Cosheaves and connectedness in formal topology

Let us call such a point t strongly top. We shall prove (Theorem 39) that
a lower powerpoint is a positivity predicate iff it is strongly top in the lower
powerlocale. This result is already known [5], but we shall give a proof that is
more categorical as a pattern for the corresponding discussion of cosheaves in
Section 7.2.
Lemma 37. Let X = (P, ≤, ⊳ 0 ) be a flat site, and let F be a lower powerpoint.
Then F is a positivity predicate iff RestF = X.
Proof. ⇒: The covers a ⊳ {a} ∩ F presenting RestF already hold in X.
⇐: For any F , the sublocale RestF is overt, so X is overt. The positivity
predicate for RestF is F , which is thus also the positivity predicate for X.
peer-00821313, version 1 - 9 May 2013
Lemma 38. Let X = (P, ≤, ⊳ 0 ) be a flat site, and let F, G : W → P L X be
two generalized lower powerpoints. Then F ⊑ G iff Rest W F ≤ Rest W G (as
subspaces of X × W ).
Proof. Note that Rest W F ≤ Rest W G iff there is a (unique) map Rest W F →
Rest W G over X ×W , and by definition of Rest W G as comma object this holds if
p F ; ↓ ⊑ q F ; G : Rest W F → P L X. (p F and q F are the projections from Rest W F
to X and W .)
⇒: This is the easy way round. If F ⊑ G then p F ; ↓ ⊑ q F ; F ⊑ q F ; G.
⇐: We first prove this in the case W = 1. Since Rest W F ≤ Rest W G, the
positivity predicate for Rest W F is included in that for Rest W G: in other words,
F ⊑ G. But that argument is constructive and hence applies also for internal
locales in SW , i.e. maps with codomain W . Let us apply it to X × W , with its
projection down to W . The geometricity of P L [6] tells us that, constructing P L
internally in SW , we have (P L ) W (X ×W ) ∼ = P L X ×W . In this context we have
global lower powerpoints of X × W , namely 〈F, W 〉, 〈G, W 〉 : W → P L X × W .
Let us calculate the Rest sublocale for 〈F, W 〉, the comma object
q
C −→ W
↓ p ⊑ W ↓ 〈F, W 〉
X × W P L X × W
−→
↓×W
(The subscript W in ⊑ W denotes that the order between maps to P L X × W
has to restrict to equality when projected down to W .) A point of C is a
triple 〈x, w 1 , w 2 〉 such that ↓ x ⊑ F (w 2 ) and w 1 = w 2 , so C ∼ = Rest W F as
defined above. Hence our working in SW shows that if Rest W F ≤ Rest W G
then 〈F, W 〉 ⊑ 〈G, W 〉 over W , i.e. F ⊑ G.
Theorem 39. Let X = (P, ≤, ⊳ 0 ) be a flat site, and F a lower powerpoint.
Then the following are equivalent.
1. X is overt, with F its positivity predicate.
2. RestF = X.
3. F is strongly top in P L X.
26