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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.Coherent Geometries
References

#### 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 ZX --> Z and ZY --> Z of x and y along f exist, and the induced map ZX + ZY --> 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 me 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 0X: 0 --> X as zero and 1X: 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. 0Z = 1Z ) iff it is initial.

Suppose f: Y --> X is a map. 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(0X) = 0Y and f-1(1X) = 1Y (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 [Diers 1986 and 1992]). Note that we have only covered the most elementary part of the theory of Zariski geometries (up to the first three chapters of [Diers 1992]).

#### 2. Distributive Properties

Let A 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 f1: Y1 --> X1 and f2: Y2 --> X2 be two maps. Then f1 + f2 is epic (resp. monic, resp. regular monic) iff f1 and f2 are so.

#### 3. Coflat Maps

A map f: Y --> X is coflat if the pullback functor A/X --> A/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 {fi: Yi --> X} is a finite family of coflat maps. Then f = fi: Y = Yi --> 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: X1 --> 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-1f+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+1n-1 = f- 1g+1.

#### 4. Analytic Monos

A mono uc: Uc --> X is a complement of a mono u: U --> X if u and uc are disjoint, and any map v: T --> X such that u and v are disjoint factors through uc (uniquely). The complement uc 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 uc vc = (u v)c.
(4.4) Finite intersections and finite sums of analytic monos are analytic monos.
(4.5) Suppose any strong map is regular. Then A 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 red(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 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 A to the (meta)category of topological spaces. For instance, if A 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 A 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 A 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 A 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.

(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 A 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.
(8.10) Suppose A 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 A is a locally disjunctable analytic category. Then
(a) Any regular object is reduced.
(b) A regular object is integral iff it is simple.

#### 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 A 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( V) = rad(U) rad(V).
(9.4) Denote by S(X) the set of reduced strong subobjects of X. The radical mapping rad: R(X) --> S(X) is the right adjoint of the inclusion S(X) --> R(X), which preserves finite joins.
(9.5) The dual Loc(X) =  S(X)op of the lattice S(X) is a locale; a reduced strong subobject is integral if and only if it is a prime element of Loc(X).
(9.6) The spectrum Spec(X) of an object X is homeomorphic to the space of points of the locale Loc(X) (therefore is a sober space); an analytic geometry is spatial iff Loc(X) is a spatial locale for each object X.
(9.7) The functor sending each object X to Loc(X) is equivalent to the analytic topology on C (cf. [L2]).
(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 U1, U2, ..., Un are strong subobjects of X such that Ui, Uj are disjoint for all j, then the induced map Ui --> Ui is an isomorphism.

#### 10. Coherent Analytic 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 a coherent analytic category 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 the terminal object with itself is finitely copresentable.

It is easy to see that a coherent analytic category is an analytic category. A coherent analytic geometry (resp. Stone geometry) is a locally disjunctable (resp. locally decidable) coherent category.

Note that a category is a coherent analytic category (resp. Stone geometry) iff its opposite is a locally indecomposable category (resp. locally simple category) in the sense of Diers . A locally finitely copresentable category is a coherent analytic category (resp. Stone geometry) iff its full subcategory of finitely copresentable objects is lextensive (resp. lextensive and decidable) These important facts were proved by Diers in [Diers 1983] in the dual situation.

Let A be a coherent analytic 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 coherent geometry whose opposite is a regular category, and its analytic topology coincides with the extensive topology.

Let A be a coherent analytic 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 (10.12) below implies that our definitions of reduced and integral objects coincide with those of Diers's in a Zariski category.

(10.7) A coherent analytic 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.
(10.8) If f: Y --> X is a finitely copresentable (i.e. f is a finitely copresentable object in A/X ) local isomorphism, then Spec(f): Spec(Y) --> Spec(X) is an open map.
(10.9) 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.
(10.10) (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).
(10.11) Any colimits and cofiltered limits of reduced objects is reduced; the full subcategory of reduced objects is a reduced coherent analytic geometry.
(10.12) An object is integral (resp. reduced) iff it is a quotient of a simple object (resp. a coproduct of simple objects).
(10.13) A coherent analytic geometry is strict iff any finite analytic cover is not contained in any proper subobject. Suppose A 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.

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