sheaf (mathematics)#Subsheaves, quotient sheaves

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

Let $X$ be a topological space. A ''presheaf $\backslash mathcal$

{F} of sets'' on $X$ consists of the following data:

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

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

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

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 $U$, one can assign the set $C^0(U)$ of continuous real-valued functions on $U$. The restriction maps are then just given by restricting a continuous function on $U$ to a smaller open subset $V\backslash subseteq\; U$, 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 $\backslash mathcal\{H\}(-)$ and a presheaf of smooth functions $C^\backslash infty(-)$.

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

Given a presheaf, a natural question to ask is to what extent its sections over an open set $U$ are specified by their restrictions to open subsets of $U$. 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 $U$ is an open set, $\backslash \{\; U\_i\; \backslash \}\_\{i\; \backslash in\; I\}$ is an open cover of $U$ with $U\_i\; \backslash subseteq\; U$ for all $i\; \backslash in\; I$, and $s,\; t\; \backslash in\; \backslash mathcal\{F\}(U)$ are sections. If $s|\_\{\; U\_i\}\; =\; t|\_\{\; U\_i\}$ for all $i\; \backslash in\; I$, then $s\; =\; t$.
2. (Gluing) Suppose $U$ is an open set, $\backslash \{\; U\_i\; \backslash \}\_\{i\; \backslash in\; I\}$ is an open cover of $U$ with $U\_i\; \backslash subseteq\; U$ for all $i\; \backslash in\; I$, and $\backslash \{\; s\_i\; \backslash in\; \backslash mathcal\{F\}(U\_i)\; \backslash \}\_\{i\; \backslash in\; I\}$ is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if $s\_i|\_\{U\_i\backslash cap\; U\_j\}\; =\; s\_j|\_\{U\_i\; \backslash cap\; U\_j\}$ for all $i,\; j\; \backslash in\; I$, then there exists a section $s\; \backslash in\; \backslash mathcal\{F\}(U)$ such that $s|\_\{U\_i\}\; =\; s\_i$ for all $i\; \backslash in\; I$.{{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 = GTM | date = 6 April 2006 | pages = 11–18}}

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

The section $s$ 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 $s\_i$ and $s\_j$ 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.{{Citation | last1=Tennison | first1=B. R. | title=Sheaf theory | publisher=Cambridge University Press | mr=0404390 | year=1975}}

The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions $f\_i\; :\; U\_i\; \backslash to\; \backslash R$ which agree on the intersections $U\_i\; \backslash cap\; U\_j$, there is a unique continuous function $f:\; U\; \backslash to\; \backslash R$ whose restriction equals the $f\_i$. 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, $F$ being particularly common, presumably for the French word for sheaf, faisceau. Use of calligraphic letters such as $\backslash mathcal\{F\}$ 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 $R$, whose points are the prime ideals $\backslash mathfrak\{p\}$ in $R$. The open sets $D\_f\; :=\; \backslash \{\; \backslash mathfrak\{p\}\; \backslash subseteq\; R,\; f\; \backslash notin\; \backslash mathfrak\{p\}\backslash \}$ form a basis for the Zariski topology on this space. Given an $R$-module $M$, there is a sheaf, denoted by $\backslash tilde\; M$ on the $\backslash operatorname\{Spec\}R$, that satisfies

:$\backslash tilde\; M(D\_f)\; :=\; M[1/f],$ the localization of $M$ at $f$.

There is another characterization of sheaves that is equivalent to the previously discussed.

A presheaf $\backslash mathcal\{F\}$ is a sheaf if and only if for any open $U$ and any open cover $\backslash \{U\_a\backslash \}$ of $U$, $\backslash mathcal\{F\}(U)$ is the fibre product $\backslash mathcal\{F\}(U)\backslash cong\backslash mathcal\{F\}(U\_a)\backslash times\_\{\backslash mathcal\{F\}(U\_a\backslash cap\; U\_b)\}\backslash mathcal\{F\}(U\_b)$. This characterization is useful in construction of sheaves, for example, if $\backslash mathcal\{F\},\backslash mathcal\{G\}$ are abelian sheaves, then the kernel of sheaves morphism $\backslash mathcal\{F\}\backslash to\backslash mathcal\{G\}$ 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.

Any continuous map $f:Y\backslash to\; X$ of topological spaces determines a sheaf $\backslash Gamma(Y/X)$ on $X$ by setting

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

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

:$\backslash C\; \backslash stackrel\{\backslash exp\}\{\backslash longrightarrow\}\; \backslash C\backslash setminus\; \backslash \{0\backslash \}$

is the sheaf which assigns to any $U\backslash subseteq\; \backslash mathbb\{C\}\backslash setminus\backslash \{0\backslash \}$ the set of branches of the complex logarithm on $U$.

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

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

The assignment sending $U$ to the compactly supported functions on $U$ 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.{{harvtxt|Bredon|1997|loc=Chapter V, §1}} However, taking the dual of these vector spaces does give a sheaf, the sheaf of distributions.

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 $X$ be the two-point topological space $\backslash \{x,y\backslash \}$ with the discrete topology. Define a presheaf $F$ as follows: $$F(\backslash varnothing)\; =\; \backslash \{\backslash varnothing\backslash \},\backslash \; F(\backslash \{x\backslash \})\; =\; \backslash R,\backslash \; F(\backslash \{y\backslash \})\; =\; \backslash R,\backslash \; F(\backslash \{x,\; y\backslash \})\; =\; \backslash R\backslash times\backslash R\backslash times\backslash R$$The restriction map $F(\backslash \{x,\; y\backslash \})\; \backslash to\; F(\backslash \{x\backslash \})$ is the projection of $\backslash R\; \backslash times\backslash R\backslash times\backslash R$ onto its first coordinate, and the restriction map $F(\backslash \{x,\; y\backslash \})\; \backslash to\; F(\backslash \{y\backslash \})$ is the projection of $\backslash R\; \backslash times\backslash R\backslash times\backslash R$ onto its second coordinate. $F$ is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over $\backslash \{x\backslash \}$ and $\backslash \{y\backslash \}$ determine only two of those numbers. So while we can glue any two sections over $\backslash \{x\backslash \}$ and $\backslash \{y\backslash \}$, we cannot glue them uniquely.
• Let $X\; =\; \backslash R$ be the real line, and let $F(U)$ be the set of bounded continuous functions on $U$. This is not a sheaf because it is not always possible to glue. For example, let $U\_i$ be the set of all $x$ such that

One of the historical motivations for sheaves have come from studying complex manifolds,{{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}} complex analytic geometry,{{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}} and scheme theory from algebraic geometry. This is because in all of the previous cases, we consider a topological space $X$ together with a structure sheaf $\backslash mathcal\{O\}$ 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).

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 $X$ (like complex projective space or the vanishing locus in projective space of a homogeneous polynomial), the only holomorphic functions

$f:X\; \backslash to\; \backslash C$

Another complexity when considering the ring of holomorphic functions on a complex manifold $X$ is given a small enough open set $U\; \backslash subseteq\; X$, the holomorphic functions will be isomorphic to $\backslash mathcal\{H\}(U)\; \backslash cong\; \backslash mathcal\{H\}(\backslash C^n)$. 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 $X$ on arbitrary open subsets $U\; \backslash subseteq\; X$. This means as $U$ becomes more complex topologically, the ring $\backslash mathcal\{H\}(U)$ can be expressed from gluing the $\backslash mathcal\{H\}(U\_i)$. Note that sometimes this sheaf is denoted $\backslash mathcal\{O\}(-)$ or just $\backslash mathcal\{O\}$, or even $\backslash mathcal\{O\}\_X$ when we want to emphasize the space the structure sheaf is associated to.

Another common example of sheaves can be constructed by considering a complex submanifold $Y\; \backslash hookrightarrow\; X$. There is an associated sheaf $\backslash mathcal\{O\}\_Y$ which takes an open subset $U\; \backslash subseteq\; X$ and gives the ring of holomorphic functions on $U\; \backslash cap\; Y$. 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.

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

:$\backslash begin\{array\}\{rcl\}$

\mathcal{F}(U) & \xrightarrow{\quad\varphi_U\quad} & \mathcal{G}(U)\\

r^{U}_{V}\Biggl\downarrow & & \Biggl\downarrow {r'}^{U}_{V}\\

\mathcal{F}(V) & \xrightarrow[{\quad\varphi_V\quad}]{} & \mathcal{G}(V)

\end{array}

For example, taking the derivative gives a morphism of sheaves on $\backslash R$,

$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}x\}\backslash colon\backslash mathcal\; O^n\_\{\backslash R\}\; \backslash to\; \backslash mathcal\; O^\{n-1\}\_\{\backslash R\}.$

Indeed, given an ($n$-times continuously differentiable) function $f\; :\; U\; \backslash to\; \backslash R$ (with $U$ in $\backslash R$ open), the restriction (to a smaller open subset $V$) of its derivative equals the derivative of $f|\_V$.

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

A morphism $\backslash varphi\backslash colon\; \backslash mathcal\{F\}\backslash rightarrow\; \backslash mathcal\{G\}$ of sheaves on $X$ is an isomorphism (respectively monomorphism) if and only if there exists an open cover $\backslash \{U\_\backslash alpha\backslash \}$ of $X$ such that $\backslash varphi|\_\{U\_\backslash alpha\}\backslash colon\; \backslash mathcal\{F\}(U\_\backslash alpha)\backslash rightarrow\; \backslash mathcal\{G\}(U\_\backslash alpha)$ are isomorphisms (respectively injective morphisms) of sets (respectively abelian groups, rings, etc.) for all $\backslash alpha$. 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 $\backslash varphi\_U\; \backslash colon\; \backslash mathcal\{F\}(U)\backslash rightarrow\; \backslash mathcal\{G\}(U)$ 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.

{{Main|Stalk (sheaf)}}

The stalk $\backslash mathcal\{F\}\_x$ of a sheaf $\backslash mathcal\{F\}$ captures the properties of a sheaf "around" a point $x\backslash in\; X$, 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

:$\backslash mathcal\{F\}\_x\; =\; \backslash varinjlim\_\{U\backslash ni\; x\}\; \backslash mathcal\{F\}(U),$

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

The natural morphism $\backslash mathcal\{F\}(U)\backslash to\; \backslash mathcal\{F\}\_x$ takes a section $s$ in $\backslash mathcal\{F\}(U)$ to its germ $s\_x$ at $x$. 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 $\backslash mathcal\; F(X)$ on the whole space $X$, typically carry less information. For example, for a compact complex manifold $X$, the global sections of the sheaf of holomorphic functions are just $\backslash C$, since any holomorphic function

:$X\; \backslash to\; \backslash C$

is constant by Liouville's theorem.

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 $\backslash mathcal\{F\}$ and produces a new sheaf $a\backslash mathcal\{F\}$ called the sheafification or sheaf associated to the presheaf $\backslash mathcal\{F\}$. 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 $a\backslash mathcal\{F\}$ can be constructed using the étalé space $E$ of $\backslash mathcal\{F\}$, namely as the sheaf of sections of the map

:$E\; \backslash to\; X.$

Another construction of the sheaf $a\backslash mathcal\{F\}$ proceeds by means of a functor $L$ from presheaves to presheaves that gradually improves the properties of a presheaf: for any presheaf $\backslash mathcal\{F\}$, $L\backslash mathcal\{F\}$ is a separated presheaf, and for any separated presheaf $\backslash mathcal\{F\}$, $L\backslash mathcal\{F\}$ is a sheaf. The associated sheaf $a\backslash mathcal\{F\}$ is given by $LL\backslash mathcal\{F\}$.SGA 4 II 3.0.5

The idea that the sheaf $a\backslash mathcal\{F\}$ is the best possible approximation to $\backslash mathcal\{F\}$ by a sheaf is made precise using the following universal property: there is a natural morphism of presheaves $i\backslash colon\; \backslash mathcal\{F\}\backslash to\; a\backslash mathcal\{F\}$ so that for any sheaf $\backslash mathcal\{G\}$ and any morphism of presheaves $f\backslash colon\; \backslash mathcal\{F\}\backslash to\; \backslash mathcal\{G\}$, there is a unique morphism of sheaves $\backslash tilde\; f\; \backslash colon\; a\backslash mathcal\{F\}\; \backslash rightarrow\; \backslash mathcal\{G\}$ such that $f\; =\; \backslash tilde\; f\; i$. In fact, $a$ is the left adjoint functor to the inclusion functor (or forgetful functor) from the category of sheaves to the category of presheaves, and $i$ 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.

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

:$0\; \backslash to\; K\; \backslash to\; F\; \backslash to\; Q\; \backslash to\; 0.$

(this is also called a sheaf extension.)

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

:$\backslash mathcal\{Hom\}(F,\; G)$

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

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

{{Main|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 $f:X\backslash to\; Y$ between two topological spaces, pushforward and pullback relate sheaves on $X$ to those on $Y$ and vice versa.

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

:$(f\_*\; \backslash mathcal\; F)(V)\; =\; \backslash mathcal\; F(f^\{-1\}(V)).$

Here $V$ is an open subset of $Y$, so that its preimage is open in $X$ by the continuity of $f$.

This construction recovers the skyscraper sheaf $S\_x$ mentioned above:

:$S\_x\; =\; i\_*\; (S)$

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

For a map between locally compact spaces, the direct image with compact support is a subsheaf of the direct image.{{harvtxt|Iversen|1986|loc=Chapter VII}} By definition, $(f\_!\; \backslash mathcal\; F)(V)$ consists of those $s\; \backslash in\; \backslash mathcal\; F(f^\{-1\}(V))$ whose support is mapped properly. If $f$ is proper itself, then $f\_!\; \backslash mathcal\; F\; =\; f\_*\; \backslash mathcal\; F$, but in general they disagree.

The pullback or inverse image goes the other way: it produces a sheaf on $X$, denoted $f^\{-1\}\; \backslash mathcal\; G$ out of a sheaf $\backslash mathcal\; G$ on $Y$. If $f$ is the inclusion of an open subset, then the inverse image is just a restriction, i.e., it is given by $(f^\{-1\}\; \backslash mathcal\; G)(U)\; =\; \backslash mathcal\; G(U)$ for an open $U$ in $X$. A sheaf $\backslash mathcal\{F\}$ (on some space $X$) is called locally constant if $X=\; \backslash bigcup\_\{i\; \backslash in\; I\}\; U\_i$ by some open subsets $U\_i$ such that the restriction of $\backslash mathcal\; F$ to all these open subsets is constant. On a wide range of topological spaces $X$, such sheaves are equivalent to representations of the fundamental group $\backslash pi\_1(X)$.

For general maps $f$, the definition of $f^\{-1\}\; \backslash mathcal\; G$ 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 $i$ is as above:

:$\backslash mathcal\; G\_x\; =\; i^\{-1\}\backslash mathcal\{G\}(\backslash \{x\backslash \}).$

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

{{redirect|Extension by zero|uses in analysis|Sobolev space#Extension by zero|and|Extension of a function}}

For the inclusion $j\; :\; U\; \backslash to\; X$ of an open subset, the extension by zero $j\_!\; \backslash mathcal\; F$ (pronounced "j lower shriek of F") of a sheaf $\backslash mathcal\; F$ of abelian groups on $U$ is the sheafification of the presheaf defined by

:$V\; \backslash mapsto\; \backslash mathcal\; F(V)$ if $V\; \backslash subseteq\; U$ and $V\; \backslash mapsto\; 0$ otherwise.

For a sheaf $\backslash mathcal\; G$ on $X$, this construction is in a sense complementary to $i\_*$, where $i\; :\; X\; \backslash setminus\; U\; \backslash to\; X$ is the inclusion of the complement of $U$:

:$(j\_!\; j^*\; \backslash mathcal\; G)\_x\; =\; \backslash mathcal\; G\_x$ for $x$ in $U$, and the stalk is zero otherwise, while

:$(i\_*\; i^*\; \backslash mathcal\; G)\_x\; =\; 0$ for $x$ in $U$, and equals $\backslash mathcal\; G\_x$ otherwise.

More generally, if $A\; \backslash subset\; X$ is a locally closed subset, then there exists an open $U$ of $X$ containing $A$ such that $A$ is closed in $U$. Let $f\; :\; A\; \backslash to\; U$ and $j\; :\; U\; \backslash to\; X$ be the natural inclusions. Then the extension by zero of a sheaf $\backslash mathcal\; F$ on $A$ is defined by $j\_!\; f\_*\; F$.

Due to its nice behavior on stalks, the extension by zero functor is useful for reducing sheaf-theoretic questions on $X$ to ones on the strata of a stratification, i.e., a decomposition of $X$ into smaller, locally closed subsets.

In addition to (pre-)sheaves as introduced above, where $\backslash mathcal\; F(U)$ 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 $C$ are defined by first considering the category of open sets on $X$ to be the posetal category $O(X)$ whose objects are the open sets of $X$ and whose morphisms are inclusions. Then a $C$-valued presheaf on $X$ is the same as a contravariant functor from $O(X)$ to $C$. 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 $C$ admits all limits, a $C$-valued presheaf is a sheaf if the following diagram is an equalizer for every open cover

$\backslash mathcal\{U\}\; =\; \backslash \{U\_i\backslash \}\_\{i\; \backslash in\; I\}$ of any open set $U$:

:$F(U)\; \backslash rightarrow\; \backslash prod\_\{i\}\; F(U\_i)\; \{\{\{\}\; \backslash atop\; \backslash longrightarrow\}\backslash atop\{\backslash longrightarrow\; \backslash atop\; \{\}\}\}\; \backslash prod\_\{i,\; j\}\; F(U\_i\; \backslash cap\; U\_j).$

Here the first map is the product of the restriction maps

:$\backslash operatorname\{res\}\_\{U\_i,\; U\}\; \backslash colon\; F(U)\; \backslash rightarrow\; F(U\_i)$

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

:$\backslash operatorname\{res\}\_\{U\_i\; \backslash cap\; U\_j,\; U\_i\}\; \backslash colon\; F(U\_i)\; \backslash rightarrow\; F(U\_i\; \backslash cap\; U\_j)$

and

:$\backslash operatorname\{res\}\_\{U\_i\; \backslash cap\; U\_j,\; U\_j\}\; \backslash colon\; F(U\_j)\; \backslash rightarrow\; F(U\_i\; \backslash cap\; U\_j).$

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

:$0\; \backslash to\; F(U)\; \backslash to\; \backslash prod\_i\; F(U\_i)\; \backslash xrightarrow\{\backslash operatorname\{res\}\_\{U\_i\; \backslash cap\; U\_j,\; U\_i\}\; -\; \backslash operatorname\{res\}\_\{U\_i\; \backslash cap\; U\_j,\; U\_j\}\}\; \backslash prod\_\{i,j\}\; F(U\_i\; \backslash cap\; U\_j).$

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

{{Main|Ringed space|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 $\backslash mathcal\{O\}\_X$. Such a pair $(X,\; \backslash mathcal\; O\_X)$ is called a ringed space. Many types of spaces can be defined as certain types of ringed spaces. Commonly, all the stalks $\backslash mathcal\; O\_\{X,\; x\}$ of the structure sheaf are local rings, in which case the pair is called a locally ringed space.

For example, an $n$-dimensional $C^k$ manifold $M$ is a locally ringed space whose structure sheaf consists of $C^k$-functions on the open subsets of $M$. The property of being a locally ringed space translates into the fact that such a function, which is nonzero at a point $x$, is also non-zero on a sufficiently small open neighborhood of $x$. 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 $\backslash R^n$ (respectively $\backslash C^n$) together with the sheaf of $C^k$ (respectively holomorphic) functions.{{harvtxt|Ramanan|2005}} 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 $\backslash mathcal\{M\}$ such that on every open set $U$ of $X$, $\backslash mathcal\{M\}(U)$ is an $\backslash mathcal\{O\}\_X(U)$-module and for every inclusion of open sets $V\backslash subseteq\; U$, the restriction map $\backslash mathcal\{M\}(U)\; \backslash to\; \backslash mathcal\{M\}(V)$ is compatible with the restriction map $\backslash mathcal\{O\}(U)\; \backslash to\; \backslash mathcal\{O\}(V)$: the restriction of $fs$ is the restriction of $f$ times that of $s$ for any $f$ in $\backslash mathcal\{O\}(U)$ and $s$ in $\backslash mathcal\{M\}(U)$.

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 $\backslash mathcal\{O\}\_X$-modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where $\backslash mathcal\; O$ consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are $D$-modules, that is, modules over the sheaf of differential operators. On any topological space, modules over the constant sheaf $\backslash underline\{\backslash mathbf\{Z\}\}$ 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 $f^*$ and it is distinct from $f^\{-1\}$. See inverse image functor.

Finiteness conditions for module over commutative rings give rise to similar finiteness conditions for sheaves of modules: $\backslash mathcal\{M\}$ is called finitely generated (respectively finitely presented) if, for every point $x$ of $X$, there exists an open neighborhood $U$ of $x$, a natural number $n$ (possibly depending on $U$), and a surjective morphism of sheaves $\backslash mathcal\{O\}\_X^n|\_U\; \backslash to\; \backslash mathcal\{M\}|\_U$ (respectively, in addition a natural number $m$, and an exact sequence $\backslash mathcal\{O\}\_X^m|\_U\; \backslash to\; \backslash mathcal\{O\}\_X^n|\_U\; \backslash to\; \backslash mathcal\{M\}|\_U\; \backslash to\; 0$.) Paralleling the notion of a coherent module, $\backslash mathcal\{M\}$ is called a coherent sheaf if it is of finite type and if, for every open set $U$ and every morphism of sheaves $\backslash phi\; :\; \backslash mathcal\{O\}\_X^n\; \backslash to\; \backslash mathcal\{M\}$ (not necessarily surjective), the kernel of $\backslash phi$ is of finite type. $\backslash mathcal\{O\}\_X$ 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.

{{anchor|Etale space}}

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 {{IPA|fr|etale|}}, meaning roughly "spread out". If $F\; \backslash in\; \backslash text\{Sh\}(X)$ is a sheaf over $X$, then the étalé space (sometimes called the étale space) of $F$ is a topological space $E$ together with a local homeomorphism $\backslash pi:\; E\; \backslash to\; X$ such that the sheaf of sections $\backslash Gamma(\backslash pi,\; -)$ of $\backslash pi$ is $F$. The space $E$ is usually very strange, and even if the sheaf $F$ arises from a natural topological situation, $E$ may not have any clear topological interpretation. For example, if $F$ is the sheaf of sections of a continuous function $f:\; Y\; \backslash to\; X$, then $E=Y$ if and only if $f$ is a local homeomorphism.

The étalé space $E$ is constructed from the stalks of $F$ over $X$. As a set, it is their disjoint union and $\backslash pi$ is the obvious map that takes the value $x$ on the stalk of $F$ over $x\; \backslash in\; X$. The topology of $E$ is defined as follows. For each element $s\; \backslash in\; F(U)$ and each $x\; \backslash in\; U$, we get a germ of $s$ at $x$, denoted $[s]\_x$ or $s\_x$. These germs determine points of $E$. For any $U$ and $s\; \backslash in\; F(U)$, the union of these points (for all $x\; \backslash in\; U$) is declared to be open in $E$. Notice that each stalk has the discrete topology as subspace topology. A morphism between two sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.

The construction above determines an equivalence of categories between the category of sheaves of sets on $X$ and the category of étalé spaces over $X$. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf.

This construction makes all sheaves into representable functors on certain categories of topological spaces. As above, let $F$ be a sheaf on $X$, let $E$ be its étalé space, and let $\backslash pi:E\; \backslash to\; X$ be the natural projection. Consider the overcategory $\backslash text\{Top\}/X$ of topological spaces over $X$, that is, the category of topological spaces together with fixed continuous maps to $X$. Every object of this category is a continuous map $f:Y\backslash to\; X$, and a morphism from $Y\backslash to\; X$ to $Z\backslash to\; X$ is a continuous map $Y\backslash to\; Z$ that commutes with the two maps to $X$. There is a functor

$\backslash Gamma:\backslash text\{Top\}/X\; \backslash to\; \backslash text\{Sets\}$

$\backslash Gamma(i)\; =\; f^\{-1\}\; F(U)\; =\; F(U)\; =\; \backslash Gamma(F,\; U)$

$\backslash Gamma(i)\; =\; f^\{-1\}\; F(\backslash \{x\backslash \})\; =\; F|\_x$

$(f^\{-1\}F)(Y)\; \backslash cong\; \backslash operatorname\{Hom\}\_\{\backslash mathbf\{Top\}/X\}(f,\; \backslash pi)$,

$E$ is constructed so that the projection map $\backslash pi$ is a covering map. In algebraic geometry, the natural analog of a covering map is called an étale morphism. Despite its similarity to "étalé", the word étale {{IPA|fr|etal|}} has a different meaning in French. It is possible to turn $E$ into a scheme and $\backslash pi$ into a morphism of schemes in such a way that $\backslash pi$ retains the same universal property, but $\backslash pi$ is not in general an étale morphism because it is not quasi-finite. It is, however, formally étale.

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as mathematical analysis.

{{Main|Sheaf cohomology}}

In contexts where the open set $U$ is fixed, and the sheaf is regarded as a variable, the set $F(U)$ is also often denoted $\backslash Gamma(U,\; F).$

As was noted above, this functor does not preserve epimorphisms. Instead, an epimorphism of sheaves $\backslash mathcal\; F\; \backslash to\; \backslash mathcal\; G$ is a map with the following property: for any section $g\; \backslash in\; \backslash mathcal\; G(U)$ there is a covering $\backslash mathcal\{U\}\; =\; \backslash \{U\_i\backslash \}\_\{i\; \backslash in\; I\}$ where

$U\; =\; \backslash bigcup\_\{i\; \backslash in\; I\}\; U\_i$

:$\backslash mathcal\; O\; \backslash stackrel\{\backslash exp\}\; \backslash to\; \backslash mathcal\; O^\backslash times$

between the sheaf of holomorphic functions and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function $g$ (on some open subset in $\backslash C$, say), admits a complex logarithm locally, i.e., after restricting $g$ to appropriate open subsets. However, $g$ need not have a logarithm globally.

Sheaf cohomology captures this phenomenon. More precisely, for an exact sequence of sheaves of abelian groups

:$0\; \backslash to\; \backslash mathcal\; F\_1\; \backslash to\; \backslash mathcal\; F\_2\; \backslash to\; \backslash mathcal\; F\_3\; \backslash to\; 0,$

(i.e., an epimorphism $\backslash mathcal\; F\_2\; \backslash to\; \backslash mathcal\; F\_3$ whose kernel is $\backslash mathcal\; F\_1$), there is a long exact sequence$$0\; \backslash to\; \backslash Gamma(U,\; \backslash mathcal\; F\_1)\; \backslash to\; \backslash Gamma(U,\; \backslash mathcal\; F\_2)\; \backslash to\; \backslash Gamma(U,\; \backslash mathcal\; F\_3)\; \backslash to\; H^1(U,\; \backslash mathcal\; F\_1)\; \backslash to\; H^1(U,\; \backslash mathcal\; F\_2)\; \backslash to\; H^1(U,\; \backslash mathcal\; F\_3)\; \backslash to\; H^2(U,\; \backslash mathcal\; F\_1)\; \backslash to\; \backslash dots$$By means of this sequence, the first cohomology group $H^1(U,\; \backslash mathcal\; F\_1)$ is a measure for the non-surjectivity of the map between sections of $\backslash mathcal\; F\_2$ and $\backslash mathcal\; F\_3$.

There are several different ways of constructing sheaf cohomology. {{harvtxt|Grothendieck|1957}} introduced them by defining sheaf cohomology as the derived functor of $\backslash Gamma$. This method is theoretically satisfactory, but, being based on injective resolutions, of little use in concrete computations. Godement resolutions are another general, but practically inaccessible approach.

Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). For example, a partition of unity argument shows that the sheaf of smooth functions on a manifold is soft. The higher cohomology groups $H^i(U,\; \backslash mathcal\; F)$ for $i\; >\; 0$ vanish for soft sheaves, which gives a way of computing cohomology of other sheaves. For example, the de Rham complex is a resolution of the constant sheaf $\backslash underline\{\backslash mathbf\{R\}\}$ on any smooth manifold, so the sheaf cohomology of $\backslash underline\{\backslash mathbf\{R\}\}$ is equal to its de Rham cohomology.

A different approach is by Čech cohomology. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations, such as computing the coherent sheaf cohomology of complex projective space $\backslash mathbb\{P\}^n$.Hartshorne (1977), Theorem III.5.1. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct $H^1$ but incorrect higher cohomology groups. To get around this, Jean-Louis Verdier developed hypercoverings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of Pierre Deligne's mixed Hodge structures.

Many other coherent sheaf cohomology groups are found using an embedding $i:X\; \backslash hookrightarrow\; Y$ of a space $X$ into a space with known cohomology, such as $\backslash mathbb\{P\}^n$, or some weighted projective space. In this way, the known sheaf cohomology groups on these ambient spaces can be related to the sheaves $i\_*\backslash mathcal\{F\}$, giving $H^i(Y,i\_*\backslash mathcal\{F\})\; \backslash cong\; H^i(X,\backslash mathcal\{F\})$. For example, computing the coherent sheaf cohomology of projective plane curves is easily found. One big theorem in this space is the Hodge decomposition found using a spectral sequence associated to sheaf cohomology groups, proved by Deligne.{{Cite journal|last=Deligne|first=Pierre|date=1971|title=Théorie de Hodge : II|url=http://www.numdam.org/item/?id=PMIHES_1971__40__5_0|journal=Publications Mathématiques de l'IHÉS|language=en|volume=40|pages=5–57|doi=10.1007/BF02684692 |s2cid=118967613 }}{{Cite journal|last=Deligne|first=Pierre|date=1974|title=Théorie de Hodge : III|url=http://www.numdam.org/item/PMIHES_1974__44__5_0/|journal=Publications Mathématiques de l'IHÉS|language=en|volume=44|pages=5–77|doi=10.1007/BF02685881 |s2cid=189777706 }} Essentially, the $E\_1$-page with terms

$E\_1^\{p,q\}\; =\; H^p(X,\backslash Omega^q\_X)$

Another clean approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of Lie groups. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds.

In many cases there is a duality theory for sheaves that generalizes Poincaré duality. See Grothendieck duality and Verdier duality.

The derived category of the category of sheaves of, say, abelian groups on some space X, denoted here as $D(X)$, is the conceptual haven for sheaf cohomology, by virtue of the following relation:

:$H^n(X,\; \backslash mathcal\; F)\; =\; \backslash operatorname\{Hom\}\_\{D(X)\}(\backslash mathbf\; Z,\; \backslash mathcal\; F[n]).$

The adjunction between $f^\{-1\}$, which is the left adjoint of $f\_*$ (already on the level of sheaves of abelian groups) gives rise to an adjunction

:$f^\{-1\}\; :\; D(Y)\; \backslash rightleftarrows\; D(X)\; :\; R\; f\_*$ (for $f:\; X\; \backslash to\; Y$),

where $Rf\_*$ is the derived functor. This latter functor encompasses the notion of sheaf cohomology since $H^n(X,\; \backslash mathcal\; F)\; =\; R^n\; f\_*\; \backslash mathcal\; F$ for $f:\; X\; \backslash to\; \backslash \{*\backslash \}$.

{{Images of sheaves}}

Like $f\_*$, the direct image with compact support $f\_!$ can also be derived. By virtue of the following isomorphism $R\; f\_!\; \backslash mathcal\{F\}$ parametrizes the cohomology with compact support of the fibers of $f$:

:$(R^i\; f\_!\; \backslash mathcal\; F)\_y\; =\; H^i\_c(f^\{-1\}(y),\; \backslash mathcal\; F).${{harvtxt|Iversen|1986|loc=Chapter VII, Theorem 1.4}}

This isomorphism is an example of a base change theorem. There is another adjunction

:$Rf\_!\; :\; D(X)\; \backslash rightleftarrows\; D(Y)\; :\; f^!.$

Unlike all the functors considered above, the twisted (or exceptional) inverse image functor $f^!$ is in general only defined on the level of derived categories, i.e., the functor is not obtained as the derived functor of some functor between abelian categories. If $f:\; X\; \backslash to\; \backslash \{*\backslash \}$ and X is a smooth orientable manifold of dimension n, then

:$f^!\; \backslash underline\; \backslash mathbf\; R\; \backslash cong\; \backslash underline\; \backslash mathbf\; R\; [n].${{harvtxt|Kashiwara|Schapira|1994|loc=Chapter III, §3.1}}

This computation, and the compatibility of the functors with duality (see Verdier duality) can be used to obtain a high-brow explanation of Poincaré duality. In the context of quasi-coherent sheaves on schemes, there is a similar duality known as coherent duality.

Perverse sheaves are certain objects in $D(X)$, i.e., complexes of sheaves (but not in general sheaves proper). They are an important tool to study the geometry of singularities.{{harvtxt|de Cataldo|Migliorini|2010}}

Another important application of derived categories of sheaves is with the derived category of coherent sheaves on a scheme $X$ denoted $D\_\{Coh\}(X)$. This was used by Grothendieck in his development of intersection theory{{cite web|last=Grothendieck|title=Formalisme des intersections sur les schema algebriques propres|url=http://library.msri.org/books/sga/sga/6/6t_519.html}} using derived categories and K-theory, that the intersection product of subschemes $Y\_1,\; Y\_2$ is represented in K-theory as

$[Y\_1]\backslash cdot[Y\_2]\; =\; [\backslash mathcal\{O\}\_\{Y\_1\}\backslash otimes\_\{\backslash mathcal\{O\}\_X\}^\{\backslash mathbf\{L\}\}\backslash mathcal\{O\}\_\{Y\_2\}]\; \backslash in\; K(\backslash text\{Coh(X)\})$

{{Main|Grothendieck topology|Topos}}

André Weil's Weil conjectures stated that there was a cohomology theory for algebraic varieties over finite fields that would give an analogue of the Riemann hypothesis. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf $\backslash underline\{\backslash mathbf\{C\}\}$ in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the Zariski topology, and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). Alexandre Grothendieck solved this problem by introducing Grothendieck topologies, which axiomatize the notion of covering. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to define étale cohomology and ℓ-adic cohomology, which eventually were used to prove the Weil conjectures.

A category with a Grothendieck topology is called a site. A category of sheaves on a site is called a topos or a Grothendieck topos. The notion of a topos was later abstracted by William Lawvere and Miles Tierney to define an elementary topos, which has connections to mathematical logic.

{{More citations needed section|date=January 2023}}

The first origins of sheaf theory are hard to pin down – they may be co-extensive with the idea of analytic continuation{{Clarify|date=July 2010}}. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.

• 1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering.
• 1938 Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work since J. W. Alexander and Kolmogorov first defined cochains.
• 1943 Norman Steenrod publishes on homology with local coefficients.{{cite journal |doi=10.2307/1969099|jstor=1969099 |last1=Steenrod |first1=N. E. |title=Homology with Local Coefficients |journal=Annals of Mathematics |year=1943 |volume=44 |issue=4 |pages=610–627 }}
• 1945 Jean Leray publishes work carried out as a prisoner of war, motivated by proving fixed-point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.{{Cite book | last = Dieudonné | first = Jean | author-link = Jean Dieudonné | title = A history of algebraic and differential topology 1900–1960 | publisher = Birkhäuser | year = 1989 | pages = 123–141 | isbn = 978-0-8176-3388-2}}
• 1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with André Weil (see De Rham–Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the later carapaces).
• 1948 The Cartan seminar writes up sheaf theory for the first time.
• 1950 The "second edition" sheaf theory from the Cartan seminar: the sheaf space (espace étalé) definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables.
• 1951 The Cartan seminar proves theorems A and B, based on Oka's work.
• 1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre,{{cite journal |url=https://gallica.bnf.fr/ark:/12148/bpt6k3189t/f128.item| zbl=0050.17701 | title=Un théorème de finitude concernant les variétés analytiques compactes | journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris | year=1953 | volume=237 | pages=128–130 | last1=Cartan | first1=Henri | last2=Serre | first2=Jean-Pierre }} as is Serre duality.
• 1954 Serre's paper Faisceaux algébriques cohérents{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Faisceaux algébriques cohérents | url=http://www.mat.uniroma1.it/people/arbarello/FAC.pdf | mr=0068874 | year=1955 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=61 | pages=197–278 | doi=10.2307/1969915 | jstor=1969915 | issue=2 }} (published in 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Friedrich Hirzebruch, who writes a major 1956 book on topological methods.
• 1955 Alexander Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
• 1956 Oscar Zariski's report Algebraic sheaf theory{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Scientific report on the second summer institute, several complex variables. Part III. Algebraic sheaf theory | year=1956 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=62 | pages=117–141 | doi=10.1090/S0002-9904-1956-10018-9 | issue=2| doi-access=free }}
• 1957 Grothendieck's Tohoku paper{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Sur quelques points d'algèbre homologique | mr=0102537 | year=1957 | journal=The Tohoku Mathematical Journal |series=Second Series | issn=0040-8735 | volume=9 | issue=2 | pages=119–221 | doi=10.2748/tmj/1178244839| doi-access=free |ref=none}} rewrites homological algebra; he proves Grothendieck duality (i.e., Serre duality for possibly singular algebraic varieties).
• 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them, local cohomology, derived categories (with Verdier), and Grothendieck topologies. There emerges also his influential schematic idea of 'six operations' in homological algebra.
• 1958 Roger Godement's book on sheaf theory is published. At around this time Mikio Sato proposes his hyperfunctions, which will turn out to have sheaf-theoretic nature.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology. It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke–Joyal semantics, but probably should be attributed to a number of authors).

• Coherent sheaf
• Gerbe
• Stack (mathematics)
• Sheaf of spectra
• Perverse sheaf
• Presheaf of spaces
• Constructible sheaf
• De Rham's theorem

{{Reflist}}

• {{Citation | last1=Bredon | first1=Glen E. | author1-link = Glen Bredon | title=Sheaf theory | publisher=Springer-Verlag | series=Graduate Texts in Mathematics | isbn=978-0-387-94905-5 | mr=1481706 | edition=2nd | year=1997 | volume=170}} (oriented towards conventional topological applications)
• {{cite journal

| url = http://www.ams.org/notices/201005/rtx100500632p.pdf

| last1 = de Cataldo

| first1 = Andrea Mark

| first2 = Luca

|last2=Migliorini

| title=What is a perverse sheaf?

| journal = Notices of the American Mathematical Society

| date = 2010

| volume = 57

| issue = 5

| pages = 632–4 |arxiv=1004.2983 |bibcode=2010arXiv1004.2983D

| mr = 2664042

}}

• {{Citation | last1=Godement | first1=Roger | author1-link = Roger Godement | title=Topologie algébrique et théorie des faisceaux | publisher=Hermann | location=Paris | mr=0345092 |orig-year=1973 |year=2006 |isbn=2705612521}}
• {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Sur quelques points d'algèbre homologique | mr=0102537 | year=1957 | journal=The Tohoku Mathematical Journal |series=Second Series | issn=0040-8735 | volume=9 | issue=2 | pages=119–221 | doi=10.2748/tmj/1178244839| doi-access=free }}
• {{Citation | last1=Hirzebruch | first1=Friedrich | author1-link = Friedrich Hirzebruch | title=Topological methods in algebraic geometry | publisher=Springer-Verlag | series=Classics in Mathematics | isbn=978-3-540-58663-0 | mr=1335917 | year=1995}} (updated edition of a classic using enough sheaf theory to show its power)
• {{Citation|last1=Iversen|first1=Birger|title=Cohomology of sheaves|series=Universitext|publisher=Springer|year=1986|isbn=3-540-16389-1|mr=842190|doi=10.1007/978-3-642-82783-9}}
• {{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | last2=Schapira | first2=Pierre | title=Sheaves on manifolds | publisher=Springer-Verlag | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-51861-7 | mr=1299726 | year=1994 | volume=292}} (advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces)
• {{Citation | last1=Mac Lane | first1=Saunders | author1-link = Saunders Mac Lane | last2=Moerdijk | first2=Ieke | author2-link = Ieke Moerdijk | title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory | publisher=Springer-Verlag | series=Universitext | isbn=978-0-387-97710-2 | mr=1300636 | year=1994}} (category theory and toposes emphasised)
• {{Citation | last1=Martin | first1=William T. | last2=Chern | first2=Shiing-Shen | author2-link=Shiing-Shen Chern | last3=Zariski | first3=Oscar | author3-link=Oscar Zariski | title=Scientific report on the Second Summer Institute, several complex variables | mr=0077995 | year=1956 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=62 | pages=79–141 | doi=10.1090/S0002-9904-1956-10013-X | issue=2| doi-access=free }}
• {{Citation|last1=Ramanan|first1=S.|title=Global calculus|series=Graduate Studies in Mathematics|volume=65|publisher=American Mathematical Society|year=2005|isbn=0-8218-3702-8|mr=2104612|doi=10.1090/gsm/065}}
• {{Citation |author1-link=J. Arthur Seebach |first1=J. Arthur |last1=Seebach |first2=Linda A. |last2=Seebach |author3-link=Lynn A. Steen |first3=Lynn A. |last3=Steen |year=1970 |title=What is a Sheaf |journal=American Mathematical Monthly |volume=77 |issue=7 |pages=681–703 |doi=10.1080/00029890.1970.11992563 |mr=0263073|s2cid=203043621 }}
• {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Faisceaux algébriques cohérents | url=http://www.mat.uniroma1.it/people/arbarello/FAC.pdf | mr=0068874 | year=1955 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=61 | pages=197–278 | doi=10.2307/1969915 | jstor=1969915 | issue=2 }}
• {{Citation | last1=Swan | first1=Richard G. | author-link=Richard Swan| title=The Theory of Sheaves | publisher=University of Chicago Press| year=1964 |isbn=9780226783291 |series=Chicago lectures in mathematics |edition=3 }} (concise lecture notes)
• {{Citation | last1=Tennison | first1=Barry R. | title=Sheaf theory | publisher=Cambridge University Press | mr=0404390 | year=1975 |url=https://books.google.com/books?id=JOglpKIw6kIC |isbn=978-0-521-20784-3 |volume=20 |series=London Mathematical Society Lecture Note Series }} (pedagogic treatment)
• {{Cite book |last=Rosiak |first=Daniel |url=https://direct.mit.edu/books/oa-monograph/5460/Sheaf-Theory-through-Examples |title=Sheaf theory through examples |date=2022 |isbn=978-0-262-37042-4 |location=Cambridge, Massachusetts |doi=10.7551/mitpress/12581.001.0001 |s2cid=253133215 |oclc=1333708310}} (introductory book with open access)

{{Authority control}}

*

Category:Topological methods of algebraic geometry

Category:Algebraic topology