The Language of Analytic
Categories## 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 CategoriesConsider a category with an initial object0. 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
and _{X} --> ZZ of _{Y} --> Zx and y along f
exist, and the induced map Z is
an isomorphism.
_{X} + Z_{Y} --> ZAssume the category has pullbacks. A An Consider an analytic category. For any object
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 ) iff it is initial.
_{Z}
= 1_{Z}Suppose 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 and _{Y}f(^{-1}1)_{X}
= 1 (i.e. _{Y}f is bounded). Also ^{-1}f
has a left adjoint ^{-1}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 is simply the
strong image of ^{+1}(Y)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 PropertiesLetA 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 and _{1} -->
X_{1}f:_{2} Y
be two maps. Then _{2} --> X_{2}f is epic (resp. monic,
resp. regular monic) iff _{1} + f_{2}f and _{1}f
are so.
_{2}## 3. Coflat MapsA mapf: 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 { f:_{i} Y} is a finite family of coflat maps. Then _{i}
--> Xf =
f:_{i} Y =
Y is coflat.
_{i} --> X(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 is a map
which is disjoint with a coflat map _{1} --> Xf: 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 is the identity ^{-1}f^{+1}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 MonosA monou:^{c} U is
a ^{c} --> Xcomplement of a mono u:
U --> X if u and u are disjoint, and any
map ^{c}v: T --> X such that u and v are disjoint
factors through u (uniquely). The complement ^{c}u
of ^{c}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.
(4.4) 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 ObjectsA map isunipotent 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).
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 ObjectsA non-initial object isprimary
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 (6.1) Any quotient of a primary object is primary; any
primary object is quasi-primary.
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.
A mono (or subobject) is called a (7.1) The class of fractions is closed under composition
and stable under pullback.
## 8. Local ObjectsA non-initial objectX 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 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 GeometriesAnanalytic 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 (9.1) Any object has a radical; the full subcategory of
reduced subobjects is a reduced analytic geometry.
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 (cf. [L2]).
C (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 U are strong subobjects of _{1}, U_{2},
..., U_{n}X such that U,
_{i}U are disjoint for all _{j}i
j, then the induced map
U
is an isomorphism.
_{i} --> U_{i}## 10. Coherent Analytic GeometriesMost 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 It is easy to see that a coherent analytic category is
an analytic category. A Note that a category is a coherent analytic category (resp.
Stone geometry) iff its opposite is a Let (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.
Let (10.7) A coherent analytic geometry is a spatial analytic
geometry; The spectrum |