Grothendieck topology

A Grothendieck topology is a mathematical concept that allows a sheave theory and a cohomology theory to be developed in an abstract categorical framework . A category in which a Grothendieck topology is explained is called a situs . A sheaf can be explained on a situs. The concept of the Grothendieck topology was developed by Alexander Grothendieck around 1960 in order to have a replacement for the topological cohomology theories such as singular cohomology in algebraic geometry with positive characteristics . The motivation for this came from the conjectures of André Weil , who predicted a close connection between the topological shape (such as the Betti numbers ) of a variety and the number of points on it over a finite field ( Weil conjectures ). The étal topology introduced in this context together with the étal cohomology and the l-adic cohomology finally made it possible for Pierre Deligne to prove the Weil conjectures .

introduction

The classical concept of the embossing on a topological space , which is important for algebraic geometry, assigns a set to every open set , so that the following compatibility conditions are met: ${\displaystyle X}$ ${\displaystyle U\subset X}$ ${\displaystyle P(U)}$

• For an inclusion of open sets of there is a function called restriction . ${\displaystyle V\subset U}$ ${\displaystyle X}$ ${\displaystyle r_{V,U}\colon P(U)\rightarrow P(V)}$
• ${\displaystyle r_{U,U}=\mathrm {id} _{P(U)}}$for all open . ${\displaystyle U\subset X}$
• ${\displaystyle r_{W,U}=r_{W,V}\circ r_{V,U}}$for all open . ${\displaystyle W\subset V\subset U\subset X}$

(Typical example: = set of continuous functions and = restriction of a function to .) ${\displaystyle P(U)}$ ${\displaystyle U\rightarrow \mathbb {R} }$ ${\displaystyle r_{V,U}}$ ${\displaystyle U\rightarrow \mathbb {R} }$ ${\displaystyle V}$

If one considers the system of the open sets of as objects of a category whose only morphisms are the inclusions , then the above conditions state that through the data and a contravariant functor of in the category of sets . The aim is to generalize this to situations in which one has any category instead of any. ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}(X)}$ ${\displaystyle i_{V,U}\colon V\rightarrow U}$ ${\displaystyle U\mapsto P(U)}$ ${\displaystyle i_{V,U}\mapsto P(i_{V,U})=r_{V,U}}$ ${\displaystyle {\mathcal {O}}(X)}$ ${\displaystyle {\mathcal {O}}(X)}$

Many constructions use open overlaps of space and of these the following properties: ${\displaystyle X}$

• The family consisting only of is an open cover of . ${\displaystyle X}$ ${\displaystyle X}$
• If is continuous and an open cover of , then there is an open cover of . ${\displaystyle f\colon Y\rightarrow X}$ ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle (f^{-1}(U_{i}))_{i\in I}}$ ${\displaystyle Y}$
• If there is an open coverage of and if every family is an open coverage of , then the family is an open coverage of . ${\displaystyle (U_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle (V_{i,j})_{j\in I_{i}}}$ ${\displaystyle U_{i}}$ ${\displaystyle (V_{i,j})_{i\in I,j\in I_{i}}}$ ${\displaystyle X}$

The correct (because successful) generalization of the open coverage of a set on any category is the concept of the sieve on an object, i.e. H. a set of morphisms with this object as a fixed target , so that with every morphism and every morphism that can be composed from the right is also contained in it. (In the case of topological spaces, one then has to restrict oneself to such coverages which, with every open set, also contain all open subsets contained therein.) The idea of ​​the generalization indicated now consists in defining which sieves on an object count as "cover" and what relationships should exist between them. The following definition, which is essentially a transfer of the above-mentioned overlap properties, has proven to be very far-reaching. ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f\circ g}$

definition

A Grothendieck topology on a small category is a mapping that each object of a lot of screens on maps so that the following applies: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle J}$ ${\displaystyle C}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle J(U)}$ ${\displaystyle U}$

• Maximum sieves: For each object, the maximum screen is all morphisms with target in included. ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle J(U)}$
• Axiom of stability: Is a morphism and so is . ${\displaystyle f\colon V\rightarrow U}$ ${\displaystyle S\in J(U)}$ ${\displaystyle f^{*}(S):=\{g\mid f\circ g\in S\}\in J(V)}$
• Axiom of transitivity: Is and is a sieve on , so that it is so for everyone . ${\displaystyle S\in J(C)}$ ${\displaystyle R}$ ${\displaystyle U}$ ${\displaystyle f^{*}(R)\in J(V)}$ ${\displaystyle (f\colon V\rightarrow U)\in S}$ ${\displaystyle R\in J(U)}$

A pair consisting of a small category and a Grothendieck topology defined on it is called a situs . ${\displaystyle ({\mathcal {C}},J)}$ ${\displaystyle J}$

Examples

• If a topological space is the set of all open coverages for every open set , which with every family member also contains all of its open subsets, then a Grothendieck topology is on . In this sense, every topological space becomes a situs. ${\displaystyle X}$ ${\displaystyle J(U)}$ ${\displaystyle U\subset X}$ ${\displaystyle J}$ ${\displaystyle {\mathcal {O}}(X)}$
• If there is a small category and only consists of the maximum sieve , then there is a Grothendieck topology , the so-called trivial Grothendieck topology . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle J(C)}$ ${\displaystyle C}$ ${\displaystyle J}$ ${\displaystyle {\mathcal {C}}}$
• Is a small category and consists of all non-empty Seven , there is exactly then a Grothendieck topology ago when two arrows and with the same goal in a commutative square ${\displaystyle {\mathcal {C}}}$ ${\displaystyle J(C)}$ ${\displaystyle C}$ ${\displaystyle V\rightarrow U}$ ${\displaystyle W\rightarrow U}$

${\displaystyle {\begin{array}{ccc}X&\rightarrow &V\\\downarrow &&\downarrow \\W&\rightarrow &U\end{array}}}$

can complement. (This is fulfilled, for example, with the frequently asked requirement that the category contains pullbacks .) This Grothendieck topology is called the atomic Grothendieck topology .

Basis of a Grothendieck topology

A basis for a Grothendieck topology in a category with pullbacks is given by marking out families of morphisms as overlapping families of for each object . These families must satisfy the following axioms: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle (\phi _{i}\colon V_{i}\to U)_{i\in I}}$ ${\displaystyle U}$

• An isomorphism is a covering family of . ${\displaystyle \phi _{1}\colon V_{1}\to U}$ ${\displaystyle U}$

• If there is a covering family of and a morphism, then the pullback exists for each and the induced family is a covering family for . ${\displaystyle (\phi _{i}\colon V_{i}\to U)_{i\in I}}$ ${\displaystyle U}$ ${\displaystyle f\colon V\to U}$ ${\displaystyle P_{i}=V\times _{U}V_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle (\pi _{i}\colon P_{i}\to V)_{i\in I}}$ ${\displaystyle V}$

• If there is an overlapping family of , and if for each there is an overlapping family of , then there is an overlapping family of . ${\displaystyle (\phi _{i}\colon V_{i}\to U)_{i\in I}}$ ${\displaystyle U}$ ${\displaystyle i\in I}$ ${\displaystyle (\phi _{j}^{i}\colon V_{j}^{i}\to V_{i})_{j\in J_{i}}}$ ${\displaystyle V_{i}}$ ${\displaystyle (\phi _{i}\phi _{j}^{i}\colon V_{j}^{i}\to U)_{i\in \ I,j\in J_{i}}}$ ${\displaystyle U}$

If, for an object and a sieve on : ${\displaystyle U}$ ${\displaystyle S}$ ${\displaystyle U}$

${\displaystyle S\in J(U)}$if and only if there is a family of morphisms assigned in the base that is contained in, then it is a Grothendieck topology so defined . This is what the term basis of a Grothendieck topology means. ${\displaystyle S}$ ${\displaystyle J}$

Sheaves on a Grothendieck topology

A preamble on a category C is a contravariant functor in a category A, such as the category of sets or the category of Abelian groups . If C has a Grothendieck topology, then a preamble is called a sheaf if the sequence for each covering family ${\displaystyle {\mathcal {F}}\colon C\to A}$ ${\displaystyle \{\phi _{i}\colon V_{i}\to U\}_{i\in I}}$

${\displaystyle {\mathcal {F}}(U)\to \prod {\mathcal {F}}(V_{i})\,{\begin{matrix}\to \\[-.7em]\to \end{matrix}}\,\prod {\mathcal {F}}(V_{i}\times _{U}V_{j})}$

is exact, that is, if the difference kernel of the two right arrows is. ${\displaystyle {\mathcal {F}}(U)}$

As in the case of a topological space, you can garbage marks. One can also develop various theories of cohomology , such as the Čech cohomology .

The totality of all sheaves on a site forms a topos .

Individual evidence

1. ^ Saunders Mac Lane , Ieke Moerdijk : Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , Chapter III.2: Grothendieck-Topologies , definition 1
2. ^ Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , Chapter III.2: Grothendieck-Topologies
3. ↑ Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logic , Springer-Verlag (1992), ISBN 978-0-387-97710-2 , Chapter III.2: Grothendieck-Topologies , Definition 2