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.
