Date: Thu, 16 Oct 1997 16:53:57 -0300 (ADT) Subject: abstract algebraic geometry Date: Thu, 16 Oct 1997 10:23:55 -0700 From: Zhaohua Luo The following report is based on my paper CATEGORICAL GEOMETRY. The plan is to generalize (and simplify) Diers's theory of Zariski categories (presented in his book [D]). A more detailed report (in LaTex) is available upon request. Comments and suggestions are welcome. Zack Luo ABSTRACT ALGEBRAIC GEOMETRY by Zhaohua Luo (1997) 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 [CLW]. 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 topological spaces, locales, posets, 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 [D]. 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 [D] 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 [L1] and [L2]. 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 N(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 N(S) consists of only initial map. We say S is a "normal sieve" if S = N(N(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 G from C to the category of locales. A mono u: U --> X in C (and G(u): G(U) --> G(X)) is called "open effective" if G(u) is an open embedding of locales, and any map t: T --> X in C such that G(t) factors through G(u) factors through u. If u is open effective then u or U is called an "open effective subobject" of X, and G(u) or G(U) is an "open effective sublocale" of G(X). We say G is a "framed topology" on C if an object X is initial iff G(X) is initial, and any open sublocale of G(X) is a join of open effective sublocales. If {U_i} is a set of open effective subobjects of X such that G(X) is the join of {G(U_i)}, then we say that {U_i} (resp. {G(U_i)}) is an "open effective cover" on X (resp. G(X)). The collection T(G) of open effective covers is a Grothendieck topology on C. We say G is "strict" if its Grothendieck topology T(G) is subcanonical. 3. Divisors Here is a general method to define framed topologies. A class D of maps containing isomorphisms is called a "divisor" if it is closed under compositions, and its pullback along any map exists which is also in D; we say D is "normal" if any map in D is a normal mono. If D is a divisor, a sieve with the form N(N(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 normal 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 normal divisor E(C), called the "extensive divisor". The extensive divisor E(C) determines a framed topology, called the "extensive topology". It generalizes the Stone topology on the category of Stone spaces. 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 [BP] and Diers [D1]. 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 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. A complement mono is normal. A mono v: V --> X is called "singular" if it is the complement of a strong mono u: U --> X. 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 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 normal divisor A(C), called the "analytic divisor". The analytic divisor A(C) determines a framed topology, called the "analytic topology". It generates the usual Zariski topology on affine schemes. 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 non-initial object is "integral" if any non-initial coflat map 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. 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). The functor Loc is equivalent to the analytic topology on 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 my paper [L1]. 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. The metric completion of a strict spatial analytic geometry plays the role of "schemes" in categorical geometry. 8. Zariski Geometries A cocomplete regular category with a strict analytic opposite is a Zariski category in the sense of Diers if it has a strong generating set of finitely presentable objects including the terminal object which are disjunctable in its opposite. The opposite of a Zariski category is a strict spatial analytic geometry, whose analytic topology coincides with the Zariski topology defined by Diers. We introduce a (simplified) geometric version of a Zariski category. A strict locally disjunctable analytic category is called a "Zariski geometry" if it is a locally finitely copresentable category with a finitely copresentable initial object. Any Zariski geometry is a strict spatial analytic geometry with coherent spectrums. Most of the theorems proved by Diers in [D] for a Zariski category can be extended to any Zariski geometry. 9. Examples (a) 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. (b) In a reduced coflat disjunctable analytic category, the notions of strong, normal, analytic, singular, and fractional mono are the same. (c) 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. (d) The category of locales is a reduced analytic geometry; its analytic topology is the functor sending each locale to the locale of its nuclei. (e) The category of topological spaces (resp. posets) is a reduced coflat disjunctable spatial analytic geometry; its analytic topology is the discrete topology. (f) The category of coherent spaces (resp. Stone spaces) is a reduced spatial analytic geometry; its analytic topology is the patch topology. (g) The category of Hausdorff spaces is a strict reduced disjunctable spatial analytic geometry; its analytic topology is the Hausdorff topology. (h) The opposite of the category of commutative rings is a Zariski geometry; its analytic topology is the Zariski topology. References [BP] Barr, M. and Pare, R. Molecular toposes, J. Pure Applied Algebra 17, 127 -152, 1980 [CLW] Carboni, C. Lack, S. and Walters, R. F. C. Introduction to extensive and distributive categories, Journal of Pure and Applied Algebra 84, 145-158, 1993. [D] Diers, Y. Categories of Commutative Algebras, Oxford University Press, 1992. [D1] Diers, Y. Categories of Boolean Sheaves of Simple Algebras, Lecture Notes in Mathematics Vol. 1187, Springer Verlag, Berlin, 1986. [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. Date: Wed, 5 Nov 1997 17:35:05 -0400 (AST) Subject: abstract algebraic geometry Date: Tue, 4 Nov 1997 14:17:40 -0800 From: Zhaohua Luo The following is the first part of "The language of analytic categories", which is a report on my paper CATEGORICAL GEOMETRY. The entire report (in LaTex) is available upon request. 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 Appendix: Analytic Dictionary ------------------------------------------------------------- 1. Analytic Categories Consider a category with an initial object 0. Two maps u: U - -> X and v: V --> X are "disjoint" if 0 is the pullback of (u, v). Suppose X + Y is the sum of two objects with the injections (also called "direct monos") x: X --> X + Y and y: Y --> X + Y. Then X + Y is "disjoint" if the injections x and y are disjoint and monic. The sum X + Y is "stable" if for any map f: Z --> X + Y, the pullbacks Z_X --> Z and Z_Y --> Z of x and y along f exist, and the induced map Z_X + Z_Y --> Z is an isomorphism. Assume the category has pullbacks. A "strong mono" is a map (in fact, a mono) such that any of its pullbacks is not proper (i.e. non-isomorphic) epic. The category is "perfect" if any intersection of strong monos exist. If a map f is the composite m.e of an epi e followed by a strong mono m then the pair (e, m) is called an "epi-strong-mono factorization" of f; the codomain of e is called the "strong image" of f. In a perfect category any map has an epi-strong-mono factorization. An "analytic category" is a category satisfying the following axioms: (Axiom 1) Finite limits and finite sums exist. (Axiom 2) Finite sums are disjoint and stable. (Axiom 3) Any map has an epi-strong-mono factorization. Consider an analytic category. For any object X denote by R(X) the set of strong subobjects of X. Since finite limits exist, the poset R(X) has meets. Suppose u: U --> X and v: V --> X are two strong subobjects. Suppose T = U + V is the sum of U and V and t: T --> X is the map induced by u and v. Then the strong image t(T) of T in X is the join of U and V in R(X). It follows that R(X) has joins. Thus R(X) is a lattice, with 0_X: 0 --> X as zero and 1_X: X --> X as one. If the category is prefect then R(X) is a complete lattice. An object Z has exactly one strong subobject (i.e. 0_Z = 1_Z) iff it is initial. If u: U --> X is a mono, we denote by f^{-1}(u) the pullback of u along f. Then f^{-1}: R(X) --> R(Y) is a mapping preserving meets with f^{-1}(0_X) = 0_Y and f^{-1}(1_X) = 1_Y (i.e. f^{-1} is bounded). Also f^{-1} has a left adjoint f^{+1}: R(Y) --> R(X) sending each strong subobject v: V -- > Y to the strong image of the composite f. v: V --> X. If V = Y then f^{+1}(Y) is simply the strong image of f. The theories of analytic categories and Zariski geometries (including the notions of coflat maps and analytic monos) given below are based on the works of Diers (see [D] and [D1]). Note that we have only covered the most elementary part of the theory of Zariski geometries (up to the first three chapters of [D]). ----------------------------------------------------------- 2. Distributive Properties Let C be an analytic category. Recall that a "regular mono" is a map which can be written as the equalizer of some pair of maps. (2.1) The class of strong monos is closed under composition and stable under pullback; any intersection of strong monos is a strong mono. (2.2) An epi-strong-mono factorization of a map is unique up to isomorphism. (2.3) Any regular mono is a strong mono; any pullback of a regular mono is a regular mono; any direct mono is a regular mono; finite sums commute with equalizers. (2.4) Any proper (i.e. non-isomorphic) strong subobject is contained in a proper regular subobject; a map is not epic iff it factors through a proper regular (or strong) mono. (2.5) The initial object 0 is strict (i.e. any map X --> 0 is an isomorphism); any map 0 --> X is regular (thus is not epic unless X is initial). (2.6) If the terminal object 1 is strict (i.e. any map 1 --> X is an isomorphism) then the category is equivalent to the terminal category 1 (thus the opposite of an analytic category is not analytic unless it is a terminal category). (2.7) Let f_1: Y_1 --> X_1 and f_2: Y_2 --> X_2 be two maps. Then f_1 + f_2 is epic (resp. monic, resp. regular monic) iff f_1 and f_2 are so. --------------------------------------------------------------- 3. Coflat Maps A map f: Y --> X is "coflat" if the pullback functor C/X --> C/Y along f preserves epis. More generally a map f: Y --> X is called "precoflat" if the pullback of any epi to X along f is epic. A map is coflat iff it is "stable precoflat" (i.e. any of its pullback is precoflat). An analytic category is "coflat" if any map is coflat (or equivalently, any epi is stable). (3.1) Coflat maps (or monos) are closed under composition and stable under pullback; isomorphisms are coflat; any direct mono is coflat. (3.2) Finite products of coflat maps are coflat; a finite sum of maps is coflat iff each factor is coflat. (3.4) Suppose f: Y --> X is a mono and g: Z --> Y is a map. Then g is coflat if f.g is coflat. (3.5) For any object X, the codiagonal map X + X --> X is coflat. (3.6) Suppose {f_i: Y_i --> X} is a finite family of coflat maps. Then f = \sum (f_i): Y = \sum Y_i --> X is coflat. (3.7) Suppose f: Y --> X is a coflat bimorphisms. If g: Z --> Y is a map such that f.g is an epi, then g is an epi. (3.8) Suppose f: Y --> X is a coflat mono (bimorphisms) and g: Z --> Y is any map. Then g is a coflat mono (bimorphisms) iff f.g is a coflat mono (bimorphisms). (3.9) If x: X_1 --> X is a map which is disjoint with a coflat map f: Y --> X, then the strong image of x is disjoint with f. (3.10) If f: Y --> X is a coflat map, then f^{-1}: R(X) --> R(Y) is a morphism of lattice. (3.11) If f: Y --> X is a coflat mono, then f^{-1}f^{+1} is the identity R(Y) --> R(Y). (3.12) (Beck-Chevalley condition) Suppose f: Y --> X is a coflat map and g: S --> X is a map. Let (p: T --> Y, n: T --> S) be the pullback of (f, g). Then p^{+1}n^{-1} = f^{- 1}g^{+1}. ------------------------------------------------------------------ 4. Analytic Monos A mono u^c: U^c --> X is a "complement" of a mono u: U -- > X if u and u^c are disjoint, and any map v: T --> X such that u and v are disjoint factors through u^c (uniquely). The complement u^c of u, if exists, is uniquely determined up to isomorphism. A mono is "singular" if it is the complement of a strong mono. An "analytic mono" is a coflat singular mono. A mono is "disjunctable" if it has a coflat complement. An analytic category is "disjunctable" if any strong mono is disjunctable; it is "locally disjunctable" if any strong mono is an intersection of disjunctable strong monos. (4.1) Analytic monos are closed under composition and stable under pullbacks; isomorphisms are analytic monos; a mono is analytic iff it is a coflat complement of a mono; any direct mono is analytic. (4.2) The pullback of any disjunctable mono is disjunctable. (4.3) If u: U --> X and v: V --> X are two disjunctable strong subobjects of X, then u^c \cap v^c = (u \vee v)^c. (4.4) Finite intersections and finite sums of analytic monos are analytic monos. (4.5) Suppose any strong map is regular. Then C is disjunctable iff any object is disjunctable. It is locally disjunctable if there is a set of cogenerators consisting of disjunctable objects. ------------------------------------------------------------------ 5. Reduced Objects A map is "unipotent" if any of its pullback is not proper initial. A map (in fact, a mono) is "normal" if any of its pullback is not proper unipotent. A "reduced object" is an object such that any unipotent map to it is epic. A unipotent reduced strong subobject of an object X is called the "radical" of X, denoted by rad(X). A "reduced model" of an object X is the largest reduced strong subobject of X, denoted by red(X). An analytic category is "reduced" if any object is reduced. An analytic category is "reducible" if any non-initial object has a non-initial reduced strong subobject. If f: Y --> X is an epi we simply say that X is a "quotient" of Y. A "locally direct mono" is a mono which is an intersection of direct monos. An analytic category is "decidable" (resp. "locally decidable") if any strong mono is a direct (resp. locally direct) mono. (5.1) An object is reduced iff any unipotent strong mono to it is an isomorphism (i.e. any object has no proper unipotent strong subobject). (5.2) Any stable epi is unipotent; conversely any unipotent map in a reduced analytic category is a stable epi. (5.3) A unipotent strong subobject contains each reduced subobject. (5.4) A radical is the largest reduced and the smallest unipotent strong subobject (therefore is unique). (5.5) Any quotient of a reduced object is reduced; if f: Y --> X is a map and U is a reduced strong subobject of Y then its strong image f^{+1}(U) in X is reduced. (5.6) Any reduced subobject is contained in a reduced strong subobject. (5.7) The join of a set of reduced strong subobjects of an object (in the lattice of strong subobjects) is reduced. (5.8) Any analytic subobject of a reduced object is reduced. (5.9) An analytic category is reduced iff every strong mono is normal. (5.10) Any object in a perfect analytic category has a reduced model. (5.11) If X has a reduced model red(X) then any map from a reduced object to X factors uniquely through the mono r(X) --> X. (5.12) In a perfect analytic category the full subcategory of reduced subobjects is a coreflective subcategory. (5.13) The radical of an object X is the reduced model of X. (5.14) In a reducible analytic category the reduced model of an object is unipotent (thus is the radical); any object in a perfect reducible analytic category has a radical. (5.15) Any decidable or locally decidable analytic category is reduced. ----------------------------------------------------------------- 6. Integral Objects A non-initial reduced object is "integral" if any non-initial coflat map to it is epic. An integral strong subobject of an object X is called a "prime" of X. Denote by Spec(X) the set of primes of X. An analytic category is "spatial" if any non- initial object has a non-initial prime. (6.1) Any quotient of an integral object is integral; if f: Y --> X is a map and U a prime of Y, then f^{+1}(U) is a prime of X. (6.2) If U and V are two non-initial coflat (resp. analytic) subobjects of an integral object, then the intersection of U and V is non-initial. (6.3) Any non-initial analytic subobject of an integral object is integral. (6.4) In a locally disjunctable analytic category the following are equivalent for a non-initial reduced object X: (a) X is integral; (b) Any non-initial analytic mono is epic; (c) X is not the join of two proper strong subobjects in R(X). ------------------------------------------------------------------ THE END OF THE FIRST PART Date: Sat, 22 Nov 1997 08:57:26 -0400 (AST) Subject: abstract algebraic geometry Date: Thu, 20 Nov 1997 12:40:02 -0800 From: Zhaohua Luo The following is the second part of "The language of analytic categories", which is a report on my paper CATEGORICAL GEOMETRY. Please note that Section 6 on integral objects (which was included in the first part) has been modified in order to conform with the notion of a primary object by Diers. The fact is that there are several ways to introduce a primary object in a general analytic category, and the one give by Diers (for a Zariski category) happens to be the weakest one. The new definition of an integral object given below (being a reduced primary object) is therefore weaker than the old one given in the first part of this note, but the basic properties remain the same (see (6.1) - (6.3)). On the other hand, Diers's definition of an integral object in a Zariski category (being a quotient of a simple object) is the strongest one. In practice these definitions agree in most cases (for instance, see (6.4) and (6.5) below). Z. Luo ---------------------------------------------------------------- The opposite RING^op of the category RING of commutative rings (with unit) is an analytic category, which is equivalent to the category of affine schemes. Following Diers we have the following list: RING^op RING simple field integral integral domain reduced without non-null nilpotent elements radical the residue class ring with respect to its radical pseudo-simple exactly one prime ideal quasi-primary ab = 0 => (a or b is nilpotent) primary any zero divisor is nilpotent analytically closed total ring of quotients irreducible the ideal {0} is irreducible with respect to intersection regular von Neumann regular ring local local ring generic residue quotient field ---------------------------------------------------------------- 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 Appendix: Analytic Dictionary ---------------------------------------------------------------------- SECOND PART ------------------------------------------------------------------------ 6. Integral Objects Let C be an analytic category (i.e. a lextensive category with epi-strong-mono factorizations). A non-initial object is "primary" if any non-initial analytic subobject is epic. A non-initial object is "quasi-primary" if any two non-initial analytic subobjects has a non-initial intersection. An "integral" object is a reduced primary object. A "prime" of an object is an integral strong subobject. A non-initial object is "irreducible" if it is not the join of two proper strong subobjects. For any object X denote by Spec(X) the set of primes of X. If U is any analytic subobject of X we denote by X(U) the set of primes of X which is not disjoint with U, called an "affine subset" of X. Using (4.3) one can show that the class of affine subsets is closed under intersection. Thus affine subsets form a base for a topology on Spec(X). The resulting topological space Spec(X) is called the "spectrum" of X. Since the pullback of an analytic mono is analytic, it follows from (6.2) below that Spec is naturally a functor from C to the (meta)category of topological spaces. For instance, if C is the category of affine schemes or affine varieties then Spec coincides with the classical Zariski topology. (6.1) Any quotient of a primary object is primary; any primary object is quasi-primary. (6.2) Any quotient of an integral object is integral; if f: Y --> X is a map and U a prime of Y, then f^{+1}(U) is a prime of X. (6.3) Any non-initial analytic subobject of a primary object is primary; any non-initial analytic subobject of an integral object is integral. (6.4) Suppose C is locally disjunctable. The following are equivalent for a non-initial reduced object X: (a) Any non-initial coflat map to X is epic. (b) X is primary. (c) X is quasi-primary. (d) X is irreducible. (6.5) Suppose C is locally disjunctable. Then (a) An object is integral iff it is reduced and quasi-primary. (b) An object is integral iff it is reduced and irreducible. 7. Simple Objects A mono (or subobject) is called a "fraction" if it is coflat normal. A map to an object X is called "local" (resp. "generic") if it is not disjoint with any non-initial strong subobject (resp. analytic subobject). A map to an object X is called "quasi-local" if it does not factor through any proper fraction to X. A map to an object X is called "prelocal" if it does not factor through any proper analytic mono to X. A non-initial object is called "simple" (resp. "extremal simple", resp. "unisimple", resp. "pseudo-simple", resp. "quasi- simple", resp. "presimple") if any non-initial map to it is epic (resp. extremal epic, resp. unipotent, resp. local, resp. quasi- local, resp. prelocal). (7.1) The class of fractions is closed under composition and stable under pullback. (7.2) Any local map is quasi-local; any quasi-local map is prelocal; the class of local (resp. generic, resp. quasi-local, resp. prelocal) maps is closed under composition; a quasi- local fraction (resp. prelocal analytic mono) is an isomorphism. (7.3) Any unipotent map is both local and generic; any epi is generic. (7.4) An object X is simple (resp. extremal simple, resp. unisimple, resp. quasi-simple, resp. presimple) iff it has exactly two strong subobjects (resp. subobjects, resp. normal sieves, resp. fractions, resp. analytic subobjects). (7.5) Any simple object is integral; any extremal simple object and any reduced unisimple object is simple. (7.6) A non-initial object is pseudo-simple iff any non-initial strong subobject is unipotent; any simple object, extremal simple object, and unisimple object is pseudo-simple; any pseudo-simple object is quasi-simple; any quasi-simple object is presimple; any presimple object is primary. (7.7) Any reduced pseudo-simple object is simple; the radical of any pseudo-simple object is simple. (7.8) Suppose C is locally disjunctable reducible. The following are equivalent for an object X: (a) X is pseudo-simple. (b) X is quasi-simple. (c) X is presimple. (d) The radical of X is simple. (7.9) Suppose any coflat unipotent map is regular epic and any map to a simple object is coflat. Then (a) Any coflat mono is normal. (b) Any simple object is extremal simple and unisimple. 8. Local Objects A non-initial object X is called "local" if non-initial strong subobjects of X has a non-initial intersection M. An epic simple fraction of an integral object X is called a "generic residue" of X. A mono (or subobject) p: P --> X is called a "residue" of X if P is a generic residue of a prime of X. An object is called "regular" if any disjunctable strong mono to it is direct. An object is "analytically closed" if any epic analytic mono to it is an isomorphism. (8.1) Suppose X is a local object with the strong subobject M as above. Then M is the unique simple prime of X; any proper fraction U of X is disjoint with M; M --> X is a local map. (8.2) Any integral object has at most one generic residue, which is the intersection of all the non-initial fractions; any generic residue is a generic subobject. (8.3) Any simple fraction and any simple prime is a residue; any residue of an object is a maximal simple subobject (i.e. it is not contained in any other simple subobject); any two distinct residues of an object are disjoint with each other. (8.4) Suppose p: P --> U is a residue and u: U --> X is a fraction (resp. strong mono). Then u.p: P --> X is a residue of X. (8.5) Suppose f: P --> Z is a local map with P simple. Then Z is local and f^{+1}(P) is the simple prime of Z. (8.6) Suppose f: X --> Z is a local map and X is local. Then Z is local. (8.7) Suppose f: P --> X is a map and P is simple. Then (a) f is a local epi iff X is simple. (b) f is a local strong mono iff X is local with the simple prime P. (c) f is an epic fraction iff X is integral with the generic residue P. (8.8) Suppose C is locally disjunctable reducible. (a) Suppose f: P --> Z is a prelocal map with P simple. Then f is a local map; Z is a local object with f^{+1}(P) as the simple prime of Z. (b) Suppose f: X --> Z is a prelocal map and X is local. Then f is a local map and Z is a local object. (8.9) Any sum of regular objects is regular; any extremal quotient of a regular object is regular; any regular and presimple object is analytically closed. (8.10) Suppose C is a complete and cocomplete, well- powered and co-well-powered analytic category. Then (a) The union of any family of subobjects consisting of regular objects is regular. (b) The full subcategory of regular objects is a coreflective subcategory. (8.11) Suppose C is a locally disjunctable analytic category. Then (a) Any regular object is reduced. (b) A regular object is integral iff it is simple. ------------------------------------------------------------------ END OF SECOND PART Date: Sat, 20 Dec 1997 09:54:43 -0400 (AST) Subject: abstract algebraic geometry Date: Fri, 19 Dec 1997 12:22:05 -0800 From: Zhaohua Luo 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 Date: Mon, 27 Apr 1998 08:25:03 -0400 From: Zhaohua Luo Subject: categories: abstract algebraic geometry Last year I posted the following research notes on categorical geometry: Abstract algebraic geometry (10/16/97) The language of analytic categories (three parts, 11/4/97, 11/20/97, and 12/20/97) to this list. Recently I started a small page at my home page address: www.iswest.com/~zack which, at present, only contains the slightly modified version (html files) of these notes. The paper "categorical geometry" are still under preparation, and I hope the first three chapters will be available soon. Thanks to all those who showed interests in this project. Zack Luo Date: Wed, 06 May 1998 15:41:38 -0400 From: Zhaohua Luo Subject: categories: abstract algebraic geometry Please visit my home page Categorical Geometry (www.iswest.com/~zack) for the newly posted paper Categorical Geometry: 1 Analytic Categories Regards, Z. Luo Date: Tue, 19 May 1998 14:50:49 -0400 From: Zhaohua Luo Subject: categories: abstract algebraic geometry The following paper Categorical Geometry: 2 Analytic Topologies is available on my WWW home page at the following new address: http://modigliani.brandx.net/user/zluo/ (the old address will soon cease to work). Regards, Zack Luo Date: Mon, 13 Jul 1998 14:10:21 -0400 From: Zhaohua Luo Subject: categories: abstract algebraic geometry The following short note (see the abstract below) Atomic Categories is available on Categorical Geometry Homepage at the following address: http://www.azd.com Note that to read the special symbols on these pages requires a viewer under Win95. (thanks to Vaughan Pratt for bringing this to my attention). Please let me know if you would like to have a copy in dvi format. Z. Luo ------------------------------------------------------------------------------------- Atomic Categories Zhaohua Luo Abstract: Let C be a category with a strict initial object 0. A map is called "non-initial" if its domain is not an initial object. A non-initial object T is called "unisimple" if for any two non-initial maps f: X --> T and g: Y --> T there are non-initial maps r: R --> X and s: R --> Y such that fr = gs. We say that C is an "atomic category" if any non-initial object is the codomain of a map with a unisimple domain. Many natural (left) categories arising in geometry are atomic (such as the categories of sets, topological spaces, posets, coherent spaces, Stone spaces, schemes, local ringed spaces, etc.) In this short note we show that each atomic category carries a unique functor to the category of sets, which plays the traditional role of "underlying functor" in categorical geometry Date: Fri, 17 Jul 1998 15:28:29 -0400 From: Zhaohua Luo Subject: categories: abstract algebraic geometry The following short note (see the abstract below) Uniform Functors (html) is available on Categorical Geometry Homepage at the following address: http://www.azd.com (the dvi version is under preparation) Zhaohua (Zack) Luo ------------------------------------------------------------------------------------- Uniform Functors Zhaohua Luo Abstract: In a previous note [atomic categories] we introduced the notion of an atomic category, and showed that each atomic category C carries a canonical functor to the category of sets, called the unifunctor of C. We also introduced the notion of a uniform functor between atomic categories. In this note we give an intrinsic definition of a uniform functor between any two categories with strict initials. Roughly speaking a functor is uniform if it induces isomorphisms between the complete boolean algebras of normal sieves on the objects. We show that any uniform functor to the category of sets is unique up to equivalence. A functor between Grothendieck toposes is uniform iff it induces an isomorphism between the complete boolean algebras of complemented subobjects. Since any unifunctor is uniform, this implies that a Grothendieck topos is atomic iff the complete boolean algebra of complemented subobjects of each object is atomic (or equivalently, there is a uniform functor to the category of sets). Date: Sun, 27 Sep 1998 23:29:18 -0400 From: Zhaohua Luo Subject: categories: Abstract Algebraic Geometry The following short note (see the abstract below) A Note on Reduced Categories is available on Categorical Geometry Homepage at the following address: http://www.azd.com/reduced.html Note that this file (together with most of the other files in the homepage) can be read now by any viewer capable of graphics (the symbols are included as gif. files). Z. Luo __________________________________________________________________ A Note on Reduced Categories Zhaohua Luo Abstract: In this note we introduce the notion of a reduced object for any category A with a strict initial object 0. A pair of parallel maps f, g: X --> Z is called "disjointed" if its kernel is the initial map to X. It is called "nilpotent" if any map t: T --> X such that (tf, tg) is disjointed is initial. An object X is called "reduced" if any pair of distinct parallel maps with domain X is not nilpotent. A category A is called "reduced" if any object is reduced. One can show that any epic quotient of a reduced object is reduced. A class D of objects of A is called "uni-dense" if any non-initial object is the codomain of a map with a non-initial object in D as domain. We show that any uni-dense class D of a reduced category A is a set of generators. Other properties and criterions of reduced categories are also studied. Date: Mon, 09 Nov 1998 21:01:09 -0500 From: Zhaohua Luo Subject: categories: Axioms of Algebraic Geometry Axioms of Algebraic Geometry I Zhaohua Luo (11/7/98) (a draft) The axioms of algebraic geometry given below consist of three (well known) algebraic axioms (A1) - (A3) and three geometric axioms (G1) - (G3), based on Diers's axioms of Zariski categories. The complete html version of this note (with links) is available at http://www.azd.com/axioms.html. Comments and suggestions are welcome. Consider a functor U: A --> Set from a category A to the category Set of sets. Algebraic Axioms: (Axiom A1) U has a left adjoint. (Axiom A2) Any bijective morphism in A is an isomorphism. (Axiom A3) Any pair of parallel morphisms in A has a surjective coequalizer. Recall that a functor satisfying the axioms (A1) - (A3) is an algebraic functor, and the pair (A, U) is an algebraic category (or algebraic construct, or quasivariety). An algebraic functor U is finitary if it preserves direct colimits. Any algebraic functor is faithful. In the following we shall regard A as a concrete category over Set via an algebraic functor U, and identify an object X with its underlying set U(X). A difference of an object is an ordered pair (a, b) of elements of A, denoted formally by a - b. (a) A difference a - b is a zero if a = b. (b) A difference a - b is a unit if for any morphism f: A --> B, f(a) = f(b) implies that B is terminal. (c) A difference a - b is nilpotent if for any morphism f: A --> B, f(a) - f(b) is a unit implies that B is terminal. (d) An object is reduced if it has no non-zero nilpotent difference; (A, U) is reduced if any object is reduced. (e) A morphism f: A --> B is flat if the pushout functor C/A --> C/B along it preserves monomorphisms. (f) A difference a - b is invertible (or disjunctable) if there is a flat epimorphism i: A --> A(a, b) such that i(a) - i(b) is a unit, and any morphism j: A --> B factors through i if j(a) - j(b) is a unit. Suppose U x V is the product of two objects U and V with the projections u: U x V --> U and v: U x V --> V. The product U x V is co-universal if for any morphism f: U x V --> Z, let Z --> ZU and Z --> ZV be the pushouts of u and v along f, then the induced morphism Z --> ZU x ZV is an isomorphism. Geometric Axioms: (Axiom G1) Any object has a unit difference. (Axiom G2) The product of any two objects isco-universal. (Axiom G3) Any difference of an object is invertible. We call any functor U: A --> Set satisfying the above six axioms an algebraic-geometric functor. An algebraic-geometric category is a pair (A, U) consisting of a category A and an algebraic-geometric functor U on A. Remark. (cf. [Luo, Categorical Geometry]) (a) An algebraic-geometric category is the opposite of an analytic geometry. (b) A finitary algebraic-geometric category is the opposite of a coherent analytic geometry. (c) Any finitary algebraic-geometric category satisfies the first five of the six axioms of Zariski categories defined in Diers's book Categories of Commutative Algebras, Oxford University Press, 1992. The sixth axiom simply means that the category is strict, i.e. the Grothendieck topology defined by open subsets is subcanonical. Example. The following categories are algebraic-geometric categories: (a) The category of frames (non-finitary, reduced, non-strict). (b) The category of distributive lattices (finitary, reduced, non-strict). (c) The category of Boolean algebras (finitary, reduced, strict). (d) The category of commutative rings with identity (finitary, non-reduced, strict). (e) The category of reduced commutative rings (finitary, reduced, strict). (f) The category of commutative regular rings (finitary, reduced, strict).