An abstract model of the
category
of commutative algebras (cf.
Commutative algebra)
in which basic commutative algebra and
algebraic geometry
can be performed. Zariski categories are axiomatically defined as categories
satisfying the following six axioms:
1)
is
co-complete,
i.e., has all small colimits;
2)
has a strong generating set of objects whose objects are finitely presentable and
flatly co-disjunctable;
3)
regular epimorphisms in
are
universal,
i.e., stable under pull-backs along any morphism (cf.
Morphism);
4)
the terminal object of
is finitely presentable and has no proper subobject;
5)
the product of two objects in
is
co-universal,
i.e., stable under pushouts along any morphism;
6)
for any finite sequence of co-disjunctable congruences
on any object
,
with respective co-disjunctors
,
one has
where
denotes union of congruences on
on the left-hand side and co-union of quotient objects of
on the right-hand side.
These axioms rely on the notions of limits, colimits, finitely presentable objects,
and co-disjunctors
[a1].
This latter notion describes in a
universal way the calculus of fractions, and is defined as follows. Let
be an arbitrary pair of parallel morphisms in
.
It is said to be
co-disjointed
if its co-equalizer is a strict terminal object. A morphism
co-disjoints
if
is co-disjointed. A
co-disjunctor
of
is a morphism
that co-disjoints
and through which any morphism
that co-disjoints
factors uniquely. The pair
is
co-disjunctable
if it has a co-disjunctor. The object
is
co-disjunctable
if the pair of inductions
of
into its
coproduct
by itself is co-disjunctable. It is
flatly co-disjunctable
if, moreover, the co-disjunctor
of
is a
flat morphism,
i.e., the pushout functor along
preserves monomorphisms.
The whole of basic commutative algebra and algebraic geometry can be formally developed
in any Zariski category as if its objects were commutative rings. Any result proven
in an arbitrary Zariski category has various interpretations in various concrete
categories of commutative algebras of different kinds, possible equipped with some
extra structure such as an order, lattice, gradation, filtration, differentiation,
etc.