
1.2. Analytic Categories
Consider a category with an initial object 0 and finite sums.
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 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. A category with finite stable disjoint sums is
an called an extensive
category; a lextensive
category is an extensive category with finite limits (see Carboni,
Lack and Walters [1993]).
Definition 1.2.1. An analytic
category is a category satisfying the following axioms:
(a) Finite limits and finite sums exist.
(b) Finite sums are disjoint and stable.
(c) Any map has an epistrongmono factorization.
Thus an analytic category is simply a lextensive category with epistrongmono
factorizations. Since by definition any lextensive category has pullbacks,
the following two propositions follow from (1.1.5)
immediately.
Proposition 1.2.2. A lextensive category
A has epistrongmono factorizations if any
of the following conditions are satisfied:
(a) Any intersection of strong monos exists.
(b) A is a complete wellpowered category.
(c) A or its opposite is locally presentable.
Proof. (a) follows from (1.1.5.a);
(b) from (a); (c) is true by (b) because any locally presentable category
is complete, cocomplete, wellpowered and cowellpowered (see [Adamek
and Rosicky 1994, p.45]). n
Proposition 1.2.3. A lextensive category
has epiregularmono factorizations if any of the following conditions
are satisfied:
{a} The class of regular monos is closed under composition and intersection.
{b} Any composite of regular monos is a regular mono and any map has
a cokernel pair.
Proof. This follows from (1.1.5.b)
and (c). n
Example 1.2.3.1. The categories of
locales, topological spaces, coherent spaces, Hausdorff spaces, Stone spaces,
posets, ringed spaces, local ringed spaces, affine varieties, and any elementary
topos are analytic categories by (1.2.3).
Remark 1.2.4. Recall that a category
is regular if the following axioms are satisfied:
(a) Finite limits exist.
(b) Regular epis are stable.
(c) Any map has a regepimono factorization.
If every kernel pair has a coequalizer then (c) follows from the other
axioms (see [Borceux, Vol. II, p90]).
Remark 1.2.5. The following are equivalent
on a category A by (1.2.4):
(a) A is a analytic category with a regular
opposite.
(b) A is a lextensive category with a regular
opposite.
(c) A has finite limits and finite colimits,
finite stable disjoint sums and costable regular monos.
Example 1.2.5.1. Algebraic and monadic
categories over the category of sets are regular with limits and colimits.
By (1.2.5.c) they are opposites of analytic categories
iff products are codisjoint and costable. For instance, the category of
commutative rings (with unit) is algebraic with costable codisjoint products,
thus its opposite is an analytic category, which is equivalent to the category
of affine schemes.
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