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 non-initial 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 {Ui} of X, which is not the case as the extensive topology is strict, therefore X is the colimit of {Ui  Uj}, 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 (r1, r2): X --> T of maps with T finitely copresentable. There is an direct mono u: U --> X of P such that r1u = r2u (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 {Vi} of direct monos to M. If each V  Ni is non-initial, then (V  Ni) = V  ( Ni) = V  N as 0 is finitely copresentable, which contradicts the fact that V and N are disjoint. Thus there is some Ni such that V  Ni is initial. Suppose M = O + Ni. 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):  homA(X, 1+1) --> homA(U, 1+1) is surjective.  (b) If u: U --> X is a regular mono and the induced function hom(u, 1+1): homA(X, 1+1) --> hom(U, 1+1) is bijective, then u is an isomorphism.  Proof. Note that we may identify naturally the set homA(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 {ui}. If hom(u, 1+1) is bijective, then each hom(ui, 1+1) is injective, thus is bijective by (a). It follows that each hom(ui, 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):  homA(X, 1+1) --> homA(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 + Tc)  U = S  U + Tc  U = T  U + Tc  U = (T + Tc)  U = V  U = U. This means that S + Tc is a direct factor containing U. It follows that S + Tc = 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 co-universal. 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 co-universal.   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].                          [Next Section][Content][References][Notations][Home]