It is well known that most geometric-like categories have finite limits and finite stable disjoint sums. These are lextensive categories in the sense of Carboni, Lack and Walters [1993]. We introduce the notion of an analytic category, which is a lextensive category with the property that any map factors as an epi followed by a strong mono. The class of analytic categories includes many natural categories arising in geometry, such as the categories of sets, posets, topological spaces, locales, affine schemes, as well as all the elementary toposes.
A large class of analytic categories is formed by the opposites of Zariski categories in the sense of Diers [1992]. The notion of a Zariski category captures the categorical properties of commutative rings. Many algebraic-geometric analysis carried by Diers for a Zariski category can be done for a more general analytic category in the dual situation. We show that the notion of a flat singular epi developed in [Diers 1992] can be applied to define a canonical functor from an analytic category to the category of locales, which is a framed topology in the sense of Luo [1995a and b]. This topology plays the fundamental role of Zariski topology in categorical geometry.
1. Unipotent Maps and Normal Monos
Consider a 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).
2. Framed Topologies
Consider a functor 
from C to the category of locales. A mono u: U --> X in C (and (u): (U) --> (X)) is called open effective if (u) is an open embedding of locales, and any map t: T --> X in C such that (t) factors through (u) factors through u. If u is open effective then u or U is called an open effective subobject of X, and (u) or (U) is an open effective sublocale of (X).
We say is a framed topology on C if an object X is initial iff (X) is initial, and any open sublocale of (X) is a join of open effective sublocales. If {Ui} is a set of open effective subobjects of X such that (X) is the join of
{(Ui)}, then we say that {Ui} (resp. {(Ui)}) is an open effective cover on X (resp. (X)). The collection T() of open effective covers is a Grothendieck topology on C. We say is strict if its Grothendieck topology T() is subcanonical.
3. Divisors
Here is a general method to define framed topologies. A class D of maps containing isomorphisms is called a (stable) divisor if it is closed under compositions, and its pullback along any map exists which is also in D; we say D is subnormal if any map in D is a normal mono. If D is a divisor, a sieve with the form T, where T is any set of monos to X in D, is called a D-sieve on X. One can show the set D(X) of D-sieves on X is a locale and the pullbacks of D-sieves along a map induce a morphism of locales. Thus each divisor D determine a functor L(D) to the category of locales. If D is subnormal then L(D) is a framed topology, called the framed topology determined by D.
4. Extensive Topologies.
Recall that a category with finite stable disjoint sums is an extensive category. An extensive category C has a strict initial object. An injection of a sum is simply called a direct mono. An intersection of direct monos is called a locally direct mono. The class of direct monos is a subnormal divisor E(C), called the extensive divisor. The extensive divisor E(C) determines a framed topology, called the extensive topology.
For any object X we denote by Dir(X) the set of locally direct subobjects of X, viewed as a poset with the reverse order. If any intersection of direct monos exist in C, then Dir(X) is a locale for any object X, and Dir is naturally a functor from C to the category of locales, which is equivalent to the extensive topology. Special cases of extensive topologies were considered by Barr and Pare [1980] and Diers [1986].
5. Analytic Topologies
An analytic category is a lextensive category with epi-strong-mono factorizations. In the following we consider an analytic category C. One of the most important notion introduced by Diers to categorical geometry is that of a flat singular map. We consider the dual notion. A map f: Y --> X is called coflat if the pullback functor C/X --> C/Y along it preserves epis. The main point here is that any pullback along a coflat map preserves epi-strong-mono factorizations.
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. A coflat normal mono is called a fraction (thus any analytic mono is a fraction). A fraction plays the role of local isomorphism in algebraic geometry.
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. The analytic divisor A(C) determines a framed topology, called the analytic topology. We say C is strict if its analytic divisor A(C) is strict.
6. Reduced and Integral Objects
The analytic topology can also be defined algebraically, using reduced and integral objects, as in the case of affine schemes. An object is reduced if any unipotent map to it is epic. A reduced non-initial object is integral if any non-initial analytic mono to it is epic One can show easily that any quotient of a reduced (resp. integral) object is reduced (resp. integral) (i.e. if f: Y --> X is an epi and Y is reduced or integral then so is X).
A unipotent reduced strong subobject of an object X is called the radical of X. It is the largest reduced and the smallest unipotent strong subobject of X, thus is uniquely determined by X. An analytic category is reduced if any unipotent map is epic. One can show that an analytic category is reduced iff its strong monos are normal. An analytic category is reducible (resp. spatial) if any non-initial object has a non-initial reduced (resp. integral) strong subobject. If any intersection of strong monos exist in C then the full subcategory of reduced objects is a coreflective subcategory; if moreover C is reducible then any object has a radical.
7. Spectrums
A strong mono is called disjunctive if it has an analytic complement. An object is disjunctable if its diagonal map is a disjunctable regular mono. An analytic category is called disjunctable if any strong mono is disjunctable. An analytic category is locally disjunctable if any strong subobject is an intersection of disjunctive strong subobjects. A locally disjunctable reducible analytic category in which any intersection of strong subobjects exist is called an analytic geometry.
Let C be an analytic geometry. If X is an object we denote by Loc(X) (resp. Spec(X)) the set of reduced (resp. integral) strong subobjects of X, where Loc(X) is regarded as a poset with the reverse order. Then Loc(X) is a locale with Spec(X) as the set of points. If C is spatial then Loc(X) is a spatial locale. Since any quotient of a reduced (resp. integral) object is reduced (resp. integral), Loc (resp. Spec) is naturally a functor from C to the category of locals (resp. topological spaces). Our main theorem states that the analytic topology can be determined algebraically:
Theorem. The functor Loc is equivalent to the analytic topology on an analytic geometry C.
If C is spatial then Spec determines Loc, thus in this case we simply say that Spec is the analytic topology on C. The space Spec(X) is called the spectrum of X.
A spatial analytic geometry C together with the topology Spec is a metric site defined in Luo[1995a]. Any object in C is separated (i.e. its diagonal map is universally closed). In fact Spec is the smallest separated metric topology on C if C has epi-reg-mono factorizations. The metric completion of a strict spatial analytic geometry (see [Luo 1995a]) plays the role of schemes in categorical geometry.
8. Zariski Geometries
A complete strict coregular analytic category is a Zariski geometry if it has a strong cogenerating set of disjunctable finitely copresentable objects including the initial object. The opposite of a Zariski geometry is precisely a Zariski category in the sense of Diers [1992], whose analytic topology coincides with the Zariski topology defined by Diers. A locally disjunctable analytic category is called a coherent analytic geometry if it is locally finitely copresentable such that the sum of the terminal object with itself is finitely copresentable object. Any coherent analytic geometry is a spatial analytic geometry with coherent spectrums. Most of the theorems proved by Diers in [D 1992] for a Zariski category can be extended to any coherent geometry in the dual situation.
9. Algebraic Geometries
An (abstract) algebraic geometry is a Zariski geometry with a single algebraic cogenerator. More precisely, a complete strict analytic category is an algebraic geometry if it has a disjunctable locally finitely copresentable object X such that the functor hom(~, X): Cop --> Set preserves and reflects regular maps. The opposite of an algebraic geometry is a finitary quasivarieties in the sense of universal algebra. Since in a quasivarieties the lattice of congruences is isomorphic to the dual of the lattice of regular quotients, the analytic topology of an algebraic geometry can be defined in terms of prime congruences. In the case of commutative rings we recover the fact that the Zariski topology is defined by the spectrum of prime ideals. The following equation explains the relationship between algebra and geometry:
10. Examples
An analytic category is coflat if any map is coflat (or equivalently, any epi is stable). In a coflat analytic category any epi is unipotent, any singular mono is analytic, any normal mono (thus any analytic mono) is strong, and any integral object is simple. In a reduced coflat disjunctable analytic category, the notions of strong, normal, analytic, singular, and fractional mono are the same.
(1) Any elementary topos is a coflat disjunctable analytic category; its analytic topology is determined by the double negation ; a topos is reduced iff it is boolean; a reducible Grothendieck topos is an analytic geometry.
(2) The category of locales is a reduced analytic geometry; its analytic topology is the functor sending each locale to the locale of its nuclei.
(3) The category of sets (resp. topological spaces, resp. posets) is a reduced coflat disjunctable spatial analytic geometry; its analytic topology is the discrete topology.
(4) The category of coherent spaces (resp. Stone spaces) is a reduced coherent analytic geometry; its analytic topology is the patch topology.
(5) The category of Hausdorff spaces is a strict reduced disjunctable spatial analytic geometry; its analytic topology is the Hausdorff topology.
(6) The opposite of the category of commutative rings (resp. reduced rings) is an algebraic geometry (resp. reduced algebraic geometry); its analytic topology is the Zariski topology.