In
mathematics, the gluing axiom is introduced to define what a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper s ...
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 po ...
must satisfy, given that it is a
presheaf
In mathematics, a sheaf 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 ...
, which is by definition a
contravariant functor
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 ...
:
to a category
which initially one takes to be the
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
. Here
is the
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
of
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 a ...
s of
ordered by
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
s; and considered as a category in the standard way, with a unique
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, morphis ...
:
if
is a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
of
, and none otherwise.
As phrased in the
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper s ...
article, there is a certain axiom that
must satisfy, for any
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_\alph ...
of an open set of
. For example, given open sets
and
with
union and
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
, the required condition is that
:
is the subset of
With equal image in
In less formal language, 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
over
is equally well given by a pair of sections :
on
and
respectively, which 'agree' in the sense that
and
have a common image in
under the respective restriction maps
:
and
:
.
The first major hurdle in sheaf theory is to see that this ''gluing'' or ''patching'' axiom is a correct abstraction from the usual idea in geometric situations. For example, a
vector field is a section of a
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...
on a
smooth 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 m ...
; this says that a vector field on the union of two open sets is (no more and no less than) vector fields on the two sets that agree where they overlap.
Given this basic understanding, there are further issues in the theory, and some will be addressed here. A different direction is that of the
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 c ...
, and yet another is the logical status of 'local existence' (see
Kripke–Joyal semantics).
Removing restrictions on ''C''
To rephrase this definition in a way that will work in any category
that has sufficient structure, we note that we can write the objects and morphisms involved in the definition above in a diagram which we will call (G), for "gluing":
:
Here the first map is the product of the restriction maps
:
and each pair of arrows represents the two restrictions
:
and
:
.
It is worthwhile to note that these maps exhaust all of the possible restriction maps among
, the
, and the
.
The condition for
to be a sheaf is that for any open set
and any collection of open sets
whose union is
, the diagram (G) above is an
equalizer.
One way of understanding the gluing axiom is to notice that
is the
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...
of the following diagram:
:
The gluing axiom says that
turns colimits of such diagrams into limits.
Sheaves on a basis of open sets
In some categories, it is possible to construct a sheaf by specifying only some of its sections. Specifically, let
be a topological space with
basis . We can define a category to be the full subcategory of
whose objects are the
. A B-sheaf on
with values in
is a contravariant functor
:
which satisfies the gluing axiom for sets in
. That is, on a selection of open sets of
,
specifies all of the sections of a sheaf, and on the other open sets, it is undetermined.
B-sheaves are equivalent to sheaves (that is, the category of sheaves is equivalent to the category of B-sheaves).
[Vakil]
Math 216: Foundations of algebraic geometry
2.7. Clearly a sheaf on
can be restricted to a B-sheaf. In the other direction, given a B-sheaf
we must determine the sections of
on the other objects of
. To do this, note that for each open set
, we can find a collection
whose union is
. Categorically speaking, this choice makes
the colimit of the full subcategory of
whose objects are
. Since
is contravariant, we define
to be the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of the
with respect to the restriction maps. (Here we must assume that this limit exists in
.) If
is a basic open set, then
is a terminal object of the above subcategory of
, and hence
. Therefore,
extends
to a presheaf on
. It can be verified that
is a sheaf, essentially because every element of every open cover of
is a union of basis elements (by the definition of a basis), and every pairwise intersection of elements in an open cover of
is a union of basis elements (again by the definition of a basis).
The logic of ''C''
The first needs of sheaf theory were for 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 com ...
s; so taking the category
as the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of ...
was only natural. In applications to geometry, for example
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 ...
s and
algebraic geometry, the idea of a ''sheaf of
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
s'' is central. This, however, is not quite the same thing; one speaks instead of a
locally ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
, because it is not true, except in trite cases, that such a sheaf is a functor into a
category of local rings
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) ...
. It is the ''stalks'' of the sheaf that are local rings, not the collections of ''sections'' (which are
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, but in general are not close to being ''local''). We can think of a locally ringed space
as a parametrised family of local rings, depending on
in
.
A more careful discussion dispels any mystery here. One can speak freely of a sheaf of abelian groups, or rings, because those are
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s (defined, if one insists, by an explicit
signature
A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
). Any category
having
finite products supports the idea of a
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
, which some prefer just to call a group ''in''
. In the case of this kind of purely algebraic structure, we can talk ''either'' of a sheaf having values in the category of abelian groups, or an ''abelian group in the category of sheaves of sets''; it really doesn't matter.
In the local ring case, it does matter. At a foundational level we must use the second style of definition, to describe what a local ring means in a category. This is a logical matter: axioms for a local ring require use of
existential quantification
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, wh ...
, in the form that for any
in the ring, one of
and
is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
. This allows one to specify what a 'local ring in a category' should be, in the case that the category supports enough structure.
Sheafification
To turn a given presheaf
into a sheaf
, there is a standard device called ''sheafification'' or ''sheaving''. The rough intuition of what one should do, at least for a presheaf of sets, is to introduce an equivalence relation, which makes equivalent data given by different covers on the overlaps by refining the covers. One approach is therefore to go to the
stalk
Stalk or stalking may refer to:
Behaviour
* Stalk, the stealthy approach (phase) of a predator towards its prey
* Stalking, an act of intrusive behaviour or unwanted attention towards a person
* Deer stalking, the pursuit of deer for sport
Biol ...
s and recover the
sheaf space of the ''best possible'' sheaf
produced from
.
This use of language strongly suggests that we are dealing here with
adjoint functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
. Therefore, it makes sense to observe that the sheaves on
form a
full subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...
of the presheaves on
. Implicit in that is the statement that a
morphism of sheaves
In mathematics, a sheaf 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 ...
is nothing more than a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a na ...
of the sheaves, considered as functors. Therefore, we get an abstract characterisation of sheafification as
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
to the inclusion. In some applications, naturally, one does need a description.
In more abstract language, the sheaves on
form a
reflective subcategory
In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A ...
of the presheaves (Mac Lane–
Moerdijk
Moerdijk () is a municipality and a town in the South of the Netherlands, in the province of North Brabant.
History
The municipality of Moerdijk was founded in 1997 following the merger of the municipalities of Fijnaart en Heijningen, Klunde ...
''Sheaves in Geometry and Logic'' p. 86). In
topos theory
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
, for a
Lawvere–Tierney topology and its sheaves, there is an analogous result (ibid. p. 227).
Other gluing axioms
The gluing axiom of sheaf theory is rather general. One can note that the
Mayer–Vietoris axiom
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor ''F'' on the homotopy category ''Hotc'' of pointed connected CW complexes, to the category of sets Set, to be ...
of
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
, for example, is a special case.
See also
*
Gluing schemes
In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps.
Statement
Suppose there is a (possibly infinite) family of schemes \_ and for pairs i, j, there are open subsets U_ ...
Notes
References
*
{{DEFAULTSORT:Gluing Axiom
General topology
Limits (category theory)
Homological algebra
Mathematical axioms
Differential topology