Any coherent analytic category is a spatial (4.2.2) and perfect (4.1.2.b). Thus a locally disjunctable coherent analytic category is a spatial analytic geometry, which is simply called a coherent analytic geometry. 

In the following we assume A is a coherent analytic geometry. Recall that (3.6.9) if X is an object. an open subset U of Spec(X) is affine open if it is determined by an analytic subobject of X. The set of affine open subsets is  a base for the topology on Spec(X). 

Proposition 5.1.1. (a) Spec(X) is a coherent space for any object X. 
(b} For any map f: Y ® X the continuous mapping Spec(f): Spec(Y) ® Spec(X) is a mapping of coherent spaces. 

Proof. (a) Suppose {V_i} is a set of disjunctable strong subobjects such that {(V_i)^c}_iÎI  is an analytic cover for X. Let V be the intersection of these {V_i}. Then by (3.1.10) we have X = ØØ{(V_i)^c} = ØV, so V = 0. Since 0 is finitely copresentable by (4.3.2.c), there is a finite subset J Í I such that {V_i}_iÎJ has intersection 0. Applying (3.1.10) again we see that {(V_i)^c} is a finite analytic cover for X. This shows that Spec(X) is quasi-compact. Thus any affine open subset is also quasi-compact. Since affine open subsets form a base for Spec(X), any finite intersection of quasi-compact open subsets is quasi-compact. This means that Spec(X) is coherent. 
(b) The pullback of analytic subobject of X along f is an analytic subobject of Y. Since affine open subsets form a base for Spec(X), this implies that inverse image of quasi-compact open subset is quasi-compact open. Thus Spec(f) is a mapping of coherent spaces. –n 

Proposition 5.1.2. (a) If X is quasi-primary then the fraction hull P(X) of X is the intersection of all non-initial analytic monos. 
(b) An object is primary iff its rational hull R(X) is quasi-simple; if X is primary then R(X) = Q(X) = P(X); 
(c) An object is integral iff its rational hull is simple; any integral object X has a generic residue P(X) ® X which is an epic coflat simple fraction. 

Proof. (a) Consider the intersection P of non-initial analytic subobjects of X, which is a non-initial fraction of X containing P(X). We have to prove that P is quasi-simple, or equivalently, that P is pseudo-simple (3.3.10). By (3.3.8.a) it suffices to prove that any non-initial strong subobject of P is unipotent. Since P ® X is coflat, any strong subobject of P is induced from X by (1.5.4). Since any unipotent strong subobject of X induces a unipotent strong subobject of P, we only need to prove that P is disjoint with any non-unipotent strong subobject V of X. Since A is locally disjunctable, V is an intersection of proper disjunctable strong subobjects {V_i}. Since V is non-unipotent, at least one complement (V_i)^c is non-initial, and P Í (V_i)^c. Thus P Ç V = 0. This shows that P is pseudo-simple, thus P = P(X). 
(b) If X is primary then any non-initial analytic mono is epic. Thus R(X) = P(X) by (a), and P(X) is primary by (3.3.8). Any object X is a quotient of its rational hull R(X). If R(X) is quasi-simple, it is primary by (3.3.8), thus X as a quotient of R(X) is primary by (3.2.2.a) 
(c) If X is integral then any proper strong subobject of the reduced object X is non-unipotent by (3.1.2). According to the proof of (a), P(X)  is disjoint with any non-unipotent strong subobject V of X. Since P(X) ® X is coflat, any strong subobject of P(X) is induced from X. This means that the only proper strong subobject of P(X) is 0 , thus P(X) is simple, which is a generic residue of X; and P(X) ® X is an epic coflat fraction as P(X) = R(X)  by (b) and R(X) ® X is epic coflat. Conversely if P(X) is simple the X as a quotient of integral object  R(X) = P(X) is integral by (3.2.3.a).  n 

Proposition 5.1.3. Suppose X is a quasi-primary object with the fractional hull p_X: P(X) ® X. Let s: S(X) ® X be the strong image of  p_X. 
(a) P(X) is generic quasi-simple and any generic map from a quasi-simple object to X factors through p_X uniquely. 
(b) S(X) is generic primary and any generic map from a primary object to X factors through s uniquely. 
(c) Any quasi-primary object has a unique generic primary strong subobject. 

Proof. (a)  P(X) is generic quasi-simple by  (4.2.5.c). It is easy to see that any generic map from a quasi-simple object to X factors through any non-initial analytic subobject of X, therefore it also factors through the intersection p_X of these analytic monos, which is P(X) by (5.1.2.a). 
(b) The quasi-simple object P(X) is primary (3.3.8), thus its quotient S(X) is also primary. S(X) is a generic strong subobject of X because any non-initial analytic subobject of X contains P(X), therefore not disjoint with S(X). We prove that S(X) has the required universal property. Consider a generic map t: T ® X with T primary. Let p_T: P(T) ® T be the fractional hull of T. Since p_T is generic, so is the composite t_°p_T. By (a) the generic map t_°p_T: P(X) ® X factors through p_X. Thus t_°p_T. factors through the strong mono s. Suppose m_°e = t is the epi-strong-mono factorization of t. Since p_T  is epic, m_° e_°p_T = t_°p_T is the epi-strong-mono factorization of t_°p_T. Thus m is the smallest strong mono such that t_°p_T can be factored through. Since t_°p_T factors through s, m factors throughout s, thus t =  m_°e factors through s as desired. 
(c) It follows from (b) that S(X) is the unique generic primary strong subobject of X. n 

Proposition 5.1.4. (a) Any map f: S ® X with a simple domain factors through a unique residue. 
(b) Any simple subobject is contained in a unique residue. 
(c) A simple subobject is a residue iff it is maximal. 

Proof. (a) The epi S ® f⁺¹(S) is generic by (3.3.4.f), so it factors through the generic residue of f⁺¹(S). The uniqueness follows from (3.4.4.f). 
(b) follows from (a). 
(c) It follows from (b) that any maximal simple subobject is a residue. The other direction has been noticed in (3.4.4.d). n 

Proposition 5.1.5. (a) Any colimits of reduced objects is reduced. 
(b) Any cofiltered limits of reduced object is reduced. 

Proof. (a) Since the full subcategory of reduced objects is a coreflective subobject of A , it is closed under colimits. 
(b) Let {r_i: X ® X_i}_iÎI be a cofiltered limits of reduced objects in A. We have to prove that any proper strong subobject U of X is non-unipotent. Since any proper strong subobject is contained in a proper regular subobject, and any proper regular subobject is an intersection of proper finitely cogenerated regular subobject by (4.1.3.a), we may assume that U is a finitely cogenerated regular subobject. So let us assume that U is the equalizer of a pair of distinct maps (m, n): X ® T where T is finitely copresentable. Since X is a cofiltered limits and T is finitely copresentable, we can find some t in I and a pair (m_t, n_t): X_t ® T of maps such that m_t°r_t = m and n_t°r_t = n. We may assume that t is an initial object in I. Let U_t be the equalizer of (m_t, n_t). Then the pullback of U_t along r_t: X ® X_t is U. Since the proper regular subobject U_t is an intersection of proper disjunctable strong subobject, and r_t does not factors through U_t, we can find a proper disjunctable subobject V of X_t containing U_t such that r_t does not factor through r_t. Let V be the pullback of V_t along r_t, and V_i be the pullback of V_t along X_i ® X_t. Then V_i and V are proper disjunctable strong subobjects and U Í V, and V^c is the cofiltered limit of (V_i)^c. Since each X_i is reduced and V_i is proper, each (V_i)^c is non-initial. Since the initial object is finitely copresentable, this implies that V^c is non-initial. Thus V is not unipotent, and hence U is not unipotent as desired. n 

Proposition 5.1.6. (a) An object is integral iff it is a quotient of a simple object. 
(b) A non-initial object is primary iff it is a quotient of a quasi-simple object. 
(c) An object is reduced iff it is a quotient of coproducts of simple objects. 

Proof. (a) The condition is sufficient because any quotient of a simple object is integral. Conversely any integral object is a quotient of its rational hull, which is simple (5.1.2.c). 
(b) The condition is sufficient because any quotient of a quasi-simple object is primary (3.3.8). Conversely any primary object is a quotient of its rational hull, which is quasi-simple (5.1.2.b). 
(c) Any coproducts of simple objects is reduced (5.1.5.a). Conversely, assume X is a reduced object. Let T be the coproduct of all the residues p_i: P_i ® X of X. Denote by t: T ® X the map induced by p_i. Then T is reduced (5.1.5.a). We prove that t is epic. It suffices to show that t is unipotent as by assumption X is reduced. Any map s: S ® X with a simple domain factors through a unique residue of X by (5.1.4.a). So s factors through t. Since the class of simple objects is unipotent dense, it follows that t is unipotent by (2.2.10). n