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.
Proposition 4.3.3. (a) Any map can be
factored uniquely as an indirect map followed by a locally direct mono.
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.
Proposition 4.3.4. The extensive topology on A is spatial and strict. Proof. A noninitial object is indecomposable iff it has exactly two direct subobjects. Thus a noninitial object is indecomposable iff it determines a onepointspace 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 W_{1} = U_{1}, W_{2} = V_{1} U_{2 }, ..., W_{i} = V_{1} V_{2} ... V_{i1} 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:
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 Thus B(X) is a lattice with 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)^{c}) = (W U) (W V) + (W U)^{c} (W V) + (W U) (W V)^{c} = (W U) (W V). This shows that B(X) is a distributive lattice. Clearly U^{c} is the complement U of U in the lattice B(X). Thus B(X) is a Boolean algebra. 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 finitelimitpreserving functor hom_{A} (X, ~) along J is a finitelimitpreserving 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.
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.
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.6275]).
