ABSTRACT ALGEBRAIC GEOMETRYIt 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 2. Framed Topologies Consider a functor from 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 X such that (X) is the join of {( U)}, then we say that {_{i}U} (resp. {(_{i}U)}) is an _{i}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 divisor if it is closed under compositions, and its pullback along any map exists which is also in ; we say D is Dsubnormal if any map in is a normal mono. If D is a divisor, a sieve with the form DT, where T is any set of monos to X in , is called a D on D-sieveX. One can show the set (DX) of -sieves on DX is a locale and the pullbacks of -sieves along a map induce a morphism of locales. Thus each divisor D determine a functor DL() to the category of locales. If D is subnormal then DL() is a framed topology, called the Dframed topology determined by . D4. Extensive Topologies. Recall that a category with finite stable disjoint sums is an C), called the extensive divisor. The extensive divisor (EC) determines a framed topology, called the extensive topology. For any object 5. Analytic Topologies An A 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 C), called the analytic divisor. The analytic divisor (AC) determines a framed topology, called the analytic topology. We say C is strict if its analytic divisor (AC) 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 A unipotent reduced strong subobject of an object 7. Spectrums A strong mono is called Let
If A spatial analytic geometry 8. Zariski Geometries A complete strict coregular analytic category is a 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 algebra) + categorical ^{op}geometry = algebraic geometry.10. Examples An analytic category is (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. |