From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/517 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: abstract algebraic geometry Date: Wed, 5 Nov 1997 17:34:57 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017011 26086 80.91.229.2 (29 Apr 2009 14:56:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:56:51 +0000 (UTC) To: categories Original-X-From: cat-dist Wed Nov 5 17:34:58 1997 Original-Received: by mailserv.mta.ca; id AA17010; Wed, 5 Nov 1997 17:34:57 -0400 Original-Lines: 295 Xref: news.gmane.org gmane.science.mathematics.categories:517 Archived-At: 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