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 ZX --> Z and ZY --> Z of x and y along f exist, and the induced map ZX + ZY --> 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 epi-strong-mono factorization. 
Thus an analytic category is simply a lextensive category with epi-strong-mono 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 epi-strong-mono factorizations if any of the following conditions are satisfied: 
(a) Any intersection of strong monos exists. 
(b) A is a complete well-powered 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 co-wellpowered (see [Adamek and Rosicky 1994, p.45]). n 
Proposition 1.2.3. A lextensive category has epi-regular-mono 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 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 reg-epi-mono 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 co-stable regular monos. 
Example 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 co-stable. For instance, the category of commutative rings (with unit) is algebraic with co-stable codisjoint products, thus its opposite is an analytic category, which is equivalent to the category of affine schemes. 

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