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  (u  v) = w  (u  v) = (w  u)  (w  v) = (w  u)  (w  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. –– 

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 u = (u --> 0) in the lattice R(X). 
(c) A strong subobject u: U --> X is analytic iff u = u. 

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 u^c of u exist, which is normal, thus also strong. Since u is normal, u is the complement of u^c (i.e. u = (u^c)^c ), which implies that u is analytic. 
(b) and (c) follow from the above proof for (a). –– 

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 (X) of normal sieves on object X (see (2.1.4)). 

Proposition 3.7.4. N = A and  is equivalent to the subnormal framed topology determined by N. 

Proof. We already know that N = A by (3.7.3). To prove the second assertion, 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 t = m. Thus t = m. Since U is a normal sieve, it contains t, thus it also contains the normal mono m. This indicates that U is generated by the normal mono. Thus U is a N-sieve for the normal divisor N. Since any N-sieve is normal, we see that  coincides with the set of N-sieves. 

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

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. –– 

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 the sieve sie(v) generated by v contains U. Then u is normal. and sie(u) contains U. We prove that U = sie(u). Suppose t: T --> X is a map in sie(u) which is disjoint with U. Its strong image m: S --> X satisfies the similar properties (as u is a strong mono). Thus m  U which implies that U = U  m = sie(m^c). It follows that u factors through m^c. Since m factors through u, we see that m factors through m^c. Thus m is a initial map, and t is also initial. This shows that U dominates sie(u). But U is normal, thus U = sie(u). –– 

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

Corollary 3.7.8. (X) = N(X) = A(X) = _A(X). ––