In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sheaf is a tool for systematically tracking data (such as
sets,
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s,
rings) attached to the
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
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 is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
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
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
(or
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s) from one sheaf to another; sheaves (of a specific type, such as sheaves of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s) with their
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s on a fixed topological space form a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
. On the other hand, to each
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
there is associated both a
direct image functor In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
, taking sheaves and their morphisms on the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
to sheaves and morphisms on the
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
, and an
inverse image functor In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor ...
operating in the opposite direction. These
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s, 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
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. First, geometric structures such as that of a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
or a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as
vector bundles
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
or
divisors
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. Especially in algebraic geometry and the theory of
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
s, 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 equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. In addition, generalisations of sheaves to more general settings than topological spaces, such as
Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
, have provided applications to
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
and to
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
.
Definitions and examples
In many mathematical branches, several structures defined on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
(e.g., a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
) can be naturally ''localised'' or ''restricted'' to
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s
: typical examples include
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
-valued or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued functions,
-times
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
(real-valued or complex-valued) functions,
bounded real-valued functions,
vector fields, and
sections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of any
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
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
Let
be a topological space. A ''presheaf of sets''
on
consists of the following data:
*For each open set
of
, a set
. This set is also denoted
. The elements in this set are called the ''sections'' of
over
. The sections of
over
are called the ''global sections'' of
.
*For each inclusion of open sets
, a function
. In view of many of the examples below, the morphisms
are called ''restriction morphisms''. If
, then its restriction
is often denoted
by analogy with restriction of functions.
The restriction morphisms are required to satisfy two additional (
functorial
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
) properties:
*For every open set
of
, the restriction morphism
is the identity morphism on
.
*If we have three open sets
, then the
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
Informally, the second axiom says it doesn't 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 ''
'', one can assign the set
of continuous real-valued functions on ''
''. The restriction maps are then just given by restricting a continuous function on ''
'' to a smaller open subset ''
'', 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
and a sheaf of smooth functions
.
Another common class of examples is assigning to
the set of constant real-valued functions on
. This presheaf is called the ''constant presheaf'' associated to
and is denoted
.
Sheaves
Given a presheaf, a natural question to ask is to what extent its sections over an open set ''
'' are specified by their restrictions to smaller open sets
of an
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of ''
''. A ''sheaf'' is a presheaf that satisfies both of the following two additional axioms:
# (''Locality'') Suppose
is an open set,
is an open cover of
, and
are sections. If
for all
, then
.
# (
''Gluing'') Suppose
is an open set,
is an open cover of
, and
is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if
for all
, then there exists a section
such that
for all
.
The section ''
'' 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 ''
'' and ''
'' 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.
The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions
which agree on the intersections
, there is a unique continuous function
whose restriction equals the
. 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
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific cons ...
).
Presheaves and sheaves are typically denoted by capital letters,
being particularly common, presumably for the
French word for sheaf, ''faisceau''. Use of calligraphic letters such as
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
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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 , whose points are the
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s
in
. The open sets
form a basis for the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
on this space. Given an
-module
, there is a sheaf, denoted by
on the Spec
, that satisfies
:
the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
of
at
.
Further examples
Sheaf of sections of a continuous map
Any continuous map
of topological spaces determines a sheaf
on
by setting
:
Any such
is commonly called a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
of ''
'', and this example is the reason why the elements in
are generally called sections. This construction is especially important when
is the projection of a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the
trivial bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. Another example: the sheaf of sections of
:
is the sheaf which assigns to any ''
'' the set of branches of the
complex logarithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
on ''
''.
Given a point
and an abelian group
, the skyscraper sheaf
is defined as follows: if
is an open set containing
, then
. If
does not contain
, then
, the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
. The restriction maps are either the identity on
, if both open sets contain
, or the zero map otherwise.
Sheaves on manifolds
On an
-dimensional
-manifold
, there are a number of important sheaves, such as the sheaf of
-times continuously differentiable functions
(with
). Its sections on some open
are the
-functions
. For
, this sheaf is called the ''structure sheaf'' and is denoted
. The nonzero
functions also form a sheaf, denoted
.
Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s (of degree
) also form a sheaf
. In all these examples, the restriction morphisms are given by restricting functions or forms.
The assignment sending
to the compactly supported functions on
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
In topology, a branch of mathematics, a cosheaf with values in an ∞-category ''C'' that admits colimits is a functor ''F'' from the category of open subsets of a topological space ''X'' (more precisely its nerve) to ''C'' such that
*(1) The ''F ...
, a
dual concept where the restriction maps go in the opposite direction than with sheaves. 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
be the
two-point topological space with the discrete topology. Define a presheaf
as follows:
The restriction map
is the projection of
onto its first coordinate, and the restriction map
is the projection of
onto its second coordinate.
is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over
and
determine only two of those numbers. So while we can glue any two sections over
and
, we cannot glue them uniquely.
* Let
be the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, and let
be the set of
bounded continuous functions on
. This is not a sheaf because it is not always possible to glue. For example, let
be the set of all
such that