Suppose A is a coherent analytic category. 

Recall that (1.7.4.c) a mono is locally direct if it is an intersection of direct monos. 

Proposition 4.3.1. Any composite of locally direct mono is a locally direct mono. 

Proof. Suppose f: Y --> X is an intersection of direct monos {f_i: Y_i --> X |iI}, and we may assume that f_i is a cofiltered system. Assume Y = U + V where U and V are direct subobjects of Y. Then the unique maps U --> 1 and V --> 1 induce a unique map t: Y --> 1 + 1. Since 1 + 1 is finitely copresentable, t factors through Y --> Y_r for some r in I in a map s: Y_r --> 1 + 1. The pullbacks of the injections 1 --> 1 + 1 along s induces a direct sum Y_r = U_r + V_r, and U is the pullback of U_r along Y --> Y_r. Since Y_r is a direct factor of X and U_r is a direct factor of Y_r, U_r is also a direct factor of X. This shows that any direct factor of Y is induced from a direct factor of X. Consequently any locally direct factor of Y is also a locally direct factor of X.  

Definition 4.3.2. (a) A map f: Y --> X is called indirect if it does not factors through any proper direct mono. 
(b) A non-initial object is called indecomposable if it has exactly two direct subobjects. 
(c) An indecomposable component of an object X is a locally direct indecomposable subobject of X. 

Proposition 4.3.3. (a) Any map can be factored uniquely as an indirect map followed by a locally direct mono. 
(b) The class of indirect monos is closed under composition. 
(c} If P --> X is an indirect map and P is indecomposable, then X is indecomposable. 
(d} Any non-initial object has an indecomposable component. 
(e} An indecomposable subobject is a indecomposable component iff it is a maximal indecomposable subobject. 

Proof. (a) Consider a map f: Y --> X. Let u: U --> X be the intersection of all the direct monos to X such that f factors through. Then u is a locally direct mono, and the induced map g: Y --> U is indirect by (4.3.1). The uniqueness is obvious. 
(b) The proof is similar to that of (3.5.4.a). 
(c) Suppose U + V = X is a direct sum and U is proper. Then f^-1(U) is a proper direct of P, thus f^-1(U) = 0, i.e. f is disjoint with U, thus f factors through V. Since f is indirect, we must have V = X, so U = 0 as desired. 
(d) Clearly any simple object is indecomposable. Any non-initial object has a simple subobject by (4.2.2.d), whose indirect image in X is indecomposable and locally direct by (c). 
(e) is an easy consequence of (b).  

Proposition 4.3.4. The extensive topology on A is spatial and strict. 

Proof. A non-initial object is indecomposable iff it has exactly two direct subobjects. Thus a non-initial object is indecomposable iff it determines a one-point-space in the extensive topology. Clearly any simple object is indecomposable. Thus the direct topology is spatial by (2.6.5.a) and (4.2.2.d). By (4.2.2.e) any direct cover of an object X has a finite subcover. To see that the extensive topology is strict it suffices to consider a finite direct cover {U_i}: i = 1, ..., n of X. Suppose V_i is the complement of each U_i. Let W₁ = U₁, W₂ = V₁  U₂ , ..., W_i = V₁  V₂ ...  V_i-1  U_i. Then {W_i} is a direct cover of X, with W_i  W_j = 0 for i < j, and X =  W_i. Let z: Z --> X be the sum of u_i: U_i --> X, and let s: X =  W_i --> Z be the map induced by the inclusion W_i --> U_i Then z_°s is the identity of X. Thus z is a retraction, hence a regular epi. This shows that {U_i} is a strict direct cover. Thus the direct topology is strict.  

If X is any object we denote by B(X) the poset of direct subobjects of X. The following Proposition 4.3.5 holds for any extensive category: 
 
Proposition 4.3.5. B(X) is a Boolean algebra.  

Proof. (a) First we show that B(X) is a lattice. Suppose U and V are two direct subobjects of an object X. Since finite sums are stable, we have 

X = (U + U^c)  (V + V^c) = (U  V) + (U^c  V) + (U  V^c) + (U^c  V^c). 

U  V 

(U  V) + (U^c  V) + (U  V^c)

(U  V) + (U^c  V) + (U  V^c) = (U^c  V^c)^c.

(U  V) + (U^c  V) + (U  V^c) 

U  V.

U  V

U  V^c

U  V

U^c  V

(U  V) + (U^c  V) + (U  V^c) = U  V.

Thus B(X) is a lattice with 

U  V = U  V,

(U  V) + (U^c  V) + (U  V^c) = U  V. 

X = W  U^c + W  U + W^c 

(W  U)^c = W  U^c + W^c. 

(W  U)^c  W = [(W  U^c) + W^c]  W = W  U^c W + W^c  W = W  U^c.

(W  V)^c  W = W  V^c.

    W 

U  V

= W 

U  V

U^c  V

U  V^c

= W  U  V + W  U^c  V + W  U  V^c 

= W  U  V +

W  U^c

  V + W 

U  V^c

W  U

 

W  V

W  U

^c  W

  V + U 

W 

W  V

W  U

 

W  V

W  U

^c 

W  V

W  U

 

W  V

W  U

 

W  V

Remark 4.3.6. Consider the canonical functor J: FiniteSet --> A which preservs finite limits and sums. For each object X in A the pullback of the finite-limit-preserving functor hom_A (X, ~) along J is a finite-limit-preserving functor FiniteSet --> Set, thus determines a Boolean algebra, which is precsely B(X). This argument holds for any extensive category with a terminal object 1. 

Recall that (2.6.6) the extensive topology _E  on A is the framed topology _E generated by the divisor E of direct monos. 

Proposition 4.3.7. (a) A finite set {U_i} of direct subobjects of an object X is a unipotent cover iff the join of {U_i} is X. 
(b) _E(X) is isomorphic to the frame of ideals of the Boolean algebra B(X). 
(c) There is a bijection between the set of indecomposable components and the set of prime ideals (or ultrafilters) of B(X). 

Proof. (a) Suppose {U_i} is a finite unipotent cover of X. Let V be the join of {U_i}. Then V^c is disjoint with each U_i, so V^c = 0 and V = X. Conversely, suppose the join of a finite set {U_i} of direct subobjects is X. Suppose t: T --> X is disjoint with each U_i. Then t factors through each (U_i)^c. Thus t factors through  (U_i)^c = ( U_i)^c = X^c = 0. Thus t is initial, i.e. {U_i} is a unipotent cover over X. 
(b) It follows easily from (a) that if U is an E-sieve then the collection of direct monos in U is an ideal of B(X), and the resulting map _E(X) --> B(X) is an isomorphism. 
(c) If P is an indecomposable component of X then the set V of direct subobjects containing P is a prime filter (ultrafilter) of B(X). For if U is a direct subobject not containing P, then U^c contains P as it is indecomposable. Conversely, if V is a ultrafilter of B(X) then the intersection P of direct subobjects in V is non-initial as 0 is finitely copresentable, and P is locally direct. If U is a proper direct subobject of P then U is also a locally direct subobject of X, thus there is a proper direct subobject W of X such that W  P is a proper direct subobject contains U. Since V is an ultrafilter, W is not in V implies that W^c is in V. It follows that P is contained in W^c. Thus P  W = 0, which implies that U = 0. This shows that P is an indecomposable component of X. Suppose V and V' are two different ultrafilters of B(X) and P and P' are the corresponding indecomposable components. If T is a direct subobject in V not in V' then T^c is in V'. Thus P is contained in T and P' is contained in T^c, which implies that P and P' are disjoint. This shows that _E(X) is isomorphism to B(X).  

Corollary 4.3.8. (a) The space pt(_E(X)) of _E(X) is homeomorphic to the space of prime ideals of the Boolean algebra B(X), thus is a Stone space (see [Johnstone 1982, p.62-75]). 
(b) The extensive topology on a coherent analytic category is a strict metric site, which may be interpreted as the functor sending each object X to the Stone space of the indecomposable components of X.