4.3. Extensive Topologies 

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 {fi: Yi ® X}iÎI, and we may assume that fi 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 ® Yr for some r in I in a map s: Yr ® 1 + 1. The pullbacks of the injections 1 ® 1 + 1 along s induces a direct sum Yr = Ur + Vr, and U is the pullback of Ur along Y ® Yr. Since Yr is a direct factor of X and Ur is a direct factor of Yr, Ur 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. n 

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).  n 

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 {Ui}: i = 1, ..., n of X. Suppose Vi is the complement of each Ui. Let W1 = U1, W2 = V1 Ç U2 , ..., Wi = V1 Ç V2 ... Ç Vi-1 Ç Ui. Then {Wi} is a direct cover of X, with Wi Ç Wj = 0 for i < j, and X = S Wi. Let z: Z ® X be the sum of ui: Ui ® X, and let s: X = S Wi ® Z be the map induced by the inclusion Wi ® Ui Then z°s is the identity of X. Thus z is a retraction, hence a regular epi. This shows that {Ui} is a strict direct cover. Thus the direct topology is strict. n 

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. Suppose U and V are two direct subobjects of an object X. Then 

X = (U + Uc) Ç (V + Vc) = (U Ç V) + (Uc Ç V) + (U Ç Vc) + (Uc Ç Vc). 
(U Ç V) + (Uc Ç V) + (U Ç Vc
are direct subobjects. But 
U Ç V = U Ù V,
and we have the formula 
(U Ç V) + (Uc Ç V) + (U Ç Vc) = U Ú V
in B(X) as (U Ç V) + (U Ç Vc) = U  and  (U Ç V) + (Uc Ç V) = V. Thus B(X) is a lattice. 
If W is another direct subobject of X, then 
X = W Ç Uc + W Ç U + Wc 
implies that 
(W Ç U)c = W Ç Uc + Wc
(W Ç U)c Ç W = [(W Ç Uc) + Wc] Ç W = (W Ç Uc) Ç W.
(W Ç V)c = W Ç Vc Ç W.
We have 
    W Ç (U Ú V
= W Ç [(U Ç V) + (Uc Ç V) + (U Ç Vc)]  
= W Ç U Ç V + W Ç Uc Ç V + W Ç U Ç Vc  
= (W Ç U) Ç (W Ç V) + (W Ç Uc) Ç (W Ç V) + (W Ç U) Ç (W Ç Vc)  
= (W Ç U) Ç (W Ç V) + (W Ç U)c Ç (W Ç V) + (W Ç U) Ç (W Ç V)c  
= (W Ç U) Ú (W Ç V). 
This shows that B(X) is a distributive lattice. Clearly Uc is the complement of U in B(X). Thus B(X) is a Boolean algebra. n 

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 homA (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 Fon A is the framed topology FE generated by the divisor E of direct monos

Proposition 4.3.7. (a) A finite set {Ui} of direct subobjects of an object X is a unipotent cover iff the join of {Ui} is X
(b) FE(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 {Ui} is a finite unipotent cover of X. Let V be the join of {Ui}. Then Vc is disjoint with each Ui, so Vc = 0 and V = X. Conversely, suppose the join of a finite set {Ui} of direct subobjects is X. Suppose t: T ® X is disjoint with each Ui. Then t factors through each (Ui)c. Thus t factors through Ç (Ui)c = (Ú Ui)c = Xc = 0. Thus t is initial, i.e. {Ui} 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 FE(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 Uc 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 Ç P is a proper direct subobject contains U. Since V is an ultrafilter, W is not in V implies that Wc is in V. It follows that P is contained in Wc. Thus P Ç 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 Tc is in V'. Thus P is contained in T and P' is contained in Tc, which implies that P and P' are disjoint. This shows that FE(X) is isomorphism to B(X). n 

Corollary 4.3.8. (a) The space pt(FE(X)) of FE(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. n 

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