Grothendieck Topology

The Grothendieck topology is a structure on a category that makes its objects similar to the open sets of a topological space . The category together with the Grothendieck topology is called a situs ^[1] or site ^[2] .

Grothendieck topologies axiomatize the definition of an open covering , due to which it becomes possible to define beams on categories and their cohomology , which was first carried out by Alexander Grothendik for etale cohomology of schemes .

There is a natural way to compare the topology of the Grothendieck topology, in this sense it can be considered as a generalization of ordinary topologies . Moreover, for a large class of topological spaces, it is possible to restore the topology by its Grothendieck topology, but this is not the case for an anti-discrete space .

Content

Definition

Motivation

The classical definition of a beam begins with some topological space. $X$ ${\ displaystyle X}$ . It is mapped to category $O(X)$ ${\ displaystyle O (X)}$ whose objects are open sets of topology, and the set of morphisms between two objects consists of one element if the first set is embedded in the second (these maps are called open embeddings), and empty otherwise. After this, a pre-sheaf is defined as a contravariant functor in the category of sets , and a sheaf is defined as a pre-sheaf satisfying the . The gluing axiom is formulated in terms of pointwise covering, that is, $\{U_{i}\}$ ${\ displaystyle \ {U_ {i} \}}$ covers $U$ ${\ displaystyle U}$ then and only if $\bigcup {_{i}}U_{i}=U$ ${\ displaystyle \ bigcup {_ {i}} U_ {i} = U}$ . Grothendieck topologies replace each $U_{i}$ ${\ displaystyle U_ {i}}$ a whole family of open sets; more accurately, $U_{i}$ ${\ displaystyle U_ {i}}$ replaced by a family of open attachments $V_{ij}\to U_{i}$ ${\ displaystyle V_ {ij} \ to U_ {i}}$ . Such a family is called a sieve .

Resheta

If a $c$ ${\ displaystyle c}$ - arbitrary category object ${\mathcal {C}}$ ${\ displaystyle {\ mathcal {C}}}$ then sieve on $c$ ${\ displaystyle c}$ Is a subfunctor of a functor $\mathrm {Hom} (-,c)$ ${\ displaystyle \ mathrm {Hom} (-, c)}$ . In the case of a category $O(X)$ ${\ displaystyle O (X)}$ sieve $S$ ${\ displaystyle S}$ on the open set $U$ ${\ displaystyle U}$ - this is some family of open subsets $U$ ${\ displaystyle U}$ closed relative to the operation of taking an open subset. Arbitrary open set $V$ ${\ displaystyle V}$ then $S(V)$ ${\ displaystyle S (V)}$ Is a subset $\mathrm {Hom} (V,U)$ ${\ displaystyle \ mathrm {Hom} (V, U)}$ accordingly, it is empty if $V$ ${\ displaystyle V}$ - not a subset $U$ ${\ displaystyle U}$ , and may consist of one element otherwise; if it is non-empty, we can assume that $V$ ${\ displaystyle V}$ selected by sieve. If a $W$ ${\ displaystyle W}$ - subset $V$ ${\ displaystyle V}$ there is a morphism $S(V)\to S(W)$ ${\ displaystyle S (V) \ to S (W)}$ so if $S(V)$ ${\ displaystyle S (V)}$ not empty then $S(W)$ ${\ displaystyle S (W)}$ not empty

Axioms

Grothendieck Topology $J$ ${\ displaystyle J}$ into categories ${\mathcal {C}}$ ${\ displaystyle {\ mathcal {C}}}$ - this is a choice for each object $c$ ${\ displaystyle c}$ categories ${\mathcal {C}}$ ${\ displaystyle {\ mathcal {C}}}$ recruitment sieves $c$ ${\ displaystyle c}$ denoted by $J(c)$ ${\ displaystyle J (c)}$ . Items $J(c)$ ${\ displaystyle J (c)}$ are called covering lattices on $c$ ${\ displaystyle c}$ . In particular, the sieve $S$ ${\ displaystyle S}$ on the open set $U$ ${\ displaystyle U}$ is covering if and only if the union of all $V$ ${\ displaystyle V}$ such that $S(V)$ ${\ displaystyle S (V)}$ nonexistent is everything $U$ ${\ displaystyle U}$ . This choice should satisfy the following axioms:

base replacement: if $S$ ${\ displaystyle S}$ - covering sieve on $X$ ${\ displaystyle X}$ and $f:Y\to X$ ${\ displaystyle f: Y \ to X}$ - morphism, then the sieve prototype under the action $f$ ${\ displaystyle f}$ ( $f\ast S$ ${\ displaystyle f \ ast S}$ ) is a covering sieve on $Y$ ${\ displaystyle Y}$ .
local character: if $S$ ${\ displaystyle S}$ - covering sieve on $X$ ${\ displaystyle X}$ , $T$ ${\ displaystyle T}$ - arbitrary sieve on $X$ ${\ displaystyle X}$ and for each object $Y\in \mathrm {Ob} {\mathcal {C}}$ ${\ displaystyle Y \ in \ mathrm {Ob} {\ mathcal {C}}}$ and every morphism $f:Y\to X$ ${\ displaystyle f: Y \ to X}$ owned by $S(Y)$ ${\ displaystyle S (Y)}$ sieve type $f\ast T$ ${\ displaystyle f \ ast T}$ is a covering sieve on $Y$ ${\ displaystyle Y}$ then $T$ ${\ displaystyle T}$ - covering sieve on $X$ ${\ displaystyle X}$ .
unit: $\mathrm {Hom} (-,X)$ ${\ displaystyle \ mathrm {Hom} (-, X)}$ - covering sieve on $X$ ${\ displaystyle X}$ for any object $X$ ${\ displaystyle X}$ categories ${\mathcal {C}}$ ${\ displaystyle {\ mathcal {C}}}$ .

Replacing the base matches the idea that if $\{U_{i}\}$ ${\ displaystyle \ {U_ {i} \}}$ covers $U$ ${\ displaystyle U}$ then $\{U_{i}\cap V\}$ ${\ displaystyle \ {U_ {i} \ cap V \}}$ covers $U\cap V$ ${\ displaystyle U \ cap V}$ . Local character corresponds to the fact that if $\{U_{i}\}$ ${\ displaystyle \ {U_ {i} \}}$ covers $U$ ${\ displaystyle U}$ and $\{V_{ij}\}$ ${\ displaystyle \ {V_ {ij} \}}$ covers $U_{i}$ ${\ displaystyle U_ {i}}$ for each $i$ ${\ displaystyle i}$ then all $\{V_{ij}\}$ ${\ displaystyle \ {V_ {ij} \}}$ cover $U$ ${\ displaystyle U}$ . Finally, the unit corresponds to the fact that each set can be covered by the union of all its subsets.

Situs and bundles

In category $O(X)$ ${\ displaystyle O (X)}$ one can define a bundle using the gluing axiom. It turns out that the bundle can be defined in any category with the Grothendieck topology: the bundle on the situs $({\mathcal {C}},J)$ ${\ displaystyle ({\ mathcal {C}}, J)}$ Is a bunch $F$ ${\ displaystyle F}$ such that for any object $X$ ${\ displaystyle X}$ and covering sieve $S$ ${\ displaystyle S}$ on $X$ ${\ displaystyle X}$ natural mapping $\mathrm {Hom} (\mathrm {Hom} (-,X),F)\to \mathrm {Hom} (S,F)$ ${\ displaystyle \ mathrm {Hom} (\ mathrm {Hom} (-, X), F) \ to \ mathrm {Hom} (S, F)}$ induced by embedding $S$ ${\ displaystyle S}$ in Hom (-, X ), is a bijection. The morphism between bundles, as well as the morphism between presheaves, is a natural transformation of functors. The category of all beams on a situs is called the Grothendix topos . Beams, Abelian groups, rings, modules, and other structures are defined similarly.

Using the Yoneda lemma, it is possible to prove that a sheaf in a category $O(X)$ ${\ displaystyle O (X)}$ defined in this way coincides with the bundle in a topological sense.

Situs Examples

Discrete and Anti-Discrete Topology

Discrete topology on an arbitrary category ${\mathcal {C}}$ ${\ displaystyle {\ mathcal {C}}}$ set by declaring all sieves open. To define an antidiscrete topology, only the sieve of the form should be considered open. $\mathrm {Hom} (-,X)$ ${\ displaystyle \ mathrm {Hom} (-, X)}$ . In an anti-discrete topology, any pre-sheaf is a bundle.

Canonical topology

Canonical topology on an arbitrary category ${\mathcal {C}}$ ${\ displaystyle {\ mathcal {C}}}$ Is the most subtle topology , such that all representable presheaves (functors of the form $\mathrm {Hom} (-,X))$ ${\ displaystyle \ mathrm {Hom} (-, X))}$ are bunches. A topology that is less subtle (that is, a topology such that any representable pre-sheaf is a bundle) is called subcanonical , most of the topology found in practice are subcanonical.

Small and large situs associated with topological space

To match the topological space of a small situs, in the category $O(X)$ ${\ displaystyle O (X)}$ covering such sieves are declared $S$ ${\ displaystyle S}$ that the union of all $V$ ${\ displaystyle V}$ such that $S(V)$ ${\ displaystyle S (V)}$ nonempty matches everything $U$ ${\ displaystyle U}$ .

Sieve $S$ ${\ displaystyle S}$ on the category of topological spaces $\mathbf {Top}$ ${\ displaystyle \ mathbf {Top}}$ is called a covering sieve if the following conditions are true:

for all $Y$ ${\ displaystyle Y}$ and morphisms $f:Y\to X$ ${\ displaystyle f: Y \ to X}$ owned by $S(Y)$ ${\ displaystyle S (Y)}$ object exists $V$ ${\ displaystyle V}$ and arrow $g:V\to X$ ${\ displaystyle g: V \ to X}$ such that $g$ ${\ displaystyle g}$ - open investment $g$ ${\ displaystyle g}$ belongs $S(V)$ ${\ displaystyle S (V)}$ and $f$ ${\ displaystyle f}$ sweeps through $g$ ${\ displaystyle g}$ ;
if a $W$ ${\ displaystyle W}$ - Union $f(Y)$ ${\ displaystyle f (Y)}$ where $f:Y\to X$ ${\ displaystyle f: Y \ to X}$ runs through $S(Y)$ ${\ displaystyle S (Y)}$ then $W=X$ ${\ displaystyle W = X}$ .

For the category of comma topological spaces above the fixed topological space $X$ ${\ displaystyle X}$ , topology is induced by category $\mathbf {Top}$ ${\ displaystyle \ mathbf {Top}}$ . The resulting category is called a large situs associated with the topological space. $X$ ${\ displaystyle X}$ .

Topologies in schema category

Functors between situses

Notes

↑ R. Goldblatt. Topos. Categorical analysis of logic. - M .: Mir, 1983. - 487 p.
↑ P. Johnston. The theory of topos. - M .: Science, 1986. - 440 p.

Literature

Artin, Michael. Grothendieck topologies - Harvard University, Dept. of Mathematics, 1962.
Demazure Michel, Alexandre Grothendieck. Seminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151). - Berlin; New York: Springer-Verlag, 1970. - P. xv + 564.
Michael Artin, Alexandre Grothendieck, Jean-Louis Verdier. Seminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269). - Berlin; New York: Springer-Verlag, 1972. - P. xix + 525.

[1] R. Goldblatt. Topos. Categorical analysis of logic. - M .: Mir, 1983. - 487 p.

[2] P. Johnston. The theory of topos. - M .: Science, 1986. - 440 p.