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