Editing Sheaf (mathematics) (section)

== Definitions and examples ==<!-- 
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In many mathematical branches, several structures defined on a [[topological space]] <math>X</math> (e.g., a [[differentiable manifold]]) can be naturally ''localised'' or ''restricted'' to [[open set|open]] [[subset]]s <math>U \subseteq X</math>: typical examples include [[continuous function|continuous]] [[real numbers|real]]-valued or [[complex number|complex]]-valued functions, <math>n</math>-times [[differentiable function|differentiable]] (real-valued or complex-valued) functions, [[bounded function|bounded]] real-valued functions, [[vector field]]s, and [[section (fiber bundle)|sections]] of any [[vector bundle]] on the space. The ability to restrict data to smaller open subsets gives rise to the concept of presheaves. Roughly speaking, sheaves are then those presheaves, where local data can be glued to global data.

=== Presheaves ===
{{See also|Presheaf (category theory)}}

Let <math>X</math> be a topological space. A ''presheaf <math>\mathcal 
{F}</math> of sets'' on <math>X</math> consists of the following data:
*For each open set <math>U\subseteq X</math>, there exists a set <math>\mathcal{F}(U)</math>. This set is also denoted <math>\Gamma(U, \mathcal{F})</math>. The elements in this set are called the ''sections'' of <math>\mathcal{F}</math> over <math>U</math>. The sections of <math>\mathcal{F}</math> over <math>X</math> are called the ''global sections'' of <math>\mathcal{F}</math>.
*For each inclusion of open sets <math>V \subseteq U</math>, a function <math>\operatorname{res}^U_V \colon \mathcal{F}(U) \rightarrow \mathcal{F}(V)</math>. In view of many of the examples below, the morphisms <math>\text{res}^U_V</math> are called ''restriction morphisms''. If <math>s \in \mathcal{F}(U)</math>, then its restriction <math>\text{res}^U_V(s)</math> is often denoted <math>s|_V</math> by analogy with restriction of functions.
The restriction morphisms are required to satisfy two additional ([[functorial]]) properties:
*For every open set <math>U</math> of <math>X</math>, the restriction morphism <math>\operatorname{res}^U_U \colon \mathcal{F}(U) \rightarrow \mathcal{F}(U)</math> is the identity morphism  on <math>\mathcal{F}(U)</math>.
*If we have three open sets <math>W \subseteq V \subseteq U</math>, then the [[function composition|composite]] <math>\text{res}^V_W\circ\text{res}^U_V=\text{res}^U_W</math>.
Informally, the second axiom says it does not matter whether we restrict to <math>W</math> in one step or restrict first to <math>V</math>, then to <math>W</math>. A concise functorial reformulation of this definition is given further below.

Many examples of presheaves come from different classes of functions: to any <math>U</math>, one can assign the set <math>C^0(U)</math> of continuous real-valued functions on <math>U</math>. The restriction maps are then just given by restricting a continuous function on <math>U</math> to a smaller open subset <math>V\subseteq U</math>, which again is a continuous function. The two presheaf axioms are immediately checked, thereby giving an example of a presheaf. This can be extended to a presheaf of holomorphic functions <math>\mathcal{H}(-)</math> and a presheaf of smooth functions <math>C^\infty(-)</math>.

Another common class of examples is assigning to <math>U</math> the set of constant real-valued functions on <math>U</math>. This presheaf is called the ''constant presheaf'' associated to <math>\mathbb{R}</math> and is denoted <math>\underline{\mathbb{R}}^{\text{psh}}</math>.

=== Sheaves ===
Given a presheaf, a natural question to ask is to what extent its sections over an open set ''<math>U</math>'' are specified by their restrictions to open subsets of <math>U</math>. A ''sheaf'' is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions.

Axiomatically, a ''sheaf'' is a presheaf that satisfies both of the following axioms:
# (''Locality'') Suppose <math>U</math> is an open set, <math>\{ U_i \}_{i \in I}</math> is an open cover of <math>U</math> with <math> U_i \subseteq U</math> for all <math> i \in I</math>, and <math>s, t \in \mathcal{F}(U)</math> are sections. If <math>s|_{ U_i} = t|_{ U_i}</math> for all <math>i \in I</math>, then <math>s = t</math>.
# ([[Gluing axiom|''Gluing'']]) Suppose <math>U</math> is an open set, <math>\{ U_i \}_{i \in I}</math> is an open cover of <math>U</math> with <math> U_i \subseteq U</math> for all <math> i \in I</math>, and <math>\{ s_i \in \mathcal{F}(U_i) \}_{i \in I}</math> is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if <math>s_i|_{U_i\cap U_j} = s_j|_{U_i \cap U_j}</math> for all <math>i, j \in I</math>, then there exists a section <math>s \in \mathcal{F}(U)</math> such that <math>s|_{U_i} = s_i</math> for all <math>i \in I</math>.<ref>{{Citation|title = The Geometry of Schemes | last1 = Eisenbud | last2 = Harris | first1 = David | first2 = Joe | publisher = Springer | location = New York, NY | isbn = 978-0-387-22639-2 | series = [[Graduate_Texts_in_Mathematics|GTM]] | date = 6 April 2006 | pages = 11–18}}</ref>

In both of these axioms, the hypothesis on the open cover is equivalent to the assumption that <math display="inline">\bigcup_{i \in I} U_i = U</math>.

The section ''<math>s</math>'' whose existence is guaranteed by axiom 2 is called the ''gluing'', ''concatenation'', or ''collation'' of the sections <math>s_i</math>. By axiom 1 it is unique. Sections ''<math>s_i</math>'' and ''<math>s_j</math>'' satisfying the agreement precondition of axiom 2 are often called ''compatible'' ; thus axioms 1 and 2 together state that ''any collection of pairwise compatible sections can be uniquely glued together''. A ''separated presheaf'', or ''monopresheaf'', is a presheaf satisfying axiom 1.<ref>{{Citation | last1=Tennison | first1=B. R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390  | year=1975}}</ref>

The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions <math>f_i : U_i \to \R</math> which agree on the intersections <math>U_i \cap U_j</math>, there is a unique continuous function <math>f: U \to \R</math> whose restriction equals the <math>f_i</math>. By contrast, the constant presheaf is usually ''not'' a sheaf as it fails to satisfy the locality axiom on the empty set (this is explained in more detail at [[constant sheaf]]).

Presheaves and sheaves are typically denoted by capital letters, <math>F</math> being particularly common, presumably for the [[French language|French]] word for sheaf, ''faisceau''. Use of calligraphic letters such as <math>\mathcal{F}</math> is also common.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a [[basis (topology)|basis]] for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. This observation is used to construct another example which is crucial in algebraic geometry, namely [[quasi-coherent sheaf|quasi-coherent sheaves]]. Here the topological space in question is the [[spectrum of a ring|spectrum of a commutative ring <math>R</math>]], whose points are the [[prime ideal]]s <math>\mathfrak{p}</math> in <math>R</math>. The open sets <math>D_f := \{ \mathfrak{p} \subseteq R, f \notin \mathfrak{p}\}</math> form a basis for the [[Zariski topology]] on this space. Given an <math>R</math>-module <math>M</math>, there is a sheaf, denoted by <math>\tilde M</math> on the <math>\operatorname{Spec}R</math>, that satisfies
:<math>\tilde M(D_f) := M[1/f],</math> the [[localization (commutative algebra)|localization]] of <math>M</math> at <math>f</math>.

There is another characterization of sheaves that is equivalent to the previously discussed.
A presheaf <math>\mathcal{F}</math> is a sheaf if and only if for any open <math>U</math> and any open cover <math>\{U_a\}</math> of <math>U</math>, <math>\mathcal{F}(U)</math> is the fibre product <math>\mathcal{F}(U)\cong\mathcal{F}(U_a)\times_{\mathcal{F}(U_a\cap U_b)}\mathcal{F}(U_b)</math>. This characterization is useful in construction of sheaves, for example, if <math>\mathcal{F},\mathcal{G}</math> are [[sheaf of abelian groups|abelian sheaves]], then the kernel of sheaves morphism <math>\mathcal{F}\to\mathcal{G}</math> is a sheaf, since projective limits commutes with projective limits. On the other hand, the cokernel is not always a sheaf because inductive limit not necessarily commutes with projective limits. One of the way to fix this is to consider Noetherian topological spaces; every open sets are compact so that the cokernel is a sheaf, since finite projective limits commutes with inductive limits.

=== 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.

=== Motivating sheaves from complex analytic spaces and algebraic geometry ===
One of the historical motivations for sheaves have come from studying [[complex manifold]]s,<ref>{{cite web|last=Demailly|first=Jean-Pierre|title=Complex Analytic and Differential Geometry|url=https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf|url-status=live|archive-url=https://web.archive.org/web/20200828212129/https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf|archive-date=28 August 2020}}</ref> [[complex analytic geometry]],<ref>{{cite web|last=Cartan|first=Henri|title=Variétés analytiques complexes et cohomologie|url=http://www.inp.nsk.su/~silagadz/Cartan.pdf|url-status=live|archive-url=https://web.archive.org/web/20201008164857/http://www.inp.nsk.su/~silagadz/Cartan.pdf|archive-date=8 October 2020}}</ref> and [[Scheme (mathematics)|scheme theory]] from [[algebraic geometry]]. This is because in all of the previous cases, we consider a topological space <math>X</math> together with a structure sheaf <math>\mathcal{O}</math> giving it the structure of a complex manifold, complex analytic space, or scheme. This perspective of equipping a topological space with a sheaf is essential to the theory of locally ringed spaces (see below).

==== Technical challenges with complex manifolds ====
One of the main historical motivations for introducing sheaves was constructing a device which keeps track of [[holomorphic function]]s on [[complex manifold]]s. For example, on a [[Compact space|compact]] complex manifold <math>X</math> (like [[complex projective space]] or the [[vanishing locus]] in projective space of a [[homogeneous polynomial]]), the ''only'' holomorphic functions<blockquote><math>f:X \to \C</math></blockquote>are the constant functions.<ref name="stackexchange1">{{cite web|title=differential geometry - Holomorphic functions on a complex compact manifold are only constants|url=https://math.stackexchange.com/questions/881742/holomorphic-functions-on-a-complex-compact-manifold-are-only-constants|access-date=2020-10-07|website=Mathematics Stack Exchange}}</ref><ref>{{cite journal |doi=10.2307/1969438|jstor=1969438 |last1=Hawley |first1=Newton S. |title=A Theorem on Compact Complex Manifolds |journal=[[Annals of Mathematics]] |year=1950 |volume=52 |issue=3 |pages=637–641 }}</ref> This means there exist two compact complex manifolds <math>X,X'</math> which are not isomorphic, but nevertheless their rings of global holomorphic functions, denoted <math>\mathcal{H}(X), \mathcal{H}(X')</math>, are isomorphic. Contrast this with [[smooth manifold]]s where every manifold <math>M</math> can be embedded inside some <math>\R^n</math>, hence its ring of smooth functions <math>C^\infty(M)</math> comes from restricting the smooth functions from <math>C^\infty(\R^n)</math>, of which there exist plenty. 

Another complexity when considering the ring of holomorphic functions on a complex manifold <math>X</math> is given a small enough open set <math>U \subseteq X</math>, the holomorphic functions will be isomorphic to <math>\mathcal{H}(U) \cong \mathcal{H}(\C^n)</math>. Sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of <math>X</math> on arbitrary open subsets <math>U \subseteq X</math>. This means as <math>U</math> becomes more complex topologically, the ring <math>\mathcal{H}(U)</math> can be expressed from gluing the <math>\mathcal{H}(U_i)</math>. Note that sometimes this sheaf is denoted <math>\mathcal{O}(-)</math> or just <math>\mathcal{O}</math>, or even <math>\mathcal{O}_X</math> when we want to emphasize the space the structure sheaf is associated to.

==== Tracking submanifolds with sheaves ====
Another common example of sheaves can be constructed by considering a complex submanifold <math>Y \hookrightarrow X</math>. There is an associated sheaf <math>\mathcal{O}_Y</math> which takes an open subset <math>U \subseteq X</math> and gives the ring of holomorphic functions on <math>U \cap Y</math>. This kind of formalism was found to be extremely powerful and motivates a lot of [[homological algebra]] such as [[sheaf cohomology]] since an [[intersection theory]] [[Intersection number|can be built using these kinds of sheaves]] from the Serre intersection formula.