* abstract algebraic geometry
@ 1998-04-27 12:25 Zhaohua Luo
1998-04-28 13:46 ` Book available by ftp Giuseppe Longo
0 siblings, 1 reply; 11+ messages in thread
From: Zhaohua Luo @ 1998-04-27 12:25 UTC (permalink / raw)
To: categories
Last year I posted the following research notes on categorical geometry:
Abstract algebraic geometry (10/16/97)
The language of analytic categories (three parts, 11/4/97, 11/20/97, and
12/20/97)
to this list. Recently I started a small page at my home page address:
www.iswest.com/~zack
which, at present, only contains the slightly modified version (html
files) of these notes. The paper "categorical geometry" are still under
preparation, and I hope the first three chapters will be available soon.
Thanks to all those who showed interests in this project.
Zack Luo
^ permalink raw reply [flat|nested] 11+ messages in thread
* Book available by ftp
1998-04-27 12:25 abstract algebraic geometry Zhaohua Luo
@ 1998-04-28 13:46 ` Giuseppe Longo
0 siblings, 0 replies; 11+ messages in thread
From: Giuseppe Longo @ 1998-04-28 13:46 UTC (permalink / raw)
To: categories
The book below is currently out of print. Upon kind permission of
the M.I.T. Press, it is now available by ftp, via my web page (see
the book content page in Downloadable Papers).
Andrea Asperti and Giuseppe Longo. Categories, Types and
Structures: an introduction to Category Theory for the working
computer scientist. M.I.T.- Press, 1991. (pp. 1--300).
--Giuseppe Longo
http://www.dmi.ens.fr/users/longo
e-mail: longo@dmi.ens.fr
^ permalink raw reply [flat|nested] 11+ messages in thread
* Abstract Algebraic Geometry
@ 1998-09-28 3:29 Zhaohua Luo
0 siblings, 0 replies; 11+ messages in thread
From: Zhaohua Luo @ 1998-09-28 3:29 UTC (permalink / raw)
To: categories
The following short note (see the abstract below)
A Note on Reduced Categories
is available on Categorical Geometry Homepage at the following address:
http://www.azd.com/reduced.html
Note that this file (together with most of the other files in the
homepage) can be read now by any viewer capable of graphics (the symbols
are included as gif. files).
Z. Luo
__________________________________________________________________
A Note on Reduced Categories
Zhaohua Luo
Abstract:
In this note we introduce the notion of a reduced object for any
category A with a strict initial object 0. A pair of parallel maps f, g:
X --> Z is called "disjointed" if its kernel is the initial map to X. It
is called "nilpotent" if any map t: T --> X such that (tf, tg) is
disjointed is initial. An object X is called "reduced" if any pair of
distinct parallel maps with domain X is not nilpotent. A category A is
called "reduced" if any object is reduced. One can show that any epic
quotient of a reduced object is reduced. A class D of objects of A is
called "uni-dense" if any non-initial object is the codomain of a map
with a non-initial object in D as domain. We show that any uni-dense
class D of a reduced category A is a set of generators. Other properties
and criterions of reduced categories are also studied.
--------------A7DB306CE63DF22C54674FBE
Content-Type: text/html; charset=us-ascii
Content-Transfer-Encoding: 7bit
<HTML>
<BODY TEXT="#000000" BGCOLOR="#FFFFEA" LINK="#0000EE" VLINK="#551A8B" ALINK="#FF0000">
The following short note (see the abstract below)
<P>A Note on Reduced Categories
<P>is available on Categorical Geometry Homepage at the following address:
<P><A HREF="http://www.azd.com/reduced.html">http://www.azd.com/reduced.html</A>
<P>Note that this file (together with most of the other files in the homepage)
can be read now by any viewer capable of graphics (the symbols are included
as gif. files).
<P>Z. Luo
<BR>__________________________________________________________________
<P>A Note on Reduced Categories
<P>Zhaohua Luo
<P>Abstract:
<P>In this note we introduce the notion of a reduced object for any category
<B>A</B> with a strict initial object 0. A pair of parallel maps <I>f</I>,
<I>g</I>: <I>X</I> --> <I>Z</I> is called "<FONT COLOR="#000000">disjointed"
</FONT>if its kernel is the initial map to <I>X. </I>It is called "<FONT COLOR="#000000">nilpotent"</FONT>
if any map <I>t</I>: <I>T</I> --> <I>X</I> such that (<I>tf,</I>
<I>tg</I>) is disjointed is initial. An object <I>X</I> is called "<FONT COLOR="#000000">reduced"</FONT>
if any pair of distinct parallel maps with domain <I>X</I> is not nilpotent.
A category <B>A</B> is called "<FONT COLOR="#000000">reduced"</FONT> if
any object is reduced. One can show that any epic quotient of a reduced
object is reduced. A class <B>D</B> of objects of <B>A</B> is called "<FONT COLOR="#000000">uni-dense"</FONT>
if any non-initial object is the codomain of a map with a non-initial object
in <B>D</B> as domain. We show that any uni-dense class <B>D </B>of a reduced
category <B>A</B> is a set of generators. Other properties and criterions
of reduced categories are also studied.
<BR>
<BR>
<BR>
<BR>
<BR>
</BODY>
</HTML>
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^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1998-07-17 19:28 Zhaohua Luo
0 siblings, 0 replies; 11+ messages in thread
From: Zhaohua Luo @ 1998-07-17 19:28 UTC (permalink / raw)
To: categories
[-- Attachment #1: Type: text/plain, Size: 1366 bytes --]
The following short note (see the abstract below)
Uniform Functors (html)
is available on Categorical Geometry Homepage at the following address:
http://www.azd.com
(the dvi version is under preparation)
Zhaohua (Zack) Luo
-------------------------------------------------------------------------------------
Uniform Functors
Zhaohua Luo
Abstract:
In a previous note [atomic categories] we introduced the notion of an
atomic category, and showed that each atomic category C carries a
canonical functor to the category of sets, called the unifunctor of C.
We also introduced the notion of a uniform functor between atomic
categories. In this note we give an intrinsic definition of a uniform
functor between any two categories with strict initials. Roughly
speaking a functor is uniform if it induces isomorphisms between the
complete boolean algebras of normal sieves on the objects. We show that
any uniform functor to the category of sets is unique up to equivalence.
A functor between Grothendieck toposes is uniform iff it induces an
isomorphism between the complete boolean algebras of complemented
subobjects. Since any unifunctor is uniform, this implies that a
Grothendieck topos is atomic iff the complete boolean algebra of
complemented subobjects of each object is atomic (or equivalently, there
is a uniform functor to the category of sets).
[-- Attachment #2: Type: text/html, Size: 1628 bytes --]
^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1998-07-13 18:10 Zhaohua Luo
0 siblings, 0 replies; 11+ messages in thread
From: Zhaohua Luo @ 1998-07-13 18:10 UTC (permalink / raw)
To: categories
The following short note (see the abstract below)
Atomic Categories
is available on Categorical Geometry Homepage at the following address:
http://www.azd.com
Note that to read the special symbols on these pages requires a viewer
under Win95. (thanks to Vaughan Pratt for bringing this to my
attention). Please let me know if you would like to have a copy in dvi
format.
Z. Luo
-------------------------------------------------------------------------------------
Atomic Categories
Zhaohua Luo
Abstract:
Let C be a category with a strict initial object 0. A map is called
"non-initial" if its domain is not an initial object. A non-initial
object T is called "unisimple" if for any two non-initial maps f: X -->
T and g: Y --> T there are non-initial maps r: R --> X and s: R --> Y
such that fr = gs. We say that C is an "atomic category" if any
non-initial object is the codomain of a map with a unisimple domain.
Many natural (left) categories arising in geometry are atomic (such as
the categories of sets, topological spaces, posets, coherent spaces,
Stone spaces, schemes, local ringed spaces, etc.) In this short note we
show that each atomic category carries a unique functor to the category
of sets, which plays the traditional role of "underlying functor" in
categorical geometry
--------------84037E1FA36AB214A0902978
Content-Type: text/html; charset=us-ascii
Content-Transfer-Encoding: 7bit
<HTML>
<BODY TEXT="#000000" BGCOLOR="#FFFFEA" LINK="#0000EE" VLINK="#551A8B" ALINK="#FF0000">
The following short note (see the abstract below)
<P>Atomic Categories
<P>is available on Categorical Geometry Homepage at the following address:
<P><A HREF="http://www.azd.com">http://www.azd.com</A>
<P>Note that to read the special symbols on these pages requires a viewer
under Win95. (thanks to Vaughan Pratt for bringing this to my attention).
Please let me know if you would like to have a copy in dvi format.
<P>Z. Luo
<BR>-------------------------------------------------------------------------------------
<BR>Atomic Categories
<P>Zhaohua Luo
<P>Abstract:
<P>Let C be a category with a strict initial object 0. A map is called
"<FONT COLOR="#000000">non-initial"</FONT><I> </I>if its domain is not
an initial object. A non-initial object T is called "<FONT COLOR="#000000">unisimple"</FONT><I>
</I>if for any two non-initial maps f: X --> T and g: Y --> T<I> </I>there
are non-initial maps<I> </I>r: R --> X and s: R --> Y such that fr = gs<I>.</I>
We say that<B> C </B>is an "atomic category" if any non-initial object
is the codomain of a map with a unisimple domain. Many natural (left) categories
arising in geometry are atomic (such as the categories of sets, topological
spaces, posets, coherent spaces, Stone spaces, schemes, local ringed spaces,
etc.) In this short note we show that each atomic category carries a unique
functor to the category of sets, which plays the traditional role of "underlying
functor<I>" </I>in categorical geometry
<BR>
<BR>
<BR>
</BODY>
</HTML>
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^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1998-05-19 18:50 Zhaohua Luo
0 siblings, 0 replies; 11+ messages in thread
From: Zhaohua Luo @ 1998-05-19 18:50 UTC (permalink / raw)
To: categories
The following paper
Categorical Geometry:
2 Analytic Topologies
is available on my WWW home page at the following new address:
http://modigliani.brandx.net/user/zluo/
(the old address will soon cease to work).
Regards, Zack Luo
^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1998-05-06 19:41 Zhaohua Luo
0 siblings, 0 replies; 11+ messages in thread
From: Zhaohua Luo @ 1998-05-06 19:41 UTC (permalink / raw)
To: categories
Please visit my home page
Categorical Geometry
(www.iswest.com/~zack)
for the newly posted paper
Categorical Geometry:
1 Analytic Categories
Regards,
Z. Luo
^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1997-12-20 13:53 categories
0 siblings, 0 replies; 11+ messages in thread
From: categories @ 1997-12-20 13:53 UTC (permalink / raw)
To: categories
Date: Fri, 19 Dec 1997 12:22:05 -0800
From: Zhaohua Luo <zack@iswest.com>
The following is the third part of "The language of analytic
categories", which is a report on my paper CATEGORICAL
GEOMETRY. 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
Analytic Dictionary
-------------------------------------------------------------
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 C 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(U \vee V) = rad(U) \vee rad(V).
(9.4) Denote by D(X) the set of reduced strong subobjects of
X. The radical mapping rad: R(X) --> D(X) is the right
adjoint of the inclusion D(X) --> R(X), which preserves finite
joins.
(9.5) The dual D(X)^{op} of the lattice D(X) is a locale; a
reduced strong subobject is integral if and only if it is a prime
element of D(X)^{op}.
(9.6) The spectrum Spec(X) of an object X is homeomorphic
to the space of points of the locale D(X)^{op} (therefore is a
sober space); an analytic geometry is spatial iff D(X)^{op} is
a spatial locale for each object X.
(9.7) The functor sending each object X to D(X)^{op} and
each map f: Y --> X to rad(f)^{-1} is equivalent to the
analytic topology on C (cf. [L4]).
(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_1, U_2, ..., U_n are
strong subobjects of X such that U_i, U_j are disjoint for all i
\neq j, then the induced map \sum U_i --> \vee U_i is an
isomorphism.
--------------------------------------------------------------
10. Zariski 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 "coherent" 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 its terminal object with itself is finitely
copresentable.
It is easy to see that a coherent category is an analytic
category. A "Zariski geometry" (resp. "Stone geometry") is a
locally disjunctable (resp. locally decidable) coherent
category.
Diers proved in [D1] that a locally finitely copresentable
category is a coherent category (resp. Stone geometry) iff its
full subcategory of finitely copresentable objects is lextensive
(resp. lextensive and decidable). Note that a category is a
coherent category (resp. Stone geometry) iff its opposite is a
"locally indecomposable category" (resp. "locally simple
category") in the sense of [D1].
Let C be a coherent 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 Zariski geometry
whose opposite is a regular category, and its analytic
topology coincides with the extensive topology.
Let C be a Zariski 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 (10.13) below
implies that our definitions of reduced and integral objects
coincide with those of Diers's in a Zariski geometry.
(10.8) A Zariski 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.
(10.9) If f: Y --> X is a finitely copresentable (i.e. f is a
finitely copresentable object in C/X) local isomorphism, then
Spec(f): Spec(Y) --> Spec(X) is an open map.
(10.10) 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.
(10.11) (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).
(10.12) Any colimits and cofiltered limits of reduced objects
is reduced; the full subcategory of reduced objects is a
reduced Zariski geometry.
(10.13) An object is integral (resp. reduced) iff it is a quotient
of a simple object (resp. a coproduct of simple objects).
(10.14) A Zariski geometry is strict iff any finite analytic
cover is not contained in any proper subobject. Suppose C 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.
References
[D1] Diers, Y. Categories of Boolean Sheaves of Simple
Algebras, Lecture Notes in Mathematics Vol. 1187, Springer
Verlag, Berlin, 1986.
[D2] Diers, Y. Categories of Commutative Algebras, Oxford
University Press, 1992.
[L1] Luo, Z. On the geometry of metric sites, Journal of
Algebra 176, 210-229, 1995.
[L2] Luo, Z. On the geometry of framed sites, preprint, 1995.
[L3] Luo, Z. Categorical Geometry, preprint, 1997.
[L4] Luo, Z. Abstract Algebraic Geometry, preprint, 1997.
----------------------------------------------------------
END OF THIRD PART
^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1997-11-22 12:56 categories
0 siblings, 0 replies; 11+ messages in thread
From: categories @ 1997-11-22 12:56 UTC (permalink / raw)
To: categories
Date: Thu, 20 Nov 1997 12:40:02 -0800
From: Zhaohua Luo <zack@iswest.com>
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
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* abstract algebraic geometry
@ 1997-11-05 21:34 categories
0 siblings, 0 replies; 11+ messages in thread
From: categories @ 1997-11-05 21:34 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 11+ messages in thread
* abstract algebraic geometry
@ 1997-10-16 19:53 categories
0 siblings, 0 replies; 11+ messages in thread
From: categories @ 1997-10-16 19:53 UTC (permalink / raw)
To: categories
Date: Thu, 16 Oct 1997 10:23:55 -0700
From: Zhaohua Luo <zack@iswest.com>
The following report is based on my paper CATEGORICAL
GEOMETRY. The plan is to generalize (and simplify) Diers's
theory of Zariski categories (presented in his book [D]). A
more detailed report (in LaTex) is available upon request.
Comments and suggestions are welcome.
Zack Luo
ABSTRACT ALGEBRAIC GEOMETRY
by Zhaohua Luo (1997)
It is well known that most geometric-like categories have
finite limits and finite stable disjoint sums. These are
lextensive categories in the sense of [CLW]. We introduce
the notion of an analytic category, which is a lextensive
category with the property that any map factors as an epi
followed by a strong mono. The class of analytic categories
includes many natural categories arising in geometry, such as
the categories of topological spaces, locales, posets, affine
schemes, as well as all the elementary toposes.
A large class of analytic categories is formed by the opposites
of Zariski categories in the sense of Diers [D]. The notion of
a Zariski category captures the categorical properties of
commutative rings. Many algebraic-geometric analysis carried
by Diers for a Zariski category can be done for a more
general analytic category in the dual situation. We show that
the notion of a flat singular epi developed in [D] can be
applied to define a canonical functor from an analytic
category to the category of locales, which is a framed
topology in the sense of [L1] and [L2]. This topology plays
the fundamental role of Zariski topology in categorical
geometry.
1. Unipotent Maps and Normal Monos
Consider a category C with a strict initial object. Two maps
u: U --> X and v: V --> X are "disjoint" if the initial object is
the pullback of u and v. If S is a set of maps to an object X
we denote by N(S) the sieve of maps to X which is disjoint
with each map in S. The set S is called a "unipotent cover" on
X if N(S) consists of only initial map. We say S is a "normal
sieve" if S = N(N(S)). A map is called "unipotent" if it is a
unipotent cover. A mono is called "normal" if it generates a
normal sieve. If C has pullbacks then a mono is normal iff any
of its pullback is not proper unipotent. The class of unipotent
(resp. normal) maps is closed under compositions and stable,
and any intersection of normal monos is normal.
Geometrically a unipotent map (resp. normal mono) plays the
role of a surjective map (resp. embedding).
2. Framed Topologies
Consider a functor G from C to the category of locales. A
mono u: U --> X in C (and G(u): G(U) --> G(X)) is called
"open effective" if G(u) is an open embedding of locales, and
any map t: T --> X in C such that G(t) factors through G(u)
factors through u. If u is open effective then u or U is called
an "open effective subobject" of X, and G(u) or G(U) is an
"open effective sublocale" of G(X).
We say G is a "framed topology" on C if an object X is initial
iff G(X) is initial, and any open sublocale of G(X) is a join of
open effective sublocales. If {U_i} is a set of open effective
subobjects of X such that G(X) is the join of {G(U_i)}, then
we say that {U_i} (resp. {G(U_i)}) is an "open effective
cover" on X (resp. G(X)). The collection T(G) of open
effective covers is a Grothendieck topology on C. We say G
is "strict" if its Grothendieck topology T(G) is subcanonical.
3. Divisors
Here is a general method to define framed topologies. A class
D of maps containing isomorphisms is called a "divisor" if it is
closed under compositions, and its pullback along any map
exists which is also in D; we say D is "normal" if any map in
D is a normal mono. If D is a divisor, a sieve with the form
N(N(T)), where T is any set of monos to X in D, is called a
"D-sieve" on X. One can show the set D(X) of D-sieves on X
is a locale and the pullbacks of D-sieves along a map induce a
morphism of locales. Thus each divisor D determine a functor
L(D) to the category of locales. If D is normal then L(D) is a
framed topology, called the "framed topology" determined by
D.
4. Extensive Topologies.
Recall that a category with finite stable disjoint sums is an
extensive category. An extensive category C has a strict initial
object. An injection of a sum is simply called a "direct mono".
An intersection of direct monos is called a "locally direct
mono". The class of direct monos is a normal divisor E(C),
called the "extensive divisor". The extensive divisor E(C)
determines a framed topology, called the "extensive
topology". It generalizes the Stone topology on the category
of Stone spaces.
For any object X we denote by Dir(X) the set of locally direct
subobjects of X, viewed as a poset with the reverse order. If
any intersection of direct monos exist in C, then Dir(X) is a
locale for any object X, and Dir is naturally a functor from C
to the category of locales, which is equivalent to the
extensive topology. Special cases of extensive topologies
were considered by Barr and Pare [BP] and Diers [D1].
5. Analytic Topologies
An "analytic category" is a lextensive category with epi-
strong-mono factorizations. In the following we consider an
analytic category C. One of the most important notion
introduced by Diers to categorical geometry is that of a flat
singular map. We consider the dual notion. A mono v: V -->
X is a "complement" of a mono u: U --> X if u and v are
disjoint, and any map t: T --> X such that u and t are disjoint
factors through v. A complement mono is normal. A mono v:
V --> X is called "singular" if it is the complement of a strong
mono u: U --> X. A map f: Y --> X is called "coflat " if the
pullback functor C/X --> C/Y along it preserves epis. The
main point here is that any pullback along a coflat map
preserves epi-strong-mono factorizations.
A coflat singular mono is called an "analytic mono". A coflat
normal mono is called a "fraction" (thus any analytic mono is
a fraction). A fraction plays the role of local isomorphism in
algebraic geometry. The class of coflat maps (resp. analytic
monos, resp. fractions) is closed under compositions and
stable. The class of analytic monos is a normal divisor A(C),
called the "analytic divisor". The analytic divisor A(C)
determines a framed topology, called the "analytic topology".
It generates the usual Zariski topology on affine schemes. We
say C is "strict" if its analytic divisor A(C) is strict.
6. Reduced and Integral Objects
The analytic topology can also be defined algebraically, using
reduced and integral objects, as in the case of affine schemes.
An object is "reduced" if any unipotent map to it is epic. A
non-initial object is "integral" if any non-initial coflat map to
it is epic One can show easily that any quotient of a reduced
(resp. integral) object is reduced (resp. integral) (i.e. if f: Y --
> X is an epi and Y is reduced or integral then so is X).
A unipotent reduced strong subobject of an object X is called
the "radical" of X. It is the largest reduced and the smallest
unipotent strong subobject of X, thus is uniquely determined
by X. An analytic category is "reduced" if any unipotent map
is epic. An analytic category is reduced iff its strong monos
are normal. An analytic category is "reducible" (resp.
"spatial") if any non-initial object has a non-initial reduced
(resp. integral) strong subobject. If any intersection of strong
monos exist in C then the full subcategory of reduced objects
is a coreflective subcategory; if moreover C is reducible then
any object has a radical.
7. Spectrums
A strong mono is called "disjunctive" if it has an analytic
complement. An object is disjunctable if its diagonal map is a
disjunctable regular mono. An analytic category is called
"disjunctable" if any strong mono is disjunctable. An analytic
category is "locally disjunctable" if any strong subobject is an
intersection of disjunctive strong subobjects. A locally
disjunctable reducible analytic category in which any
intersection of strong subobjects exist is called an "analytic
geometry".
Let C be an analytic geometry. If X is an object we denote by
Loc(X) (resp. Spec(X)) the set of reduced (resp. integral)
strong subobjects of X, where Loc(X) is regarded as a poset
with the reverse order. Then Loc(X) is a locale with Spec(X)
as the set of points. If C is spatial then Loc(X) is a spatial
locale. Since any quotient of a reduced (resp. integral) object
is reduced (resp. integral), Loc (resp. Spec) is naturally a
functor from C to the category of locals (resp. topological
spaces). The functor Loc is equivalent to the analytic
topology on C. If C is spatial then Spec determines Loc, thus
in this case we simply say that Spec is the analytic topology
on C. The space Spec(X) is called the "spectrum" of X.
A spatial analytic geometry C together with the topology
Spec is a metric site defined in my paper [L1]. Any object in
C is separated (i.e. its diagonal map is universally closed). In
fact Spec is the smallest separated metric topology on C. The
metric completion of a strict spatial analytic geometry plays
the role of "schemes" in categorical geometry.
8. Zariski Geometries
A cocomplete regular category with a strict analytic opposite
is a Zariski category in the sense of Diers if it has a strong
generating set of finitely presentable objects including the
terminal object which are disjunctable in its opposite. The
opposite of a Zariski category is a strict spatial analytic
geometry, whose analytic topology coincides with the Zariski
topology defined by Diers. We introduce a (simplified)
geometric version of a Zariski category. A strict locally
disjunctable analytic category is called a "Zariski geometry" if
it is a locally finitely copresentable category with a finitely
copresentable initial object. Any Zariski geometry is a strict
spatial analytic geometry with coherent spectrums. Most of
the theorems proved by Diers in [D] for a Zariski category
can be extended to any Zariski geometry.
9. Examples
(a) An analytic category is "coflat" if any map is coflat (or
equivalently, any epi is stable). In a coflat analytic category
any epi is unipotent, any singular mono is analytic, any
normal mono (thus any analytic mono) is strong, and any
integral object is simple.
(b) In a reduced coflat disjunctable analytic category, the
notions of strong, normal, analytic, singular, and fractional
mono are the same.
(c) Any elementary topos is a coflat disjunctable analytic
category; its analytic topology is determined by the double
negation; a topos is reduced iff it is boolean; a reducible
Grothendieck topos is an analytic geometry.
(d) The category of locales is a reduced analytic geometry; its
analytic topology is the functor sending each locale to the
locale of its nuclei.
(e) The category of topological spaces (resp. posets) is a
reduced coflat disjunctable spatial analytic geometry; its
analytic topology is the discrete topology.
(f) The category of coherent spaces (resp. Stone spaces) is a
reduced spatial analytic geometry; its analytic topology is the
patch topology.
(g) The category of Hausdorff spaces is a strict reduced
disjunctable spatial analytic geometry; its analytic topology is
the Hausdorff topology.
(h) The opposite of the category of commutative rings is a
Zariski geometry; its analytic topology is the Zariski
topology.
References
[BP] Barr, M. and Pare, R. Molecular toposes, J. Pure
Applied Algebra 17, 127 -152, 1980
[CLW] Carboni, C. Lack, S. and Walters, R. F. C.
Introduction to extensive and distributive categories, Journal
of Pure and Applied Algebra 84, 145-158, 1993.
[D] Diers, Y. Categories of Commutative Algebras, Oxford
University Press, 1992.
[D1] Diers, Y. Categories of Boolean Sheaves of Simple
Algebras, Lecture Notes in Mathematics Vol. 1187, Springer
Verlag, Berlin, 1986.
[L1] Luo, Z. On the geometry of metric sites, Journal of
Algebra 176, 210-229, 1995.
[L2] Luo, Z. On the geometry of framed sites, preprint, 1995.
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