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 S 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 S consists of only initial map. We say S is a normal sieve if S = S. 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 {U_{i}} are normal subobjects of an object X and U contains each U_{i}. If {U_{i}} is a unipotent cover of U then we say that U is a normal union of {U_{i}}.Note that this means that U generates the normal sieve {U_{i}}, thus the normal union is uniquely determined by {U_{i}}. 2. Divisors A stable divisor is a class
D of maps satisfies the following conditions:
Suppose D is a stable divisor; a map in D is called a Dmap; a subobject determined by a Dmono is called a Dsubobject; a unipotent cover on an object X consisting of Dmaps is called a Dcover. For any object X denote by _{D}(X) the set of Dsubobjects of X. Suppose D is a subnormal stable divisor D:
Theorem. Suppose D 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 A exist which is a normal union of Dsubobjects 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.
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 epistrongmono 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 {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:
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 Esubobject is simply called an open subscheme. Theorem. (a) Any strict analytic category has a normal completion,
which is uniquely determined up to equivalence.
Example. The opposite of the category of commutative rings (with
unit and unitpreserving homomorphisms) is a strict analytic category.
Its normal completion is equivalent to the category of schemes in the sense
of Grothendieck.
