On the Category of Schemes
Zhaohua Luo
1. Unipotent Maps and Normal Monos    

Consider a fixed category C with a strict initial object. Two maps u: U ® X and v: V ® X are disjoint if the initial object is the pullback of u and v. If S is a set of maps to an object X we denote by ØS the sieve of maps to X which is disjoint with each map in S. The set S is called a unipotent cover on X if ØS consists of only initial map. We say S is a normal sieve if S = ØØS. A map is called unipotent if it is a unipotent cover. A mono is called normal if it generates a normal sieve. If C has pullbacks then a mono is normal iff any of its pullback is not proper unipotent. The class of unipotent (resp. normal) maps is closed under compositions and stable, and any intersection of normal monos is normal. Geometrically a unipotent map (resp. normal mono) plays the role of a surjective map (resp. embedding). 

A subobject is called normal if it is determined by a normal mono. Suppose U and {Ui} are normal subobjects of an object X and U contains each Ui. If {Ui} is a unipotent cover of U then we say that U is a normal union of {Ui}.Note that this means that U generates the normal sieve  ØØ{Ui}, thus the normal union is uniquely determined by {Ui}. 

2. Divisors 

A stable divisor is a class D of maps satisfies the following conditions: 
(a) Isomorphisms and initial maps are in D
(b) D is closed under composition. 
(c) Any pullback of a map in D exists and is again in D
We say D is a subnormal divisor if any map in D is a normal mono. 

Suppose D is a stable divisor; a map in D is called a D-map; a subobject determined by a D-mono is called a D-subobject; a unipotent cover on an object X consisting of D-maps is called a D-cover. For any object X denote by WD(X) the set of D-subobjects of X

Suppose D is a subnormal stable divisor D
(a) D is effective if any set of normal monos in D has a normal union which is again in D
(b) D is strict if an object X is a normal union of {ui: Ui ® X} of D-subobjects then X is the colimit of the systems {Us Ç Ut ® X |s, t Î I}. 
(c) D is called canonical if it is effective and strict. 

Theorem. Suppose D is an effective divisor. 
(a) WD(X) is a frame for any object X
(b) For any map f: Y ® X the function WD(X) ® WD(Y)  sending each open subobject of X to its pullback along f is a morphism of frames. 
(c) The functor WD from C to the category of locales is an effective subnormal framed topology. 
(d) Conversely, the effective open maps in an effective subnormal framed topology is an effective divisor. 

Suppose A is a full subcategory of C containing an initial of C. Consider a subnormal stable divisor D on A. Consider the class E of normal monos u: U ® X in C whose pullback along any map v: V ® X with V ÎA exist which is a normal union of D-subobjects of V. It is easy to see that E is a subnormal stable divisor on C, called the normal extension of D on C

3. Complete Divisors 

Suppose D is a subnormal stable divisor on C
(a) A glueing diagram ({Xi}, {Uij}, {uij}) for D consists of a small set {Xi} of objects of C together with, for any i ¹ j, a D-subobject Uij of Xi and an isomorphism of subobjects uij: Uij ® Uji, such that 
(i) uji = uij-1
(ii) uij(Uij Ç Uik) = Uji Ç Ujk
(iii) uik = ujkuij on Uij Ç Uik
(b) A glueing colimit of a glueing diagram ({Xi}, {Uij}, {uij}) is an object X of C, together with D-maps vi: Xi ® X for each i, such that {Xi} covers X, with Uij = Xi Ç Xj as subobjects of X, and vi = vjuij on Uij (if Uij are all empty then we say that X is the disjoint joint of the Xi). Note that since {Xi} covers X and C is strict, X is a colimit of the glueing diagram  ({Xi}, {Uij}, {uij}), therefore is uniquely determined up to isomorphism. 
(c) A complete divisor is a strict effective D such that any glueing diagram for D has a glueing colimit. 

4. Analytic Categories     

A lextensive category is a category with finite limits and finite stable disjoint sums. An analytic category is a lextensive category with epi-strong-mono factorizations. In the following we consider an analytic category C. A map f: Y ® X is called coflat if the pullback functor C/X ® C/Y along it preserves epis. A mono v: V ® X is a complement of a mono u: U ® X if u and v are disjoint, and any map t: T ® X such that u and t are disjoint factors through v. This condition is equivalent to that v generates the normal sieve Ø{u}, thus a complement mono is always normal. A mono v: V ® X is called singular if it is the complement of a strong mono u: U ® X. A coflat singular mono is called an analytic mono. The class of coflat maps (resp. analytic monos, resp. fractions) is closed under compositions and stable. The class of analytic monos is a subnormal divisor A(C), called the analytic divisor. We say C is strict if its analytic divisor A(C) is strict. 

5. Normal Completion of an Analytic Category

A normal completion of a strict analytic category A is a category C containing A as a full subcategory which satisfies the following conditions: 
(a) The initial object of A is also a strict initial object of C
(b) The extension E of the analytic divisor on A to C is a complete divisor. 
(c) Any object X is the normal union of the E-subobjects in A

Suppose C is a fixed normal completion of a strict analytic category A. Suppose E is the extension of the analytic divisor on A. An object in C is called a scheme (over A). Any scheme which is isomorphic to an object in A is called an affine scheme. An E-subobject is simply called an open subscheme

Theorem. (a) Any strict analytic category has a normal completion, which is uniquely determined up to equivalence. 
(b) The normal completion of any strict analytic category is lextensive. 
(c) The class of open subschemes defines a strict framed topology on C, called the analytic topology on C

Example. The opposite of the category of commutative rings (with unit and unit-preserving homomorphisms) is a strict analytic category. Its normal completion is equivalent to the category of schemes in the sense of Grothendieck.