abstract algebraic geometry

[categories <cat-dist@mta.ca>]

Date: Tue, 4 Nov 1997 14:17:40 -0800
From: Zhaohua Luo <zack@iswest.com>

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