Cosheaves and connectedness in formal topology

and by the above argument α a has to map everything in F (b) to γ. Condition
2(a) in Definition 16 tells us that this respects all the edges in the diagram,
and so defines a function α a : F (a) → π 0 (a). In other words, α a (x) = γ where
x = F b,a (y) for some b ≤ γ a. Condition 2(c) now tells us that these functions
are natural in a, and so give a homomorphism α.
Our next aim is to show that existence of the structure for local connectedness
is homeomorphism invariant. In fact, we can be slightly more general. In a
topos-theoretic setting, local connectedness of X is equivalent to the existence
of π 0 left adjoint to ! ∗ : Set → SX. If we have f : X → Y and g : Y → X
with (f, g) an adjoint pair, then f ∗ : SY → SX is left adjoint to g ∗ , and so
π 0 ◦ f ∗ = Cosh f(π 0 ) is left adjoint to g ∗ ◦! ∗ . It follows that Y is also locally
connected. We now show this in formal topology.
peer-00821313, version 1 - 9 May 2013
Lemma 21. Let X = (P, ≤, ⊳) be locally connected, equipped with π 0 and ≤ γ .
Let Y = (Q, ≤, ⊳), and let f : X → Y and g : Y → X be maps such that (f, g)
is an adjoint pair. Then Y is locally connected, using cosheaf Cosh f(π 0 ).
Proof. Suppose b ∈ Q and δ ∈ Cosh f(π 0 )(b) = colim afb π 0 (a). Define b ′ ≤ δ b
if b ′ ≤ b and
b ′ ⊳ ⋃ {g − (≤ γ a) | afb, in a (γ) = δ}.
Now consider the conditions in Definition 16. 2(a) is obvious.
For 2(b), suppose b ′ ≤ δ b and δ ′ = in a ′(γ ′ ) ∈ Cosh f(π 0 )(b ′ ), with a ′ fb ′ .
From b ′ ≤ δ b we deduce
f − b ′ ⊳ f − ⋃ {g − (≤ γ a) | afb, in a (γ) = δ}
= ⋃ {f − (g − (≤ γ a)) | afb, in a (γ) = δ}
⊳ ⋃ {≤ γ a | afb, in a (γ) = δ} by the adjunction.
Since a ′ fb ′ we can find (a ′′ , γ ′′ ) ≤ (a ′ , γ ′ ) with a ′′ ≤ γ afb. Using the fact
that a ′ fb, we have that Cosh f(π 0 ) b′ ,b(δ ′ ) is in a ′(γ ′ ) considered as element of
Cosh f(π 0 )(b), and that is
in a ′(γ ′ ) = in a ′′(γ ′′ ) = in a ((π 0 ) a ′′ ,a(γ ′′ )) = in a (γ) = δ.
For 2(c), suppose b ′′ ≤ δ′ b ′ and (b ′ , δ ′ ) ≤ (b, δ). We want b ′′ ≤ δ b, so it
suffices to show that if a ′ fb ′ and in a (γ ′ ) = δ ′ then
g − (≤ γ′
a ′ ) ⊳ ⋃ {g − (≤ γ a) | afb, in a (γ) = δ}.
But in fact we have ⊆ here, for a ′ fb and, considered as element of Cosh f(π 0 )(b),
we have in a ′(γ ′ ) = Cosh f(π 0 ) b′ ,b(δ ′ ) = δ.
Now we look at 2(d). If b ∈ Q, then by the adjunction b ⊳ g − (f − b). Hence
it suffices to show that if afb then
⋃
b ↓ g − a ⊳
(≤ δ b).
δ∈Cosh f(π 0)(b)
16