4.4. Stone Geometries
The main results of this section are due to [Diers
1986]
Definition 4.4.1. An analytic category
is called a Stone geometry if it is
coherent and locally
decidable.
Proposition 4.4.2. (a) Any locally decidable
analytic category is a reduced locally disjunctable analytic category.
(b) Any Stone geometry is a spatial analytic geometry.
(c) Any strong mono in a Stone geometry is regular.
Proof. (a) Since any locally direct mono is normal and locally
disjunctable, any strong mono in a locally direct category is normal and
locally disjunctable, thus the category is reduced (3.1.4)
and locally disjunctable.
(b) follows from (a) and the fact that any coherent analytic category
is spatial perfect (4.2.2.a).
(c) Any intersection of regular monos in a locally finitely copresentable
category is regular. Since any direct mono is regular, any locally direct
mono is regular. Thus any strong mono in a Stone geometry is regular.
Proposition 4.4.3. Any indecomposable
object in a locally direct category is simple.
Proof. Any proper strong subobject of an indecomposable object
P is an intersection of proper direct monos, therefore is initial.
Thus P is simple.
Proposition 4.4.4. Suppose A
is a coherent analytic category. Then the following assertions are equivalent:
(a) A is a Stone geometry.
(b) Any finitely copresentable object is decidable.
(c) Any indecomposable object is simple.
Proof. Assume (a) holds. The diagonal map of any finitely copresentable
object is locally finitely generated and locally direct, therefore is an
intersection of finite direct monos, thus is direct. Thus (a) implies (b).
Assume (b) holds. Then by (4.1.3.a)
any regular mono is locally direct. Suppose P is an indecomposable
object. We prove that P is simple. Since any proper strong mono
is contained in a proper regular mono, it suffices to prove that P
has exactly two regular monos. Suppose M is a proper regular subobject
of X. Then M is an intersection of proper direct monos to
X. But X is indecomposable, any proper direct mono to it
is initial. Thus M is initial. This shows that (b) implies (c).
Finally assume (c). Any simple prime of an object is contained in an
indecomposable component, which is simple by assumption, therefore must
be itself by (3.4.4.b) and (3.4.4.d).
Thus any simple prime of an object is an indecomposable component. Next
we show that A is reduced. Consider a noninitial
object X. Since A is reducible, it is
sufficient to prove that any proper reduced strong subobject of X
is not unipotent (3.1.2). 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
subobjects by (4.1.2), it suffices to prove
the assertion for a proper finitely cogenerated regular reduced subobject
W. To see that W is not is unipotent, it suffices to show
that there is a simple prime of X not contained in W. First
we note that there is a simple prime P of X such that any
direct neighborhood U of P is not contained in W.
Otherwise W contains a direct unipotent cover {U_{i}}
of X, which is not the case as the extensive topology is strict,
therefore X is the colimit of {U_{i}
U_{j}}, and this only happens if X = W. Thus let P
be such a simple prime. Then P is the cofiltered limit of all the
direct monos containing P. Assume P is contained in W.
Suppose W is the equalizer of a pair (r_{1}, r_{2}):
X > T of maps with T finitely copresentable. There is an direct
mono u: U > X of P such that r_{1}u =
r_{2}u (see the proof of (4.1.3)).
Then U W which contradicts
to the choice of P. This shows that P is not contained in
W as desired. Now we can prove that A
is locally decidable. Consider a strong subobject V of an object
X. Let M be the intersection of all the direct monos containing
V. Then V is a strong subobject of M. If we can prove
that V is a unipotent subobject of M, then because A
is reduced, V is an epic strong subobject, therefore V = M
as desired. Thus it remains to prove that V is unipotent in M.
Assume V is not unipotent in M. Then there is a map t:
T > M from a simple object P
to X which is disjoint with V. The locally direct image N
of P in M is indecomposable, therefore is simple. Since N
is not contained in the strong mono V, it is disjoint with V.
Suppose N is the intersection {V_{i}} of direct monos
to M. If each V
N_{i} is noninitial, then (V
N_{i}) = V
( N_{i}) =
V N as 0 is finitely
copresentable, which contradicts the fact that V and N are
disjoint. Thus there is some N_{i} such that V
N_{i} is initial. Suppose M = O + N_{i}. Then
V factors through the proper direct mono O. This contradicts
the fact that V is indirect in M. This shows that V
is unipotent in M. n
Proposition 4.4.5. Suppose A
is a coherent analytic category. Then A is
a Stone geometry iff the following two conditions are satisfied:
(a) If u: U > X is a regular mono, then the
induced function hom(u, 1+1): hom_{A}(X,
1+1) > hom_{A}(U,
1+1) is surjective.
(b) If u: U > X is a regular mono and the induced
function hom(u, 1+1): hom_{A}(X,
1+1) > hom(U, 1+1) is bijective, then u
is an isomorphism.
Proof. Note that we may identify naturally the set hom_{A}(X,
1+1) with the set of direct factors of X.
First suppose A is a Stone geometry. The
regular mono u is locally direct, thus by the proof of (4.3.1)
any direct factor of U is induced from a direct factor of X,
thus hom(u, 1+1) is surjective. Assume u is
the intersection of direct factors {u_{i}}. If hom(u,
1+1) is bijective, then each hom(u_{i}, 1+1)
is injective, thus is bijective by (a). It follows that each hom(u_{i},
1+1) is an isomorphism, so hom(u, 1+1) is an
isomorphism.
Conversely, assume conditions (a) and (b) hold. By (4.4.4)
it is sufficient to prove that any indecomposable object is simple. Suppose
P is an indecomposable object. We prove that P is simple.
Since any proper strong mono is contained in a proper regular mono, it
suffices to prove that P has exactly two regular subobjects. Suppose
m: M > P is a proper regular subobject of X.
By (a) hom(m, 1+1) is surjective. Since P is
indecomposable, it has only two direct factors, so M has two or
one direct factors. If M has two direct factors, then hom(m,
1+1) is bijective, so m is an isomorphism by (b). If M has
only one direct factor it must be the initial object 0. This shows
that the only regular subobjects are M or 0.
Proposition 4.4.6. Suppose A
is a coherent analytic category. Then the following assertions are equivalent:
(a) A is a Stone geometry.
(b) If u: U > X is a map such that the induced
function hom(u, 1+1): hom_{A}(X,
1+1) > hom_{A}(U,
1+1) is injective, then u is epic.
(c) Any regular (or strong) mono is locally direct.
(d) Any finitely cogenerated regular mono is a direct mono.
Proof. Assume (a). If a map u: U > X is not epic, then
it factors through a proper regular mono v: V > X.
Since V is locally direct, it is contained in a proper direct
factor W of X. Since the pullback of W along u
is U,
hom(u, 1+1) is not injective. This proves that
(a) implies (b).
Assume (b). Consider a strong mono u: U > X.
Let v: V > X be the intersection of direct subobjects
of X containing U. Consider the induced strong mono w:
U > V. Since v: V > X is locally
direct, by the proof of (4.3.1) any direct
factor of V is induced from X, thus there is no proper direct
factor of V containing U. We prove that hom(u,
1+1) is injective. Consider two proper direct factors S,
T of V such that S
U = T U.
Then (S + T^{c})
U = S U
+ T^{c} U
= T U + T^{c}
U = (T + T^{c})
U = V U
= U. This means that S + T^{c} is a direct
factor containing U. It follows that S + T^{c}
= V, i.e. T
S. The same argument shows that S
T. Therefore S = T as desired. This shows that
hom(w, 1+1) is injective, so apply (b) we see that the strong
mono u is epic, thus an isomorphism. It follows that U =
V is locally direct.
Assume the assertion (c) that any regular mono is locally direct. Then
any finitely cogenerated regular mono u is an intersection of direct
monos. By (4.1.4.b) u is also the
intersection of a finite collection of direct monos, thus u is direct.
Finally assume (d). By (4.4.4) it suffices to
prove that any indecomposable object P is simple. Suppose m:
M > P is a proper regular subobject. By (4.1.3)
m is an intersection of finitely cogenerated regular subobjects.
Thus m is locally direct, which is contained in a proper direct
fact of P. Since the only proper direct factor of P is 0,
we see that M = 0 is initial. This shows that P is simple.
Proposition 4.4.7. The opposite of any
Stone geometry is a regular category.
Proof. If i: X > X + Y is a direct
mono and f: X > Z is any map, then Z >
Z + Y is the pushout of i along f.
Thus the pushout of any direct mono is a direct mono, i.e. direct mono
is couniversal. Since any regular mono is an intersection (i.e. a cofiltered
limit) of direct monos, and pushout commutes with cofiltered limit in any
locally finitely copresentable category (see [Johnstone
1982, p.230, Proposition 1.7 (iii)]), any regular mono is couniversal.
Proposition 4.4.8. [Diers] A locally
finitely copresentable category is a Stone geometry iff its subcategory
of finitely copresentable objects is a decidable lextensive category.
Proof. The proof of this important result is rather long; the
reader is refer to [Diers 1986, p.26, p.53
and p.56].
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