connected site in nLab

Proof

Being a locally connected site, we already know that we have a locally connected topos (Π 0⊣ΔΔΓ):Sh(C)→Set(\Pi_0 \dashv \Delta \Delta \Gamma) : Sh(C) \to Set. By the discussion there we need to check that Π 0\Pi_0 preserves the terminal object.

The terminal object in the site represents the terminal presheaf on CC, which is the presheaf constant on the point. By the discussion at locally connected site we have that every constant presheaf is a sheaf over CC, hence the terminal object of Sh(C)Sh(C) is also represented by the terminal object in the site, and we just write “**” for all these terminal objects.

By the discussion there, the left adjoint Π 0\Pi_0 in the sheaf topos over a locally connected site is given by the colimit functor lim →:[C op,Set]→Set\lim_\to : [C^{op}, Set]\to Set. The colimit over a representable functor is always the point (this is the (co)-Yoneda lemma in slight disguise), hence indeed Π 0*=*\Pi_0 * = *.