3.7. Coflat Disjunctable Analytic Categories 

Definition 3.7.1. An analytic category is coflat if any map is coflat (or equivalently, any epi is stable). 

Proposition 3.7.2. Suppose A is a coflat analytic category. Then 
(a) Any epi is unipotent
(b) Any singular mono is analytic
(c) Any normal mono (thus any analytic mono) is strong. 
(d) If f: Y ® X is a map then f-1: R(X) ® R(Y) is a morphism of bounded lattices. 
(e) R(X) is a distributive lattice for any object X
(f) If A is locally disjunctable then any integral object is simple. 

Proof. (a) is true because any stable epi is unipotent. 
(b) Any singular map in A is coflat, thus analytic. 
(c) The pullback of any normal mono is not proper unipotent, thus not proper epic by (a). 
(d) follows from (1.5.3) because any map f is coflat. 
(e) Suppose w: W ® X, u: U ® X and v: V ® X are three strong subobjects of X. Then w Ù (u Ú v) = w Ç (u Ú v) = (w Ç u) Ú (w Ç v) = (w Ù u) Ú (w Ù v) by (d). 
(f) Since any map is coflat, any non-initial map to an integral object X is epic by (3.2.6), so X is simple. n 

In the following we assume A is a coflat disjunctable analytic category. 

Proposition 3.7.3. (a) Any normal mono is analytic. 
(b) Any strong subobject u: U ® X has a negation Øu = (u ® 0) in the lattice R(X). 
(c) A strong subobject u: U ® X is analytic iff u = ØØu

Proof. (a) Suppose u: U ® X is a normal mono. It is strong by (3.7.2.c). Since any strong mono is disjunctable, the complement uc of u exist, which is normal, thus also strong. Since u is normal, u is the complement of uc (i.e. u = (uc)c ), which implies that u is analytic. 
(b) and (c) follow from the above proof for (a). n 

Recall that the class of analytic (resp. normal) monos is a subnormal divisor A (resp. N) on A (see (2.6.6)). Recall that  is the boolean functor from A to the (meta)category of complete boolean algebras, sending each object X to the set Â(X) of normal sieves on object X (see (2.1.4)). 

Proposition 3.7.4. Â = N = A

Proof. We already know that N = A by (3.7.3). To see that  = N, consider a normal sieve U on an object X. Consider any map t: T ® X in U with the strong image m: e(T) ® X. Since any map is coflat, by (1.5.2) we have Øt = Øm. Thus ØØt = ØØm. Since U is a normal sieve, it contains ØØt, thus it also contains the normal mono ØØm. This indicates that U is generated by the normal mono. Thus U is a N-sieve for the normal divisor N. n 

Corollary 3.7.5. The boolean functor is a framed topology which coincides with its generic and analytic topologies. n 

Corollary 3.7.6. Suppose A is reduced. The following notions are the same: 
(a) Strong mono. 
(b) Normal mono. 
(c) Analytic mono. 
(d) Singular mono. 
(e) Fractional mono. n 

In the following we assume X is an object such that R(X) is complete. 

Proposition 3.7.7. Any normal sieve on X is generated by a normal mono. 

Proof. Consider a normal sieve U. Let u: U ® X be the intersection of all the normal monos v to X such that J(v) contains U. Then u is normal. and J(u) contains U. We prove that U = J(u). Suppose t: T ® X is a map in J(u) which is disjoint with U. Its strong image m: S ® X satisfies the similar properties (as u is a strong mono). Thus m Î ØU which implies that U = ØØU Í Øm = J(mc). It follows that u factors through mc. Since m factors through u, we see that m factors through mc. Thus m is a initial map, and t is also initial. This shows that U dominates J(u). But U is normal, thus U = J(u). n 

If D is a divisor on A we denote by D(X) the set of D-subobjects of X

Corollary 3.7.8. Â(X) = N(X) = A(X) = FA(X). n 

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