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.
![F (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/462254eb7da92fe91d24532826f4e036cf1eeaa4)
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
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:
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![U \ subseteq X](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723)
![{\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9)
![\ rho _ {V} ^ {U} \ colon {\ mathcal F} (U) \ to {\ mathcal F} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f065a80d11beb3d0c3749a53ad4efd853d49e8)
![V \ subseteq U](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a)
![\ rho _ {U} ^ {U} = {\ mathrm {id}} _ {{{\ mathcal F} (U)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/063f5c21e592d4135d09c6fe2894e93902738a1f)
-
for open subsets .![W \ subseteq V \ subseteq U](https://wikimedia.org/api/rest_v1/media/math/render/svg/7945770f01cf7cc73bd3bd966e634c8041e0384b)
The elements of are called (local) sections from over , the elements of global sections . Instead of writing, too![{\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![{\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9)
One also writes for the restriction of a cut to an open subset .
![\ rho _ {V} ^ {U} (f)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a53c7ce9bdcb56617752eaf62d3f9d7702986f)
![\, f \ in {\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/26fe286100ea47b28f4d0ae10623cd3e1e4919e5)
![V \ subseteq U](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a)
![\, f | _ {V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b52a0ded1810fa3b89ddd91715601be3c586d5c)
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
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![\ {V_ {i} \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95e6899932d424007ae7073199094557dda5963f)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![f | _ {{V_ {i}}} = g | _ {{V_ {i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/112cb4eded12fbef55bf8b5c910afb191cdd2df2)
- for all so true .
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
![f = g](https://wikimedia.org/api/rest_v1/media/math/render/svg/795e79b6da5372a37ba3a36db68e43806232aac3)
- 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
![f_ {i} \ in {\ mathcal F} (V_ {i})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7621eb76a2a50d6587190a4fd7ab5d6a0decb85a)
![f_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79)
![f_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acc195ab3f9d65994b47774eb013601d09217aee)
![V_ {i} \ cap V_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/222e36e2144771b89dc2b4fd4ecf162f02c7be2f)
![f \ in {\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/080b42e1c695d953d3185c05455aeb0ce1134616)
![f_ {i} = f | _ {{V_ {i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bfae08769f2af23275c4779b264b795fcd3429)
- applies to all .
![i](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
From the first condition it follows that in the second condition it is uniquely determined by the .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![f_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79)
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 .
![{\ mathbf {Ouv}} (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b15ade5210498fdecf24e247ad580fa8ab3ec890)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![U \ to V](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8aade74fbc3d516467efd200969ce325f5425f4)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ mathcal F} \ colon {\ mathbf {Ouv}} (X) \ to C](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c253ac4c32ddc13d63708ce66c6a425a4654a80)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
A sheaf is called a sheaf if the following diagram is exact for every open subset and every overlap of :
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![\ {V_ {i} \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95e6899932d424007ae7073199094557dda5963f)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![{\ mathcal F} (U) \ rightarrow \ prod {\ mathcal F} (V_ {i}) \, {\ begin {matrix} {{{} \ atop \ longrightarrow} \ atop {\ longrightarrow \ atop {}} } \ end {matrix}} \, \ prod {\ mathcal F} (V_ {i} \ cap V_ {j}),](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc3fee0e7e4f8cec0c4f1bdce6e7ee116e47b19)
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 .
![{\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9)
![(i, j)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712)
![{\ displaystyle \ iota _ {1} ^ {(i, j)}: V_ {i} \ cap V_ {j} \ rightarrow V_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96004396be43f828b0f4cbdaf07c17b9a7424413)
![{\ displaystyle \ iota _ {2} ^ {(i, j)}: V_ {i} \ cap V_ {j} \ rightarrow V_ {j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eca31720183913844766d4f5f4456e7c94e06c7)
![{\ displaystyle {\ mathcal {F}} (\ iota _ {1} ^ {(i, j)}): {\ mathcal {F}} (V_ {i}) \ rightarrow {\ mathcal {F}} ( V_ {i} \ cap V_ {j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30628f9c9cf1f9ecf7aee13561324f6bbb794986)
![{\ displaystyle {\ mathcal {F}} (\ iota _ {2} ^ {(i, j)})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724e46ccde5d0fbd68460fcd56dfedbe8953e12a)
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.
![\in](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f)
![{\ mathcal F} (U) \ rightarrow \ prod {\ mathcal F} (V_ {i}) \, {{{} \ atop \ longrightarrow} \ atop {\ longrightarrow \ atop {}}} \, \ prod { \ mathcal F} (V_ {i} \ times V_ {j}),](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2deb5f11ff469a6aaa318d871e6a3a9ed7f9a8)
![{\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9)
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 .
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![\ mathcal {G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
![\ varphi \ colon {\ mathcal {F}} \ to {\ mathcal {G}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43fa5791552824aebc3809efaeb720f17d6abdc4)
![\ varphi (U) \ colon {\ mathcal {F}} (U) \ to {\ mathcal {G}} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b33adaca5de58bfaf5b94c750bf6f2df33005e15)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![V \ subseteq U](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c76555365eca037c1a0aa64b6ad5403ba32d9a)
![{\ tilde {\ rho}} _ {V} ^ {U} \ circ \ varphi (U) = \ varphi (V) \ circ \ rho _ {V} ^ {U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddbe71c0d320fbe75d5701a69eda79b67a4bd17)
![\ rho _ {V} ^ {U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2f2676e6c08ed8c62b5d572a9c4ba2a515aef0)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![{\ tilde {\ rho}} _ {V} ^ {U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79f010fc74d7484a113a162c9b91b3ef2ca46cde)
![\ mathcal {G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
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.
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
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.
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
For each point , the stalk of a preave is defined as
in the point
![{\ mathcal F} _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66d72460f7036cb595f6158fd0e9e125a595cf2a)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![{\ displaystyle {\ mathcal {F}} _ {x} = \ operatorname {colim} _ {V \ ni x} {\ mathcal {F}} (V).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da7f175d329332728b2cbf5481557c2523cbf18)
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.
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
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![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ mathbf a} {\ mathcal F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88ef2e19244dcdea638c6deb487d2b71e621ae0d)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![{\ mathrm {Hom}} _ {{{\ mathrm {(sheaves)}}}} ({\ mathbf a} {\ mathcal F}, {\ mathcal G}) = {\ mathrm {Hom}} _ {{ {\ mathrm {(Pr {\ ddot a} garben)}}}} ({\ mathcal F}, {\ mathcal G})](https://wikimedia.org/api/rest_v1/media/math/render/svg/d25a1b15f6946d1db2fcc290a9f6bcdc2bc3776b)
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
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![f \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e)
![U \ mapsto {\ mathcal F} (f ^ {{- 1}} (U)), \ quad U \ subseteq Y \ {\ mathrm {open}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b358b142a0f5acb2d12e262cb0f5badeec562915)
a sheaf in which with is called and direct image or picture sheaf of under is.
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![f _ {*} {\ mathcal F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e47cfc4575c0579289f37edf1aac078462741b99)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
If a sheaf is open , the associated sheaf is closed
![{\ mathcal G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
![Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
![U \ mapsto \ operatorname {colim} _ {{V \ supseteq f (U)}} {\ mathcal G} (V)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9a97f18c8aa6a28d8c4c96d88ff5019ef3ff7a)
a sheaf , the archetype , which is referred to with.
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![f ^ {{- 1}} {\ mathcal G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a36121bcf46bedbfe1e77c280a27dacc246bca22)
If there is another continuous map, then are the functors
![g \ colon Y \ to Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/6268a0bd7f3016093edf824725f1ffcba89f8064)
-
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
![\, f _ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/433ffa8abe894abd38067aad3d1e5c91d5b88961)
![\, f ^ {{- 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad450d89085dd5a2e505f55b45bdb929ae344d84)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ mathcal G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
![\, Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/db3a708f29d3a9598de5bc4466cf8fc471140f31)
![{\ mathrm {Hom}} (f ^ {{- 1}} {\ mathcal G}, {\ mathcal F}) = {\ mathrm {Hom}} ({\ mathcal G}, f _ {*} {\ mathcal F}).](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e502227086fe248b6c22c47543a0264894f4d3)
Straws are special cooking archetypes: Refers to the inclusion of a point, is so
![i_ {y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1949859e360e01347e3ad283bf8dcda5a1ec4b97)
![\ {y \} \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd474b71bb20ec197f91ec5b6dd6667a91de1ad)
![{\ mathcal G} _ {y} = i_ {y} ^ {{- 1}} {\ mathcal G};](https://wikimedia.org/api/rest_v1/media/math/render/svg/589d7ed6ce62ba65f162a86bec045a66be424dec)
The sheaf was identified in the single-point space with its global sections. As a result, the sheaf primitive is compatible with straws:
![i_ {y} ^ {{- 1}} {\ mathcal G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/199374a3a2d7399f2350f38a590ebb062dfb0ee3)
![\, \ {y \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44a3ae62a68c9e894e526b8891b6ab10c2be50)
![\, (f ^ {{- 1}} {\ mathcal G}) _ {x} = {\ mathcal G} _ {{f (x)}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/53285cdab1af5a385bd9718a91f160057efd7af2)
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 .
![\, f ^ {{- 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad450d89085dd5a2e505f55b45bdb929ae344d84)
![\, f _ {*}](https://wikimedia.org/api/rest_v1/media/math/render/svg/433ffa8abe894abd38067aad3d1e5c91d5b88961)
The étalé room of a sheaf
A sheaf of sets is a topological space is about defined as follows:
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
- The underlying set is the disjoint union of all stalks of ; the picture fancy on from.
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![E \ to X](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e385963fc9a729e22866cd3c05d5002c361bbf)
![{\ mathcal F} _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66d72460f7036cb595f6158fd0e9e125a595cf2a)
![x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d)
- The topology on is the strongest topology for which the mappings
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![U \ to E, \ quad x \ mapsto f_ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/468b158f2401be4ff7fe06bc1c8ec84d9bd505ce)
- are continuous for every cut over an open set .
![f \ in {\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/080b42e1c695d953d3185c05455aeb0ce1134616)
![U \ subseteq X](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723)
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 .
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
![U \ subseteq X](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723)
![\ pi \ colon E \ to X](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9d6e10a97de4dd6d790d2ef4a13adf0e38052f)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
![s \ colon U \ to E](https://wikimedia.org/api/rest_v1/media/math/render/svg/285f7ec4952d1b85aea1db7e2375d7b856fe64ce)
![\ pi \ circ s](https://wikimedia.org/api/rest_v1/media/math/render/svg/aea41c995f08ae2f07111c8dd3091182eb09e1a9)
![U \ subseteq X](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723)
This space is called the étalé space ( French étalé = expanded) or, in German-language literature, written without accents , the etale space.
![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
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 .
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![\ mathbb {Z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc)
![U = U_ {1} \ cup U_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8a929d71b3a7eff52488b00d048edb446eef06)
![U_ {1} = (1,2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5b01fc239f3da490530614aa18d9a2718045ad)
![U_ {2} = (3,4)](https://wikimedia.org/api/rest_v1/media/math/render/svg/82697a1cdfe590c78506fbf9cfe9e75a38998c7f)
![5](https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b)
![U_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9e7f892894bc50c32ce1b9f9a68a15562146ac)
![7th](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee716ec61382a6b795092c0edd859d12e64cbba8)
![U_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590fa6a550fbe2866a28243a733d54245d218b9d)
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
- 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 ).
![{\ mathcal O}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f)
![{\ mathbb C} \ {z \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09f44495516f49ee629b7f5d0fafd7214c61915c)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![za](https://wikimedia.org/api/rest_v1/media/math/render/svg/7de244374720ea1ce9df3f3191b9e8b547da130b)
- 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
![X = \ {\ eta, s \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5792d228035265f6d9e0edd1cf51935b532ffa)
![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632)
![\ eta](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64)
![M = \ Gamma (X, {\ mathcal F})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8138a94cb0f4e72b9279356c250d44d416e6a167)
![N = \ Gamma (\ {\ eta \}, {\ mathcal F})](https://wikimedia.org/api/rest_v1/media/math/render/svg/56a411f68b5b35a94599bf9eb21bace418a3d49f)
![\ rho \ colon M \ to N](https://wikimedia.org/api/rest_v1/media/math/render/svg/619fbeab4ccd6eff18c6eeec1c12d62520d39edf)
![{\ mathcal {F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676)
-
and .![{\ mathcal F} _ {s} = M](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba76c4cecd45d1db804fb5021994d72e1553ae9)
- 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.
![X = {\ mathbb {R}} / {\ mathbb {Z}} \ cong S ^ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b84ef97646de94866da1f1cccfc4eb5a2689c56d)
![U \ subseteq X](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4b6027fd83ac32e7b4f4113c60a69041292723)
![{\ mathcal F} (U)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089afc31cc951f7745f3ecfa80118767b291de9)
![f \ colon U \ to {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d71097bf5f00c4116453f42be6ed7f74e74b3be8)
![f (x + \ varepsilon + {\ mathbb {Z}}) = f (x + {\ mathbb {Z}}) + \ varepsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/03461bada6d5e578bb3fbac031b66b721efbf411)
![| \ varepsilon |](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa40e1fc28e545dbc83f767a496b23373bc020f)
![{\ mathcal F} _ {x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66d72460f7036cb595f6158fd0e9e125a595cf2a)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![\ Gamma (U, {\ mathcal F}) \ cong {\ mathbb R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41476e50938e80d243c2e7c03b270dcc620bafc1)
![U \ subsetneq X](https://wikimedia.org/api/rest_v1/media/math/render/svg/69450bfe9354caa3f3a887f1f2426ae9c3d0b6a7)
![\ Gamma (X, {\ mathcal F}) = \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0b5bf3f72be6c26ffc1f5b87ee99e638f6ef2d)
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