A sheaf is a term used in various areas of mathematics such as algebraic geometry and function theory . A sheaf of Abelian groups over a topological space consists of one Abelian group for each open subset of the base space and compatible constraint homomorphisms between these Abelian groups. Similarly, a sheaf of rings consists of one ring for each open subset and ring homomorphisms. The simplest example of a sheaf is the sheaf of continuous real-valued functions on open subsets of a topological space together with the restriction of the functions to smaller open subsets. Markings can be defined on any category . Sheaves can be defined on any site (that is a category on which a Grothendieck topology is explained).
Definitions
In order to understand the definition of the sheaf, it is advisable to keep the example of the sheaf of continuous functions in mind: if then the set of continuous functions , the constraint maps (pictures of the inclusion maps below the functor ) are simply the constraints of the functions smaller areas.
Marking on a topological space
A preamble on a topological space assigns a set (or an Abelian group, a module, a ring) to each open subset together with restriction maps for all inclusions of open subsets . The restriction mappings (in the case of Abelian groups, modules or rings must be corresponding homomorphisms and) fit together in the "obvious" way:
-
for open subsets .
The elements of are called (local) sections from over , the elements of global sections . Instead of writing, too
One also writes for the restriction of a cut to an open subset .
Sheaf on a topological space
A sheaf is one where the data is "local"; H. the following two conditions are met:
- Local agreement implies global agreement: are and intersections of over and an open coverage of , and holds
- for all so true .
- Matching local data can be "glued": If cuts are given so that the restrictions from and to coincide, then there is a cut so that
- applies to all .
From the first condition it follows that in the second condition it is uniquely determined by the .
Category-theoretical definition of a sheaf in a topological space
It is a topological space. The category has as objects the open subsets of with a morphism for each inclusion of open sets. A Prägarbe on with values in a category is a contravariant functor . own products .
A sheaf is called a sheaf if the following diagram is exact for every open subset and every overlap of :
d. That is, that the difference kernel is the two right arrows, which can be explained as follows. For each index pair there are two inclusions and . One of the arrows is the product of , the other is the product of .
Sheaf on a category, sheaf on a site
A preamble on a category C is a contravariant functor : C A into a category A , say the category of sets or the category of Abelian groups. If C has a Grothendieck topology, then a pregroove is called a sheaf if for every covering family {φ i : V i U } i I the sequence: is exact, i.e. H. when the difference kernel of the two right arrows is.
As in the case of a topological space, you can garbage marks. You can also develop different theories of cohomology, such as Čech cohomology .
The totality of all sheaves on a site forms a topos .
Morphisms
Just as a sheaf is a collection of objects, an inter-sheaf morphism is a collection of morphisms of those objects. This must be compatible with the restriction images.
Let and sheaves on with values in the same category. A morphism consists of a collection of morphisms , one for each open subset of , so that for each inclusion of open subsets the condition is met. Here, the restriction mapping from and from denotes .
If one understands the sheaves as functors, as described above, then a morphism between the sheaves is the same as a natural transformation of the functors.
For each category , the -valent sheaves with this morphism concept form a category.
Stalks and germs
Let it be a category of algebraic structures that are defined by finite projective limits, e.g. B. (Abelian) groups, rings, modules. In particular, pseudofiltration colimites exist in , and their underlying quantities match the colimites of the underlying quantities of the individual objects.
For each point , the stalk of a preave is defined as
in the point
Elements of the stalk are called germs .
Seeds are thus equivalence classes of local sections over open environments of , where sections are equivalent if they become equal when restricted to a smaller environment.
Yellowing
If a sheaf is on a topological space , there is a sheaf , the sheaf of or associated sheaf to , so that for each sheaf
applies. is thus left adjoint to the forget function
There is no uniform notation for the Vergarbungsfunktor.
Direct images and archetypes
If a sheaf on a topological space and a continuous mapping, then is
a sheaf in which with is called and direct image or picture sheaf of under is.
If a sheaf is open , the associated sheaf is closed
a sheaf , the archetype , which is referred to with.
If there is another continuous map, then are the functors
-
and
as well as the functors
-
and
of course equivalent.
The functors and are adjoint : If a sheaf is open and a sheaf is open , then is
Straws are special cooking archetypes: Refers to the inclusion of a point, is so
The sheaf was identified in the single-point space with its global sections. As a result, the sheaf primitive is compatible with straws:
This relationship is also the reason why, despite the more complicated definition, the functor is easier to understand: in a sense, cohomology is the study of the functor .
The étalé room of a sheaf
A sheaf of sets is a topological space is about defined as follows:
- The underlying set is the disjoint union of all stalks of ; the picture fancy on from.
- The topology on is the strongest topology for which the mappings
- are continuous for every cut over an open set .
Then there is a bijection between the cuts from over an open lot and the cuts from over , i.e. H. the continuous mappings for which the inclusion equals .
This space is called the étalé space ( French étalé = expanded) or, in German-language literature, written without accents , the etale space.
Examples
- The continuous functions with a compact carrier do not form a marker, because the restriction of a function with a compact carrier to an open subset generally does not have compact carriers again.
- The sheaf that assigns each open subset of the Abelian group is not a sheaf: If with and , then the cut over and the cut over cannot be "glued" to a cut over .
- The sheaf of holomorphic functions on is a sheaf of rings (a sheaf of rings): the stalk at the zero point can be identified with the ring of convergent power series , i.e. H. of the power series whose radius of convergence is not zero. The other stalks are created by changing coordinates (i.e. replace with ).
- Let it be the topological space with two points, one of which is closed and not, i.e. H. the Sierpiński room . Then a sheaf is determined by the two sets and together with an image , and vice versa, these data can be given as desired and a sheaf is obtained. The stalks of are
-
and .
- It should be and to open is the set of all functions that local slope 1 have, these are all with , if both sides are defined and is sufficiently small. This is a sheaf in which every stalk is isomorphic to and also for every connected open real subset . However, there is no global cuts . As a result, this is “only” a quantity-valued sheaf and not an Abelian-group value.
generalization
The term sheaf can be understood more generally in the context of Grothendieck topologies .
See also
literature
- Francisco Miraglia: An Introduction to Partially Ordered Structures and Sheaves. Polimetrica, Milan 2006, ISBN 88-7699-035-6 ( Contemporary Logic ).
Individual evidence
-
↑ F. Constantinescu, HF de Groote: Geometrical and algebraic methods of physics: Supermanifolds and Virasoro algebras, Teubner Study Books 1994, ISBN 978-3-519-02087-5