The following is the third part of "The language of analytic categories", which is a report on my paper CATEGORICAL GEOMETRY. Again comments and suggestions are welcome. Z. Luo _____________________________________________ THE LANGUAGE OF ANALYTIC CATEGORIES By Zhaohua Luo (1997) ------------------------------------------------------------ Content 1. Analytic Categories 2. Distributive Properties 3. Coflat Maps 4. Analytic Monos 5. Reduced Objects 6. Integral Objects 7. Simple objects 8. Local Objects 9. Analytic Geometries 10. Zariski Geometries References Analytic Dictionary ------------------------------------------------------------- 9. Analytic Geometries An "analytic geometry" is an analytic category satisfying the following axioms: (Axiom 4) Any intersection of strong subobjects exists. (Axiom 5) Any non-initial object has a non-initial reduced strong subobject. (Axiom 6) Any strong subobject is an intersection of disjunctable strong subobjects. Thus an analytic geometry is a perfect, reducible, and locally disjunctable analytic category. Suppose C is an analytic geometry. (9.1) Any object has a radical; the full subcategory of reduced subobjects is a reduced analytic geometry. (9.2) If X is the join of two strong subobjects U and V in R(X), then {U, V} is a unipotent cover on X. (9.3) If U and V are two strong subobjects of an object X, then rad(U \vee V) = rad(U) \vee rad(V). (9.4) Denote by D(X) the set of reduced strong subobjects of X. The radical mapping rad: R(X) --> D(X) is the right adjoint of the inclusion D(X) --> R(X), which preserves finite joins. (9.5) The dual D(X)^{op} of the lattice D(X) is a locale; a reduced strong subobject is integral if and only if it is a prime element of D(X)^{op}. (9.6) The spectrum Spec(X) of an object X is homeomorphic to the space of points of the locale D(X)^{op} (therefore is a sober space); an analytic geometry is spatial iff D(X)^{op} is a spatial locale for each object X. (9.7) The functor sending each object X to D(X)^{op} and each map f: Y --> X to rad(f)^{-1} is equivalent to the analytic topology on C (cf. [L4]). (9.8) If V is a strong subobject of a non-initial object X in a spatial analytic geometry then the join of all the primes contained in V is the radical of V. (9.9) A non-initial reduced object X in a spatial analytic geometry is integral iff its spectrum is irreducible. (9.10) Suppose f: Y --> X is a mono in a spatial analytic geometry. If f is coflat then Spec(f) is a topological embedding; if f is analytic then Spec(f) is an open embedding; if f is strong then Spec(f) is a closed embedding. (9.11) (Chinese remainder theorem) Let X be an object in a strict analytic geometry. Suppose U_1, U_2, ..., U_n are strong subobjects of X such that U_i, U_j are disjoint for all i \neq j, then the induced map \sum U_i --> \vee U_i is an isomorphism. -------------------------------------------------------------- 10. Zariski Geometries Most of the results stated in this section are due to Diers (in the dual situation). Our purpose is to present a geometric approach using the language of analytic categories developed above. A category is "coherent" if the following three axioms are satisfied: (Axiom 7) It is locally finitely copresentable. (Axiom 8) Finite sums are disjoint and stable. (Axiom 9) The sum of its terminal object with itself is finitely copresentable. It is easy to see that a coherent category is an analytic category. A "Zariski geometry" (resp. "Stone geometry") is a locally disjunctable (resp. locally decidable) coherent category. Diers proved in [D1] that a locally finitely copresentable category is a coherent category (resp. Stone geometry) iff its full subcategory of finitely copresentable objects is lextensive (resp. lextensive and decidable). Note that a category is a coherent category (resp. Stone geometry) iff its opposite is a "locally indecomposable category" (resp. "locally simple category") in the sense of [D1]. Let C be a coherent category. A map f: Y --> X is called "indirect" if it does not factor through any proper direct mono to X. A non-initial object is "indecomposable" if it has exactly two direct subobjects. A maximal indecomposable subobject is called an "indecomposable component". (10.1) Any non-initial object has a simple prime and an extremal simple subobject; a coherent category is a spatial reducible perfect analytic category. (10.2) Cofiltered limits and products of coflat maps are coflat; intersections of coflat monos are coflat monos; intersections of fractions are fractions; any map can be factored uniquely as a quasi-local map followed by a fraction. (10.3) Any composite of locally direct mono is locally direct; any map can be factored uniquely as an indirect map followed by a locally direct mono. (10.4) Any non-initial object has an indecomposable component; an indecomposable subobject is an indecomposable component iff it is a locally direct subobject. (10.5) The extensive topology is naturally a strict metric topology, which is determined by the canonical functor to the category of Stone spaces (preserving cofiltered limits and colimits whose right adjoint preserving sums). (10.6) A Stone geometry is a strict reduced Zariski geometry whose opposite is a regular category, and its analytic topology coincides with the extensive topology. Let C be a Zariski geometry. A "locality" is a fraction with a local object as domain. A "local isomorphism" is a map f: Y --
X such that, for any locality v: V --> Y, the composite f.v: V --> X is a locality. A complement of a set of strong monos is called a "semisingular mono". Note that (10.13) below implies that our definitions of reduced and integral objects coincide with those of Diers's in a Zariski geometry.
(10.8) A Zariski geometry is a spatial analytic geometry; The spectrum Spec(X) of any object is a coherent space for any object X; if f: Y --> X is a unipotent map then Spec(f) is surjective. (10.9) If f: Y --> X is a finitely copresentable (i.e. f is a finitely copresentable object in C/X) local isomorphism, then Spec(f): Spec(Y) --> Spec(X) is an open map. (10.10) A simple subobject on an object is a residue iff it is maximal (i.e. it is not contained in any larger simple subobject); any integral object X has a unique generic residue. (10.11) (Going Up Theorem) If f: Y --> X is a coflat map and V is in the image of Spec(f), any prime of X containing V is also in the image of Spec(f) (i.e. the image of Spec(f) is closed under generalizations). (10.12) Any colimits and cofiltered limits of reduced objects is reduced; the full subcategory of reduced objects is a reduced Zariski geometry. (10.13) An object is integral (resp. reduced) iff it is a quotient of a simple object (resp. a coproduct of simple objects). (10.14) A Zariski geometry is strict iff any finite analytic cover is not contained in any proper subobject. Suppose C is strict. A mono is analytic iff it is singular (resp. a finitely copresentable fraction); a mono is a fraction iff it is semisingular (resp. a local isomorphism); a mono is direct iff it is strong and analytic. References [D1] Diers, Y. Categories of Boolean Sheaves of Simple Algebras, Lecture Notes in Mathematics Vol. 1187, Springer Verlag, Berlin, 1986. [D2] Diers, Y. Categories of Commutative Algebras, Oxford University Press, 1992. [L1] Luo, Z. On the geometry of metric sites, Journal of Algebra 176, 210-229, 1995. [L2] Luo, Z. On the geometry of framed sites, preprint, 1995. [L3] Luo, Z. Categorical Geometry, preprint, 1997. [L4] Luo, Z. Abstract Algebraic Geometry, preprint, 1997. ---------------------------------------------------------- END OF THIRD PART