1. Grothendieck Topology
Let C be a category. Given f :X → Z and g : Y → Z two morphisms in C, the fiber product off andgconsists of an object P and two morphismsp1:P →Xand p2 :P →Y such that the diagram
P −−−−→p2 Y
p1
y g
y X −−−−→f Z commutes and given any other commuting diagram Q −−−−→q2 Y
q1
y g
y X −−−−→f Z
there exists a unique morphism u : Q → P such that p1 ◦u = q1 and p2 ◦u = q2. In other words, the fiber product (P, p1, p2) is universal with respect to the above commuting diagrams. The fiber product of f and g will be denoted by X×ZY.A category C is called a category with fiber products if given any two morphisms f :X→Z and g:Y →Z inC, their fiber product exists.
A Grothendieck topology on a categoryCwith fiber product is a functionT which assigns to each objectU a setT(U) consisting of families {ϕi:Ui→U :i∈I} of morphisms with targetU such that
(1) ifU0 →U is an isomorphism, then {U0 →U} belongs to T(U);
(2) if{ϕi:Ui→U :i∈I}is in T(U),and ifU0 →U is any morphism, then the family {Ui×UU0 →U0:i∈I}is in T(U0),
(3) if {ϕi :Ui → U :i ∈I} is in T(U), and if for each i, one has a family {Vij → Vi : j∈Ji},then{Vij →Ui→U :i∈I, j ∈Ji} is inT(U).
The 3family of T(U) are called the covering families for U in the T-topology. A site is a category with a Grothendieck topology.
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