Grothendieck topology

In mathematics,Grothendieck topology isstructure defined on an arbitrary category C which allowsdefinitionsheaves on C,with thatdefinitiongeneral cohomology theories. A category together withGrothendieck topology on itcalledsite. This toolusedalgebraic number theoryalgebraic geometry, mainlydefine étale cohomologyschemess, but alsoflat cohomologycrystalline cohomology. Note thatGrothendieck topologynottopology inclassical sense.

Historyidea

Attime when cohomologysheaves on topological spaces was well established, Alexander Grothendieck wanteddefine cohomology theoriesother structures, his schemess. He thought ofsheaf ontopological space as"measuring rod"that space, andcohomologysuchmeasuring rod asrough measure forunderlying space. His goal was thusproducestructure which would allowdefinitionmore general sheaves or "measuring rods"; once that was done,modeltopological cohomology theories could be followed almost verbatim.

Motivating example

Start withtopological space Xconsidersheafall continuous real-valued functions defined on X. This associatesevery open set UXset F(U)real-valued continuous functions defined on U. Whenver U issubsetV, we have"restriction map" from F(V)F(U). If we interprettopological space X ascategory, withopen sets beingobjects andmorphism from UV ifonly if U issubsetV, then Frevealed ascontravariant functor from this category intocategorysets. In general, every contravariant functor fromcategory C tocategorysetstherefore calledpre-sheafsets on C. Our functor F hasspecial property: if you have an open covering (V_i) ofset U,yougiven mutually compatible elementsF(V_i), then there exists precisely one elementF(U) which restrictsallgiven ones. This isdefining property ofsheaf, andGrothendieck topology on Can attemptcaptureessencewhatneededdefine sheaves on C.

Formal definition

Formally,Grothendieck topology on Cgiven by specifyingeach object UC familiesmorphisms {φ_i : V_i -> U}_iI, called covering familiesU, such thatfollowing axiomssatisfied:

• if φ₁ : U₁ -> Uan isomorphism, then {φ₁ : U₁ -> U} iscovering familyU.
• if {φ_i : V_i -> U}_iI iscovering familyUf : U₁ -> U ismorphism, thenpullback P_i = U₁ ×_UV_i existsevery iI, andinduced family {π_i : P_i -> U₁}_iI iscovering familyU₁.
• if {φ_i : V_i -> U}_iI iscovering familyU'\',ifevery iI, {φij : Vij -> Vi}jJi iscovering familyV_i, then {φiφij : Vij -> U}iIjJi iscovering familyU''.

A presheafsets oncategory C iscontravariant functor F : C -> Set. If Cequipped withGrothendieck topology, thenpresheafcalledsheaf on C if,every covering family {φ_i : V_i -> U}_iI,induced map F(U) -> Π_iI F(V_i) isequalizer oftwo natural maps Π_iI F(V_i) -> Π_{(i, j)I x I} F(V_i ×_U V_j).

In analogy, one can also define presheavessheavesabelian groups, by considering contravariant functors F : C -> Ab.

Oncesite (a category C withGrothendieck topology)given, one can considercategoryall sheaves on this site. This istopos,in factnotiontopos originated here. The categorysheavesabelian groupsalsoGrothendieck category, which essentially means that one can define cohomology theoriesthese sheaves —reason forwhole construction.