Sheaf (math)

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. ${\displaystyle F(U)}$ ${\displaystyle U\to \mathbb {R} }$ ${\displaystyle F}$

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: ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle \rho _{V}^{U}\colon {\mathcal {F}}(U)\to {\mathcal {F}}(V)}$ ${\displaystyle V\subseteq U}$

• ${\displaystyle \rho _{U}^{U}=\mathrm {id} _{{\mathcal {F}}(U)}}$
• ${\displaystyle \rho _{W}^{V}\circ \rho _{V}^{U}=\rho _{W}^{U}}$for open subsets . ${\displaystyle W\subseteq V\subseteq U}$

The elements of are called (local) sections from over , the elements of global sections . Instead of writing, too ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {F}}(X)}$ ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle \Gamma (U,{\mathcal {F}}).}$

One also writes for the restriction of a cut to an open subset . ${\displaystyle \rho _{V}^{U}(f)}$ ${\displaystyle \,f\in {\mathcal {F}}(U)}$ ${\displaystyle V\subseteq U}$ ${\displaystyle \,f|_{V}}$

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 ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle U}$ ${\displaystyle \{V_{i}\}}$ ${\displaystyle U}$

${\displaystyle f|_{V_{i}}=g|_{V_{i}}}$

for all so true . ${\displaystyle i}$ ${\displaystyle f=g}$

• 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 ${\displaystyle f_{i}\in {\mathcal {F}}(V_{i})}$ ${\displaystyle f_{i}}$ ${\displaystyle f_{j}}$ ${\displaystyle V_{i}\cap V_{j}}$ ${\displaystyle f\in {\mathcal {F}}(U)}$

${\displaystyle f_{i}=f|_{V_{i}}}$

applies to all . ${\displaystyle i}$

From the first condition it follows that in the second condition it is uniquely determined by the . ${\displaystyle f}$ ${\displaystyle f_{i}}$

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 . ${\displaystyle X}$ ${\displaystyle \mathbf {Ouv} (X)}$ ${\displaystyle X}$ ${\displaystyle U\to V}$ ${\displaystyle U\subseteq V}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {F}}\colon \mathbf {Ouv} (X)\to C}$ ${\displaystyle C}$

A sheaf is called a sheaf if the following diagram is exact for every open subset and every overlap of : ${\displaystyle {\mathcal {F}}}$ ${\displaystyle U\subseteq X}$ ${\displaystyle \{V_{i}\}}$ ${\displaystyle U}$

${\displaystyle {\mathcal {F}}(U)\rightarrow \prod {\mathcal {F}}(V_{i})\,{\begin{matrix}{{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\end{matrix}}\,\prod {\mathcal {F}}(V_{i}\cap V_{j}),}$

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 . ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle (i,j)}$ ${\displaystyle \iota _{1}^{(i,j)}:V_{i}\cap V_{j}\rightarrow V_{i}}$ ${\displaystyle \iota _{2}^{(i,j)}:V_{i}\cap V_{j}\rightarrow V_{j}}$ ${\displaystyle {\mathcal {F}}(\iota _{1}^{(i,j)}):{\mathcal {F}}(V_{i})\rightarrow {\mathcal {F}}(V_{i}\cap V_{j})}$ ${\displaystyle {\mathcal {F}}(\iota _{2}^{(i,j)})}$

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. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle \rightarrow }$ ${\displaystyle \rightarrow }$ _{${\displaystyle \in }$} ${\displaystyle {\mathcal {F}}(U)\rightarrow \prod {\mathcal {F}}(V_{i})\,{{{} \atop \longrightarrow } \atop {\longrightarrow \atop {}}}\,\prod {\mathcal {F}}(V_{i}\times V_{j}),}$ ${\displaystyle {\mathcal {F}}(U)}$

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 . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle X}$ ${\displaystyle \varphi \colon {\mathcal {F}}\to {\mathcal {G}}}$ ${\displaystyle \varphi (U)\colon {\mathcal {F}}(U)\to {\mathcal {G}}(U)}$ ${\displaystyle U}$ ${\displaystyle X}$ ${\displaystyle V\subseteq U}$ ${\displaystyle {\tilde {\rho }}_{V}^{U}\circ \varphi (U)=\varphi (V)\circ \rho _{V}^{U}}$ ${\displaystyle \rho _{V}^{U}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\tilde {\rho }}_{V}^{U}}$ ${\displaystyle {\mathcal {G}}}$

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. ${\displaystyle C}$ ${\displaystyle C}$

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. ${\displaystyle C}$ ${\displaystyle C}$

For each point , the stalk of a preave is defined as in the point ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {F}}_{x}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle x}$

${\displaystyle {\mathcal {F}}_{x}=\operatorname {colim} _{V\ni x}{\mathcal {F}}(V).}$

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. ${\displaystyle x}$

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 ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle \mathbf {a} {\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {G}}}$

${\displaystyle \mathrm {Hom} _{\mathrm {(Garben)} }(\mathbf {a} {\mathcal {F}},{\mathcal {G}})=\mathrm {Hom} _{\mathrm {(Pr{\ddot {a}}garben)} }({\mathcal {F}},{\mathcal {G}})}$

applies. is thus left adjoint to the forget function ${\displaystyle \mathbf {a} }$ ${\displaystyle \mathrm {(Garben)} \to \mathrm {(Pr{\ddot {a}}garben)} .}$

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 ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle f\colon X\to Y}$

${\displaystyle U\mapsto {\mathcal {F}}(f^{-1}(U)),\quad U\subseteq Y\ \mathrm {offen} }$

a sheaf in which with is called and direct image or picture sheaf of under is. ${\displaystyle Y}$ ${\displaystyle f_{*}{\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle f}$

If a sheaf is open , the associated sheaf is closed ${\displaystyle {\mathcal {G}}}$ ${\displaystyle Y}$

${\displaystyle U\mapsto \operatorname {colim} _{V\supseteq f(U)}{\mathcal {G}}(V)}$

a sheaf , the archetype , which is referred to with. ${\displaystyle X}$ ${\displaystyle f^{-1}{\mathcal {G}}}$

If there is another continuous map, then are the functors ${\displaystyle g\colon Y\to Z}$

${\displaystyle \,(gf)_{*}}$ and ${\displaystyle \,g_{*}f_{*}}$

as well as the functors

${\displaystyle \,(gf)^{-1}}$ and ${\displaystyle \,f^{-1}g^{-1}}$

of course equivalent.

The functors and are adjoint : If a sheaf is open and a sheaf is open , then is ${\displaystyle \,f_{*}}$ ${\displaystyle \,f^{-1}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \,Y}$

${\displaystyle \mathrm {Hom} (f^{-1}{\mathcal {G}},{\mathcal {F}})=\mathrm {Hom} ({\mathcal {G}},f_{*}{\mathcal {F}}).}$

Straws are special cooking archetypes: Refers to the inclusion of a point, is so ${\displaystyle i_{y}}$ ${\displaystyle \{y\}\to Y}$

${\displaystyle {\mathcal {G}}_{y}=i_{y}^{-1}{\mathcal {G}};}$

The sheaf was identified in the single-point space with its global sections. As a result, the sheaf primitive is compatible with straws: ${\displaystyle i_{y}^{-1}{\mathcal {G}}}$ ${\displaystyle \,\{y\}}$

${\displaystyle \,(f^{-1}{\mathcal {G}})_{x}={\mathcal {G}}_{f(x)}.}$

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 . ${\displaystyle \,f^{-1}}$ ${\displaystyle \,f_{*}}$

The étalé room of a sheaf

A sheaf of sets is a topological space is about defined as follows: ${\displaystyle {\mathcal {F}}}$ ${\displaystyle E}$ ${\displaystyle X}$

• The underlying set is the disjoint union of all stalks of ; the picture fancy on from. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle E\to X}$ ${\displaystyle {\mathcal {F}}_{x}}$ ${\displaystyle x\in X}$
• The topology on is the strongest topology for which the mappings ${\displaystyle E}$

${\displaystyle U\to E,\quad x\mapsto f_{x}}$

are continuous for every cut over an open set . ${\displaystyle f\in {\mathcal {F}}(U)}$ ${\displaystyle U\subseteq X}$

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 . ${\displaystyle {\mathcal {F}}}$ ${\displaystyle U\subseteq X}$ ${\displaystyle \pi \colon E\to X}$ ${\displaystyle U}$ ${\displaystyle s\colon U\to E}$ ${\displaystyle \pi \circ s}$ ${\displaystyle U\subseteq X}$

This space is called the étalé space ( French étalé = expanded) or, in German-language literature, written without accents , the etale space. ${\displaystyle E}$

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 . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle U=U_{1}\cup U_{2}}$ ${\displaystyle U_{1}=(1,2)}$ ${\displaystyle U_{2}=(3,4)}$ ${\displaystyle 5}$ ${\displaystyle U_{1}}$ ${\displaystyle 7}$ ${\displaystyle U_{2}}$ ${\displaystyle U}$

• 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 ). ${\displaystyle {\mathcal {O}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} \{z\}}$ ${\displaystyle z}$ ${\displaystyle z-a}$

• 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 ${\displaystyle X=\{\eta ,s\}}$ ${\displaystyle s}$ ${\displaystyle \eta }$ ${\displaystyle M=\Gamma (X,{\mathcal {F}})}$ ${\displaystyle N=\Gamma (\{\eta \},{\mathcal {F}})}$ ${\displaystyle \rho \colon M\to N}$ ${\displaystyle {\mathcal {F}}}$

${\displaystyle {\mathcal {F}}_{\eta }=N}$and . ${\displaystyle {\mathcal {F}}_{s}=M}$

• 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. ${\displaystyle X=\mathbb {R} /\mathbb {Z} \cong S^{1}}$ ${\displaystyle U\subseteq X}$ ${\displaystyle {\mathcal {F}}(U)}$ ${\displaystyle f\colon U\to \mathbb {R} }$ ${\displaystyle f(x+\varepsilon +\mathbb {Z} )=f(x+\mathbb {Z} )+\varepsilon }$ ${\displaystyle |\varepsilon |}$ ${\displaystyle {\mathcal {F}}_{x}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \Gamma (U,{\mathcal {F}})\cong \mathbb {R} }$ ${\displaystyle U\subsetneq X}$ ${\displaystyle \Gamma (X,{\mathcal {F}})=\emptyset }$

generalization

The term sheaf can be understood more generally in the context of Grothendieck topologies .

See also

• Sheaf cohomology

literature

• Francisco Miraglia: An Introduction to Partially Ordered Structures and Sheaves. Polimetrica, Milan 2006, ISBN 88-7699-035-6 ( Contemporary Logic ).

Individual evidence

1. ↑ 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