Grothendieck topology

In category theory, a branch of mathematics, a Grothendieck topology is a structure defined in an arbitrary category C that allows the definition of bundles in C, and with that the definition of the general theories of cohomology. A category together with a Grothendieck topology in it is called a site. This tool is used in algebraic number theory and algebraic geometry, mainly to define the cohomology étale of schemata, but also for shallow cohomology and crystalline cohomology. Note that a Grothendieck topology is not a topology in the classical sense.

History and idea

At a time when cohomology for bundles in topological spaces was being established, Alexander Grothendieck wanted to define cohomology theories for other structures, their schemata. He thought of a beam in a topological space as a "standard meter"; for that space, and the cohomology of that pattern as a rough measure for the underlying space. His goal was thus to produce a structure that would allow the definition of more general beams; once that was done, the model of the topological theories of cohomology could be followed almost verbatim.

Motivation example

Start with a topological space X and consider the bundle of all continuous real-valued functions defined on X. This associates to each open set U in X the set F(U) of continuous real-valued functions defined in U. Since U is a subset of V, we have a "restriction function" from F(V) to F(U). If we interpret the topological space X as a category, with the open sets being the objects, and a map from U to V if and only if U is a subset of V, so F is revealed as a contravariant functor of this category in the category of sets. In general, every contravariant functor from a category C to the category of sets is therefore called a pre-bundle of sets in C. Our functor F has a special property: if we have an open covering (V_i) of the set U, and mutually compatible elements of F (V_i) are given, then there exists exactly one F(U) element that is constrained to all dice. This is the defining property of a bundle, and a Grothendieck topology on C is an attempt to capture the essence of what is necessary to define bundles on C.

Formal definition

Formally, a topology of Grothendieck in C is given specifying for each object U of C Morphismos families {φ_i: V_i → → {displaystyle rightarrow } U! _{i한 한 {displaystyle in }I}, calls U covering familiessuch that the following axioms are satisfied:

• Yes.₁: U₁ → → {displaystyle rightarrow }U is an isomorphism, then {φ₁: U₁→ → {displaystyle rightarrow }UIt's a covering family. U.

• Yes_i: V_i → → {displaystyle rightarrow }U! _{i한 한 {displaystyle in }I} It's a covering family. U and f: U ₁ → → {displaystyle rightarrow }U is a morphism, then the pullback P _i = U₁ ×_U V_i exists for each i in I, and the induced family {π_i: P_i→ → {displaystyle rightarrow }U₁!_{i한 한 {displaystyle in }I} It's a covering family. U₁.

• Yes_i: V_i → → {displaystyle rightarrow } U! _{i한 한 {displaystyle in }I} It's a covering family. Uand if for each i in I, { φⁱ_j: V ⁱ _j → → {displaystyle rightarrow }V _i ! _{j한 한 {displaystyle in }Ji} It's a covering family. V_i, then {φ_i 'φij: V i j → → {displaystyle rightarrow }U!i한 한 {displaystyle in }I and j한 한 {displaystyle in }Ji is a covering family for U.

A setback Category C It's a countervailing funtor. F: C → → {displaystyle rightarrow } Set. Yeah. C is equipped with a topology of Grothendieck, then a prehaz is called a Do it. in C Yes, for every covering family {φ_i: V_i → → {displaystyle rightarrow } U!_{i한 한 {displaystyle in }I}, the induced function F(U) → → {displaystyle rightarrow }Русский_{i한 한 {displaystyle in }I}F(V_i) is the natural equalizer of two ENTE functions_{i한 한 {displaystyle in }I}F(V_i)→ → {displaystyle rightarrow } Русский_{(i,j)한 한 {displaystyle in }IxI} F (V_i ×_U V_j).

In analogy, you can also define prehaces and beams of Abelian groups, considering countervailing funtors F: C → → {displaystyle rightarrow }Ab.

Given a site (a category C with a Grothendieck topology), the category of all beams in that site can be considered. This is a topos, and in fact the notion of topos originated here. The category of bundles of abelian groups is also a Grothendieck category, which essentially means that cohomology theories can be defined for these bundles — the reason for the whole construction.