the elements in <math>\mathcal{F}(U)</math> are generally called sections. This construction is especially important when <math>f</math> is the projection of a [[fiber bundle]] onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the [[trivial bundle]]. 

Another example: the sheaf of sections of
:<math>\C \stackrel{\exp}{\longrightarrow} \C\setminus \{0\}</math>
is the sheaf which assigns to any ''<math>U\subseteq \mathbb{C}\setminus\{0\}</math>'' the set of branches of the [[complex logarithm]] on ''<math>U</math>''.

Given a point <math>x</math> and an abelian group <math>S</math>, the skyscraper sheaf <math>S_x</math> is defined as follows: if <math>U</math> is an open set containing <math>x</math>, then <math>S_x(U)=S</math>. If <math>U</math> does not contain <math>x</math>, then <math>S_x(U)=0</math>, the [[trivial group]]. The restriction maps are either the identity on <math>S</math>, if both open sets contain <math>x</math>, or the zero map otherwise.

==== Sheaves on manifolds ====
On an <math>n</math>-dimensional <math>C^k</math>-manifold <math>M</math>, there are a number of important sheaves, such as the sheaf of <math>j</math>-times continuously differentiable functions <math>\mathcal{O}^j_M</math> (with <math>j \leq k</math>). Its sections on some open <math>U</math> are the <math>C^j</math>-functions <math>U \to \R</math>. For <math>j = k</math>, this sheaf is called the ''structure sheaf'' and is denoted <math>\mathcal{O}_M</math>. The nonzero <math>C^k</math> functions also form a sheaf, denoted <math>\mathcal{O}_X^\times</math>. [[Differential form]]s (of degree <math>p</math>) also form a sheaf <math>\Omega^p_M</math>. In all these examples, the restriction morphisms are given by restricting functions or forms.

The assignment sending <math>U</math> to the compactly supported functions on <math>U</math> is not a sheaf, since there is, in general, no way to preserve this property by passing to a smaller open subset. Instead, this forms a [[cosheaf]], a [[duality (mathematics)|dual]] concept where the restriction maps go in the opposite direction than with sheaves.<ref>{{harvtxt|Bredon|1997|loc=Chapter V, §1}}</ref> However, taking the [[dual vector space|dual]] of these vector spaces does give a sheaf, the sheaf of [[Distribution (mathematics)|distributions]].

==== Presheaves that are not sheaves ====
In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves:
* Let <math>X</math> be the [[discrete two-point space|two-point topological space]] <math>\{x,y\}</math> with the discrete topology. Define a presheaf <math>F</math> as follows: <math display="block">F(\varnothing) = \{\varnothing\},\ F(\{x\}) = \R,\ F(\{y\}) = \R,\ F(\{x, y\}) = \R\times\R\times\R</math>The restriction map <math>F(\{x, y\}) \to F(\{x\})</math> is the projection of <math>\R \times\R\times\R</math> onto its first coordinate, and the restriction map <math>F(\{x, y\}) \to F(\{y\}) </math> is the projection of <math>\R \times\R\times\R</math> onto its second coordinate. <math>F</math> is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over <math>\{x\}</math> and <math>\{y\}</math> determine only two of those numbers. So while we can glue any two sections over <math>\{x\}</math> and <math>\{y\}</math>, we cannot glue them uniquely.
* Let <math>X = \R</math> be the [[real line]], and let <math>F(U)</math> be the set of [[bounded function|bounded]] continuous functions on <math>U</math>. This is not a sheaf because it is not always possible to glue. For example, let <math>U_i</math> be the set of all <math>x</math> such that <math>|x|<i</math>. The identity function <math>f(x)=x</math> is bounded on each <math>U_i</math>. Consequently, we get a section <math>s_i</math> on <math>U_i</math>. However, these sections do not glue, because the function <math>f</math> is not bounded on the real line. Consequently <math>F</math> is a presheaf, but not a sheaf. In fact, <math>F</math> is separated because it is a sub-presheaf of the sheaf of continuous functions.