From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/498 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: abstract algebraic geometry Date: Thu, 16 Oct 1997 16:53:13 -0300 (ADT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017001 26014 80.91.229.2 (29 Apr 2009 14:56:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:56:41 +0000 (UTC) To: categories Original-X-From: cat-dist Thu Oct 16 16:53:28 1997 Original-Received: by mailserv.mta.ca; id AA11501; Thu, 16 Oct 1997 16:53:13 -0300 Original-Lines: 263 Xref: news.gmane.org gmane.science.mathematics.categories:498 Archived-At: 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.