Sheaf (mathematics)

This article is about sheaves on topological spaces. For sheaves on a site, see Grothendieck topology and Topos.

Look up sheaf in Wiktionary, the free dictionary.

In mathematics, a sheaf (pl.: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data).

The field of mathematics that studies sheaves is called sheaf theory.

Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets.

There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-modules, which provide applications to the theory of differential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as Grothendieck topology, have provided applications to mathematical logic and to number theory.

Definitions and examples

In many mathematical branches, several structures defined on a topological space ${\displaystyle X}$  (e.g., a differentiable manifold) can be naturally localised or restricted to open subsets ${\displaystyle U\subset X}$ : typical examples include continuous real-valued or complex-valued functions, ${\displaystyle n}$ -times differentiable (real-valued or complex-valued) functions, bounded real-valued functions, vector fields, and 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 ${\displaystyle X}$  be a topological space. A presheaf of sets ${\displaystyle F}$  on ${\displaystyle X}$  consists of the following data:

• For each open set ${\displaystyle U}$  of ${\displaystyle X}$ , a set ${\displaystyle F(U)}$ . This set is also denoted ${\displaystyle \Gamma (U,F)}$ . The elements in this set are called the sections of ${\displaystyle F}$  over ${\displaystyle U}$ . The sections of ${\displaystyle F}$  over ${\displaystyle X}$  are called the global sections of ${\displaystyle F}$ .
• For each inclusion of open sets ${\displaystyle V\subseteq U}$ , a function ${\displaystyle \operatorname {res} _{V,U}\colon F(U)\rightarrow F(V)}$ . In view of many of the examples below, the morphisms ${\displaystyle {\text{res}}_{V,U}}$  are called restriction morphisms. If ${\displaystyle s\in F(U)}$ , then its restriction ${\displaystyle {\text{res}}_{V,U}(s)}$  is often denoted ${\displaystyle s|_{V}}$  by analogy with restriction of functions.

The restriction morphisms are required to satisfy two additional (functorial) properties:

• For every open set ${\displaystyle U}$  of ${\displaystyle X}$ , the restriction morphism ${\displaystyle \operatorname {res} _{U,U}\colon F(U)\rightarrow F(U)}$  is the identity morphism on ${\displaystyle F(U)}$ .
• If we have three open sets ${\displaystyle W\subseteq V\subseteq U}$ , then the composite ${\displaystyle {\text{res}}_{W,V}\circ {\text{res}}_{V,U}={\text{res}}_{W,U}}$ 

Informally, the second axiom says it does not matter whether we restrict to W in one step or restrict first to V, then to W. A concise functorial reformulation of this definition is given further below.

Many examples of presheaves come from different classes of functions: to any ${\displaystyle U}$ , one can assign the set ${\displaystyle C^{0}(U)}$  of continuous real-valued functions on ${\displaystyle U}$ . The restriction maps are then just given by restricting a continuous function on ${\displaystyle U}$  to a smaller open subset ${\displaystyle V}$ , 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 sheaf of holomorphic functions ${\displaystyle {\mathcal {H}}(-)}$  and a sheaf of smooth functions ${\displaystyle C^{\infty }(-)}$ .

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

Sheaves

Given a presheaf, a natural question to ask is to what extent its sections over an open set ${\displaystyle U}$  are specified by their restrictions to open subsets of ${\displaystyle U}$ . 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:

1. (Locality) Suppose ${\displaystyle U}$  is an open set, ${\displaystyle \{U_{i}\}_{i\in I}}$  is an open cover of ${\displaystyle U}$  with ${\displaystyle U_{i}\subseteq U}$  for all ${\displaystyle i\in I}$ , and ${\displaystyle s,t\in F(U)}$  are sections. If ${\displaystyle s|_{U_{i}}=t|_{U_{i}}}$  for all ${\displaystyle i\in I}$ , then ${\displaystyle s=t}$ .
2. (Gluing) Suppose ${\displaystyle U}$  is an open set, ${\displaystyle \{U_{i}\}_{i\in I}}$  is an open cover of ${\displaystyle U}$  with ${\displaystyle U_{i}\subseteq U}$  for all ${\displaystyle i\in I}$ , and ${\displaystyle \{s_{i}\in F(U_{i})\}_{i\in I}}$  is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if ${\displaystyle s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}}$  for all ${\displaystyle i,j\in I}$ , then there exists a section ${\displaystyle s\in F(U)}$  such that ${\displaystyle s|_{U_{i}}=s_{i}}$  for all ${\displaystyle i\in I}$ .^[1]

In both of these axioms, the hypothesis on the open cover is equivalent to the assumption that ${\textstyle \bigcup _{i\in I}U_{i}=U}$ .

The section ${\displaystyle s}$  whose existence is guaranteed by axiom 2 is called the gluing, concatenation, or collation of the sections s_i. By axiom 1 it is unique. Sections ${\displaystyle s_{i}}$  and ${\displaystyle s_{j}}$  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.^[2]

The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions ${\displaystyle f_{i}:U_{i}\to \mathbb {R} }$  which agree on the intersections ${\displaystyle U_{i}\cap U_{j}}$ , there is a unique continuous function ${\displaystyle f:U\to \mathbb {R} }$  whose restriction equals the ${\displaystyle f_{i}}$ . 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, ${\displaystyle F}$  being particularly common, presumably for the French word for sheaf, faisceau. Use of calligraphic letters such as ${\displaystyle {\mathcal {F}}}$  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 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 sheaves. Here the topological space in question is the spectrum of a commutative ring ${\displaystyle R}$ , whose points are the prime ideals ${\displaystyle p}$  in ${\displaystyle R}$ . The open sets ${\displaystyle D_{f}:=\{p\subset R,f\notin p\}}$  form a basis for the Zariski topology on this space. Given an ${\displaystyle R}$ -module ${\displaystyle M}$ , there is a sheaf, denoted by ${\displaystyle {\tilde {M}}}$  on the Spec ${\displaystyle R}$ , that satisfies

${\displaystyle {\tilde {M}}(D_{f}):=M[1/f],}$  the localization of ${\displaystyle M}$  at ${\displaystyle f}$ .

There is another characterization of sheaves that is equivalent to the previously discussed. A presheaf ${\displaystyle {\mathcal {F}}}$  is a sheaf if and only if for any open ${\displaystyle U}$  and any open cover ${\displaystyle (U_{a})}$  of ${\displaystyle U}$ , ${\displaystyle {\mathcal {F}}(U)}$  is the fibre product ${\displaystyle {\mathcal {F}}(U)\cong {\mathcal {F}}(U_{a})\times _{{\mathcal {F}}(U_{a}\cap U_{b})}{\mathcal {F}}(U_{b})}$ . This characterization is useful in construction of sheaves, for example, if ${\displaystyle {\mathcal {F}},{\mathcal {G}}}$  are abelian sheaves, then the kernel of sheaves morphism ${\displaystyle {\mathcal {F}}\to {\mathcal {G}}}$  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 ${\displaystyle f:Y\to X}$  of topological spaces determines a sheaf ${\displaystyle \Gamma (Y/X)}$  on ${\displaystyle X}$  by setting

${\displaystyle \Gamma (Y/X)(U)=\{s:U\to Y,f\circ s=\operatorname {id} _{U}\}.}$ 

Any such ${\displaystyle s}$  is commonly called a section of ${\displaystyle f}$ , and this example is the reason why the elements in ${\displaystyle F(U)}$  are generally called sections. This construction is especially important when ${\displaystyle f}$  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

${\displaystyle \mathbb {C} {\stackrel {\exp }{\to }}\mathbb {C} \setminus \{0\}}$ 

is the sheaf which assigns to any ${\displaystyle U}$  the set of branches of the complex logarithm on ${\displaystyle U}$ .

Given a point ${\displaystyle x}$  and an abelian group ${\displaystyle S}$ , the skyscraper sheaf ${\displaystyle S_{x}}$  is defined as follows: if ${\displaystyle U}$  is an open set containing ${\displaystyle x}$ , then ${\displaystyle S_{x}(U)=S}$ . If ${\displaystyle U}$  does not contain ${\displaystyle x}$ , then ${\displaystyle S_{x}(U)=0}$ , the trivial group. The restriction maps are either the identity on ${\displaystyle S}$ , if both open sets contain ${\displaystyle x}$ , or the zero map otherwise.

Sheaves on manifolds

On an ${\displaystyle n}$ -dimensional ${\displaystyle C^{k}}$ -manifold ${\displaystyle M}$ , there are a number of important sheaves, such as the sheaf of ${\displaystyle j}$ -times continuously differentiable functions ${\displaystyle {\mathcal {O}}_{M}^{j}}$  (with ${\displaystyle j\leq k}$ ). Its sections on some open ${\displaystyle U}$  are the ${\displaystyle C^{j}}$ -functions ${\displaystyle U\to \mathbb {R} }$ . For ${\displaystyle j=k}$ , this sheaf is called the structure sheaf and is denoted ${\displaystyle {\mathcal {O}}_{M}}$ . The nonzero ${\displaystyle C^{k}}$  functions also form a sheaf, denoted ${\displaystyle {\mathcal {O}}_{X}^{\times }}$ . Differential forms (of degree ${\displaystyle p}$ ) also form a sheaf ${\displaystyle \Omega _{M}^{p}}$ . In all these examples, the restriction morphisms are given by restricting functions or forms.

The assignment sending ${\displaystyle U}$  to the compactly supported functions on ${\displaystyle U}$  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 dual concept where the restriction maps go in the opposite direction than with sheaves.^[3] However, taking the dual of these vector spaces does give a sheaf, the sheaf of 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 ${\displaystyle X}$  be the two-point topological space ${\displaystyle \{x,y\}}$  with the discrete topology. Define a presheaf ${\displaystyle F}$  as follows:
  $${\displaystyle F(\varnothing )=\{\varnothing \},\ F(\{x\})=\mathbb {R} ,\ F(\{y\})=\mathbb {R} ,\ F(\{x,y\})=\mathbb {R} \times \mathbb {R} \times \mathbb {R} }$$

   
  The restriction map ${\displaystyle F(\{x,y\})\to F(\{x\})}$  is the projection of ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }$  onto its first coordinate, and the restriction map ${\displaystyle F(\{x,y\})\to F(\{y\})}$  is the projection of ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} }$  onto its second coordinate. ${\displaystyle F}$  is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over ${\displaystyle \{x\}}$  and ${\displaystyle \{y\}}$  determine only two of those numbers. So while we can glue any two sections over ${\displaystyle \{x\}}$  and ${\displaystyle \{y\}}$ , we cannot glue them uniquely.
• Let ${\displaystyle X=\mathbb {R} }$  be the real line, and let ${\displaystyle F(U)}$  be the set of bounded continuous functions on ${\displaystyle U}$ . This is not a sheaf because it is not always possible to glue. For example, let ${\displaystyle U_{i}}$  be the set of all ${\displaystyle x}$  such that ${\displaystyle |x|<i}$ . The identity function ${\displaystyle f(x)=x}$  is bounded on each ${\displaystyle U_{i}}$ . Consequently, we get a section ${\displaystyle s_{i}}$  on ${\displaystyle U_{i}}$ . However, these sections do not glue, because the function ${\displaystyle f}$  is not bounded on the real line. Consequently ${\displaystyle F}$  is a presheaf, but not a sheaf. In fact, ${\displaystyle F}$  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 manifolds,^[4] complex analytic geometry,^[5] and scheme theory from algebraic geometry. This is because in all of the previous cases, we consider a topological space ${\displaystyle X}$  together with a structure sheaf ${\displaystyle {\mathcal {O}}}$  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 functions on complex manifolds. For example, on a compact complex manifold ${\displaystyle X}$  (like complex projective space or the vanishing locus in projective space of a homogeneous polynomial), the only holomorphic functions

${\displaystyle f:X\to \mathbb {C} }$ 

are the constant functions.^[6][7] This means there exist two compact complex manifolds ${\displaystyle X,X'}$  which are not isomorphic, but nevertheless their ring of global holomorphic functions, denoted ${\displaystyle {\mathcal {H}}(X),{\mathcal {H}}(X')}$ , are isomorphic. Contrast this with smooth manifolds where every manifold ${\displaystyle M}$  can be embedded inside some ${\displaystyle \mathbb {R} ^{n}}$ , hence its ring of smooth functions ${\displaystyle C^{\infty }(M)}$  comes from restricting the smooth functions from ${\displaystyle C^{\infty }(\mathbb {R} ^{n})}$ . Another complexity when considering the ring of holomorphic functions on a complex manifold ${\displaystyle X}$  is given a small enough open set ${\displaystyle U\subset X}$ , the holomorphic functions will be isomorphic to ${\displaystyle {\mathcal {H}}(U)\cong {\mathcal {H}}(\mathbb {C} ^{n})}$ . 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 ${\displaystyle X}$  on arbitrary open subsets ${\displaystyle U\subset X}$ . This means as ${\displaystyle U}$  becomes more complex topologically, the ring ${\displaystyle {\mathcal {H}}(U)}$  can be expressed from gluing the ${\displaystyle {\mathcal {H}}(U_{i})}$ . Note that sometimes this sheaf is denoted ${\displaystyle {\mathcal {O}}(-)}$  or just ${\displaystyle {\mathcal {O}}}$ , or even ${\displaystyle {\mathcal {O}}_{X}}$  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 ${\displaystyle Y\hookrightarrow X}$ . There is an associated sheaf ${\displaystyle {\mathcal {O}}_{Y}}$  which takes an open subset ${\displaystyle U\subset X}$  and gives the ring of holomorphic functions on ${\displaystyle U\cap Y}$ . 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 can be built using these kinds of sheaves from the Serre intersection formula.

Operations with sheaves

Morphisms

Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which is simply an assignment of outputs to inputs, morphisms of sheaves are also required to be compatible with the local-global structures of the underlying sheaves. This idea is made precise in the following definition.

Let ${\displaystyle F}$  and ${\displaystyle G}$  be two sheaves of sets (resp. abelian groups, rings, etc.) on ${\displaystyle X}$ . A morphism ${\displaystyle \varphi :F\to G}$  consists of a morphism ${\displaystyle \varphi _{U}:F(U)\to G(U)}$  of sets (resp. abelian groups, rings, etc.) for each open set ${\displaystyle U}$  of ${\displaystyle X}$ , subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset ${\displaystyle V}$  of an open set ${\displaystyle U}$ , the following diagram is commutative.

${\displaystyle {\begin{array}{rcl}F(U)&\xrightarrow {\quad \varphi _{U}\quad } &G(U)\\r_{V,U}{\Biggl \downarrow }&&{\Biggl \downarrow }r'_{V,U}\\F(V)&{\xrightarrow[{\quad \varphi _{V}\quad }]{}}&G(V)\end{array}}}$ 

For example, taking the derivative gives a morphism of sheaves on ${\displaystyle \mathbb {R} }$ : ${\displaystyle {\mathcal {O}}_{\mathbb {R} }^{n}\to {\mathcal {O}}_{\mathbb {R} }^{n-1}.}$  Indeed, given an (${\displaystyle n}$ -times continuously differentiable) function ${\displaystyle f:U\to \mathbb {R} }$  (with ${\displaystyle U}$  in ${\displaystyle \mathbb {R} }$  open), the restriction (to a smaller open subset ${\displaystyle V}$ ) of its derivative equals the derivative of ${\displaystyle f|_{V}}$ .

With this notion of morphism, sheaves of sets (resp. abelian groups, rings, etc.) on a fixed topological space ${\displaystyle X}$  form a category. The general categorical notions of mono-, epi- and isomorphisms can therefore be applied to sheaves.

A morphism ${\displaystyle \varphi \colon F\rightarrow G}$  of sheaves on ${\displaystyle X}$  is an isomorphism (resp. monomorphism) if and only if there exists an open cover ${\displaystyle \{U_{\alpha }\}}$  of ${\displaystyle X}$  such that ${\displaystyle \varphi |_{U_{\alpha }}\colon F(U_{\alpha })\rightarrow G(U_{\alpha })}$  are isomorphisms (resp. injective morphisms) of sets (resp. abelian groups, rings, etc.) for all ${\displaystyle \alpha }$ . These statements give examples of how to work with sheaves using local information, but it's important to note that we cannot check if a morphism of sheaves is an epimorphism in the same manner. Indeed the statement that maps on the level of open sets ${\displaystyle \varphi _{U}\colon F(U)\rightarrow G(U)}$  are not always surjective for epimorphisms of sheaves is equivalent to non-exactness of the global sections functor — or equivalently, to non-triviality of sheaf cohomology.

Stalks of a sheaf

Main article: Stalk (sheaf)

The stalk ${\displaystyle {\mathcal {F}}_{x}}$  of a sheaf ${\displaystyle {\mathcal {F}}}$  captures the properties of a sheaf "around" a point ${\displaystyle x\in X}$ , generalizing the germs of functions. Here, "around" means that, conceptually speaking, one looks at smaller and smaller neighborhoods of the point. Of course, no single neighborhood will be small enough, which requires considering a limit of some sort. More precisely, the stalk is defined by

${\displaystyle {\mathcal {F}}_{x}=\varinjlim _{U\ni x}{\mathcal {F}}(U),}$ 

the direct limit being over all open subsets of ${\displaystyle X}$  containing the given point ${\displaystyle x}$ . In other words, an element of the stalk is given by a section over some open neighborhood of ${\displaystyle x}$ , and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism ${\displaystyle F(U)\to F_{x}}$  takes a section ${\displaystyle x}$  in ${\displaystyle F(U)}$  to its germ at ${\displaystyle x}$ . This generalises the usual definition of a germ.

In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. In this sense, a sheaf is determined by its stalks, which are a local data. By contrast, the global information present in a sheaf, i.e., the global sections, i.e., the sections ${\displaystyle {\mathcal {F}}(X)}$  on the whole space ${\displaystyle X}$ , typically carry less information. For example, for a compact complex manifold ${\displaystyle X}$ , the global sections of the sheaf of holomorphic functions are just ${\displaystyle \mathbb {C} }$ , since any holomorphic function

${\displaystyle X\to \mathbb {C} }$ 

is constant by Liouville's theorem.^[6]

Turning a presheaf into a sheaf

It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf ${\displaystyle F}$  and produces a new sheaf ${\displaystyle aF}$  called the sheafification or sheaf associated to the presheaf ${\displaystyle F}$ . For example, the sheafification of the constant presheaf (see above) is called the constant sheaf. Despite its name, its sections are locally constant functions.

The sheaf ${\displaystyle aF}$  can be constructed using the étalé space of ${\displaystyle F}$ , namely as the sheaf of sections of the map

${\displaystyle \mathrm {Spe} (F)\to X.}$ 

Another construction of the sheaf ${\displaystyle aF}$  proceeds by means of a functor ${\displaystyle L}$  from presheaves to presheaves that gradually improves the properties of a presheaf: for any presheaf ${\displaystyle F}$ , ${\displaystyle LF}$  is a separated presheaf, and for any separated presheaf ${\displaystyle F}$ , ${\displaystyle LF}$  is a sheaf. The associated sheaf ${\displaystyle aF}$  is given by ${\displaystyle LLF}$ .^[8]

The idea that the sheaf ${\displaystyle aF}$  is the best possible approximation to ${\displaystyle F}$  by a sheaf is made precise using the following universal property: there is a natural morphism of presheaves ${\displaystyle i\colon F\to aF}$  so that for any sheaf ${\displaystyle G}$  and any morphism of presheaves ${\displaystyle f\colon F\to G}$ , there is a unique morphism of sheaves ${\displaystyle {\tilde {f}}\colon aF\rightarrow G}$  such that ${\displaystyle f={\tilde {f}}i}$ . In fact ${\displaystyle a}$  is the left adjoint functor to the inclusion functor (or forgetful functor) from the category of sheaves to the category of presheaves, and ${\displaystyle i}$  is the unit of the adjunction. In this way, the category of sheaves turns into a Giraud subcategory of presheaves. This categorical situation is the reason why the sheafification functor appears in constructing cokernels of sheaf morphisms or tensor products of sheaves, but not for kernels, say.

Subsheaves, quotient sheaves

If ${\displaystyle K}$  is a subsheaf of a sheaf ${\displaystyle F}$  of abelian groups, then the quotient sheaf ${\displaystyle Q}$  is the sheaf associated to the presheaf ${\displaystyle U\mapsto F(U)/K(U)}$ ; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;

${\displaystyle 0\to K\to F\to Q\to 0.}$ 

(this is also called a sheaf extension.)

Let ${\displaystyle F,G}$  be sheaves of abelian groups. The set ${\displaystyle \operatorname {Hom} (F,G)}$  of morphisms of sheaves from ${\displaystyle F}$  to ${\displaystyle G}$  forms an abelian group (by the abelian group structure of ${\displaystyle G}$ ). The sheaf hom of ${\displaystyle F}$  and ${\displaystyle G}$ , denoted by,

${\displaystyle {\mathcal {Hom}}(F,G)}$ 

is the sheaf of abelian groups ${\displaystyle U\mapsto \operatorname {Hom} (F|_{U},G|_{U})}$  where ${\displaystyle F|_{U}}$  is the sheaf on ${\displaystyle U}$  given by ${\displaystyle (F|_{U})(V)=F(V)}$  (note sheafification is not needed here). The direct sum of ${\displaystyle F}$  and ${\displaystyle G}$  is the sheaf given by ${\displaystyle U\mapsto F(U)\oplus G(U)}$ , and the tensor product of ${\displaystyle F}$  and ${\displaystyle G}$  is the sheaf associated to the presheaf ${\displaystyle U\mapsto F(U)\otimes G(U)}$ .

All of these operations extend to sheaves of modules over a sheaf of rings ${\displaystyle A}$ ; the above is the special case when ${\displaystyle A}$  is the constant sheaf ${\displaystyle {\underline {\mathbf {Z} }}}$ .

Basic functoriality

Main article: Image functors for sheaves

Since the data of a (pre-)sheaf depends on the open subsets of the base space, sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them. However, given a continuous map ${\displaystyle f:X\to Y}$  between two topological spaces, pushforward and pullback relate sheaves on ${\displaystyle X}$  to those on ${\displaystyle Y}$  and vice versa.

Direct image

The pushforward (also known as direct image) of a sheaf ${\displaystyle {\mathcal {F}}}$  on ${\displaystyle X}$  is the sheaf defined by

${\displaystyle (f_{*}{\mathcal {F}})(V)={\mathcal {F}}(f^{-1}(V)).}$ 

Here ${\displaystyle V}$  is an open subset of ${\displaystyle Y}$ , so that its preimage is open in ${\displaystyle X}$  by the continuity of ${\displaystyle f}$ . This construction recovers the skyscraper sheaf ${\displaystyle S_{x}}$  mentioned above:

${\displaystyle S_{x}=i_{*}(S)}$ 

where ${\displaystyle i:\{x\}\to X}$  is the inclusion, and ${\displaystyle S}$  is regarded as a sheaf on the singleton (by ${\displaystyle S(\{*\})=S,S(\emptyset )=\emptyset }$ .

For a map between locally compact spaces, the direct image with compact support is a subsheaf of the direct image.^[9] By definition, ${\displaystyle (f_{!}{\mathcal {F}})(V)}$  consists of those ${\displaystyle f\in {\mathcal {F}}(f^{-1}(V))}$  whose support is proper map over ${\displaystyle V}$ . If ${\displaystyle f}$  is proper itself, then ${\displaystyle f_{!}{\mathcal {F}}=f_{*}{\mathcal {F}}}$ , but in general they disagree.

Inverse image

The pullback or inverse image goes the other way: it produces a sheaf on ${\displaystyle X}$ , denoted ${\displaystyle f^{-1}{\mathcal {G}}}$  out of a sheaf ${\displaystyle {\mathcal {G}}}$  on ${\displaystyle Y}$ . If ${\displaystyle f}$  is the inclusion of an open subset, then the inverse image is just a restriction, i.e., it is given by ${\displaystyle (f^{-1}{\mathcal {G}})(U)={\mathcal {G}}(U)}$  for an open ${\displaystyle U}$  in ${\displaystyle X}$ . A sheaf ${\displaystyle F}$  (on some space ${\displaystyle X}$ ) is called locally constant if ${\displaystyle X=\bigcup _{i\in I}U_{i}}$  by some open subsets ${\displaystyle U_{i}}$  such that the restriction of ${\displaystyle F}$  to all these open subsets is constant. On a wide range of topological spaces ${\displaystyle X}$ , such sheaves are equivalent to representations of the fundamental group ${\displaystyle \pi _{1}(X)}$ .

For general maps ${\displaystyle f}$ , the definition of ${\displaystyle f^{-1}{\mathcal {G}}}$  is more involved; it is detailed at inverse image functor. The stalk is an essential special case of the pullback in view of a natural identification, where ${\displaystyle i}$  is as above:

${\displaystyle {\mathcal {G}}_{x}=i^{-1}{\mathcal {G}}(\{x\}).}$ 

More generally, stalks satisfy ${\displaystyle (f^{-1}{\mathcal {G}})_{x}={\mathcal {G}}_{f(x)}}$ .

Extension by zero

For the inclusion ${\displaystyle j:U\to X}$  of an open subset, the extension by zero of a sheaf of abelian groups on ${\displaystyle U}$  is defined as

${\displaystyle (j_{!}{\mathcal {F}})(V)={\mathcal {F}}(V)}$  if ${\displaystyle V\subset U}$  and ${\displaystyle (j_{!}{\mathcal {F}})(V)=0}$  otherwise.

For a sheaf ${\displaystyle {\mathcal {G}}}$  on ${\displaystyle X}$ , this construction is in a sense complementary to ${\displaystyle i_{*}}$ , where ${\displaystyle i}$  is the inclusion of the complement of ${\displaystyle U}$ :

${\displaystyle (j_{!}j^{*}{\mathcal {G}})_{x}={\mathcal {G}}_{x}}$  for ${\displaystyle x}$  in ${\displaystyle U}$ , and the stalk is zero otherwise, while

${\displaystyle (i_{*}i^{*}{\mathcal {G}})_{x}=0}$  for ${\displaystyle x}$  in ${\displaystyle U}$ , and equals ${\displaystyle {\mathcal {G}}_{x}}$  otherwise.

These functors are therefore useful in reducing sheaf-theoretic questions on ${\displaystyle X}$  to ones on the strata of a stratification, i.e., a decomposition of ${\displaystyle X}$  into smaller, locally closed subsets.

Complements

Sheaves in more general categories

In addition to (pre-)sheaves as introduced above, where ${\displaystyle {\mathcal {F}}(U)}$  is merely a set, it is in many cases important to keep track of additional structure on these sections. For example, the sections of the sheaf of continuous functions naturally form a real vector space, and restriction is a linear map between these vector spaces.

Presheaves with values in an arbitrary category ${\displaystyle C}$  are defined by first considering the category of open sets on ${\displaystyle X}$  to be the posetal category ${\displaystyle O(X)}$  whose objects are the open sets of ${\displaystyle X}$  and whose morphisms are inclusions. Then a ${\displaystyle C}$ -valued presheaf on ${\displaystyle X}$  is the same as a contravariant functor from ${\displaystyle O(X)}$  to ${\displaystyle C}$ . Morphisms in this category of functors, also known as natural transformations, are the same as the morphisms defined above, as can be seen by unraveling the definitions.

If the target category ${\displaystyle C}$  admits all limits, a ${\displaystyle C}$ -valued presheaf is a sheaf if the following diagram is an equalizer for every open cover ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}$  of any open set ${\displaystyle U}$ :

${\displaystyle F(U)\rightarrow \prod _{i}F(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\prod _{i,j}F(U_{i}\cap U_{j}).}$ 

Here the first map is the product of the restriction maps

${\displaystyle \operatorname {res} _{U_{i},U}\colon F(U)\rightarrow F(U_{i})}$ 

and the pair of arrows the products of the two sets of restrictions

${\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{i}}\colon F(U_{i})\rightarrow F(U_{i}\cap U_{j})}$ 

and

${\displaystyle \operatorname {res} _{U_{i}\cap U_{j},U_{j}}\colon F(U_{j})\rightarrow F(U_{i}\cap U_{j}).}$ 

If ${\displaystyle C}$  is an abelian category, this condition can also be rephrased by requiring that there is an exact sequence

${\displaystyle 0\to F(U)\to \prod _{i}F(U_{i})\xrightarrow {\operatorname {res} _{U_{i}\cap U_{j},U_{i}}-\operatorname {res} _{U_{i}\cap U_{j},U_{j}}} \prod _{i,j}F(U_{i}\cap U_{j}).}$ 

A particular case of this sheaf condition occurs for ${\displaystyle U}$  being the empty set, and the index set ${\displaystyle I}$  also being empty. In this case, the sheaf condition requires ${\displaystyle {\mathcal {F}}(\emptyset )}$  to be the terminal object in ${\displaystyle C}$ .

Ringed spaces and sheaves of modules

Main articles: Ringed space and Sheaf of modules

In several geometrical disciplines, including algebraic geometry and differential geometry, the spaces come along with a natural sheaf of rings, often called the structure sheaf and denoted by ${\displaystyle {\mathcal {O}}_{X}}$ . Such a pair ${\displaystyle (X,{\mathcal {O}}_{X})}$  is called a ringed space. Many types of spaces can be defined as certain types of ringed spaces. Commonly, all the stalks ${\displaystyle {\mathcal {O}}_{X,x}}$  of the structure sheaf are local rings, in which case the pair is called a locally ringed space.

For example, an ${\displaystyle n}$ -dimensional ${\displaystyle C^{k}}$  manifold ${\displaystyle M}$  is a locally ringed space whose structure sheaf consists of ${\displaystyle C^{k}}$ -functions on the open subsets of ${\displaystyle M}$ . The property of being a locally ringed space translates into the fact that such a function, which is nonzero at a point ${\displaystyle x}$ , is also non-zero on a sufficiently small open neighborhood of ${\displaystyle x}$ . Some authors actually define real (or complex) manifolds to be locally ringed spaces that are locally isomorphic to the pair consisting of an open subset of ${\displaystyle \mathbb {R} ^{n}}$  (resp. ${\displaystyle \mathbb {C} ^{n}}$ ) together with the sheaf of ${\displaystyle C^{k}}$  (resp. holomorphic) functions.^[10] Similarly, schemes, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the spectrum of a ring.

Given a ringed space, a sheaf of modules is a sheaf ${\displaystyle {\mathcal {M}}}$  such that on every open set ${\displaystyle U}$  of ${\displaystyle X}$ , ${\displaystyle {\mathcal {M}}(U)}$  is an ${\displaystyle {\mathcal {O}}_{X}(U)}$ -module and for every inclusion of open sets ${\displaystyle V\subseteq U}$ , the restriction map ${\displaystyle {\mathcal {M}}(U)\to {\mathcal {M}}(V)}$  is compatible with the restriction map ${\displaystyle {\mathcal {O}}(U)\to {\mathcal {O}}(V)}$ : the restriction of fs is the restriction of ${\displaystyle f}$  times that of ${\displaystyle s}$  for any ${\displaystyle f}$  in ${\displaystyle {\mathcal {O}}(U)}$  and ${\displaystyle s}$  in ${\displaystyle {\mathcal {M}}(U)}$ .

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between vector bundles and locally free sheaves of ${\displaystyle {\mathcal {O}}_{X}}$ -modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where ${\displaystyle {\mathcal {O}}}$  consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are ${\displaystyle D}$ -modules, that is, modules over the sheaf of differential operators. On any topological space, modules over the constant sheaf ${\displaystyle {\underline {\mathbf {Z} }}}$  are the same as sheaves of abelian groups in the sense above.

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted ${\displaystyle f^{*}}$  and it is distinct from ${\displaystyle f^{-1}}$ . See inverse image functor.

Finiteness conditions for sheaves of modules

Finiteness conditions for module over commutative rings give rise to similar finiteness conditions for sheaves of modules: ${\displaystyle {\mathcal {M}}}$  is called finitely generated (resp. finitely presented) if, for every point ${\displaystyle x}$  of ${\displaystyle X}$ , there exists an open neighborhood ${\displaystyle U}$  of ${\displaystyle x}$ , a natural number ${\displaystyle n}$  (possibly depending on ${\displaystyle U}$ ), and a surjective morphism of sheaves ${\displaystyle {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}}$  (respectively, in addition a natural number ${\displaystyle m}$ , and an exact sequence ${\displaystyle {\mathcal {O}}_{X}^{m}|_{U}\to {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}\to 0}$ .) Paralleling the notion of a coherent module, ${\displaystyle {\mathcal {M}}}$  is called a coherent sheaf if it is of finite type and if, for every open set ${\displaystyle U}$  and every morphism of sheaves ${\displaystyle \phi :{\mathcal {O}}_{X}^{n}\to {\mathcal {M}}}$  (not necessarily surjective), the kernel of ${\displaystyle \phi }$  is of finite type. ${\displaystyle {\mathcal {O}}_{X}}$  is coherent if it is coherent as a module over itself. Like for modules, coherence is in general a strictly stronger condition than finite presentation. The Oka coherence theorem states that the sheaf of holomorphic functions on a complex manifold is coherent.

The étalé space of a sheaf

In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called the étalé space, from the French word étalé [etale], meaning roughly "spread out". If ${\displaystyle F\in {\text{Sh}}(X)}$  is a sheaf over ${\displaystyle X}$ , then the étalé space (sometimes called the étale space) of ${\displaystyle F}$  is a topological space ${\displaystyle E}$  together with a local homeomorphism ${\displaystyle \pi :E\to X}$  such that the sheaf of sections ${\displaystyle \Gamma (\pi ,-)}$  of ${\displaystyle \pi }$  is ${\displaystyle F}$ . The space ${\displaystyle E}$  is usually very strange, and even if the sheaf ${\displaystyle F}$  arises from a natural topological situation, ${\displaystyle E}$  may not have any clear topological interpretation. For example, if

December 15, 2023