nLab connected site

Idea

A connected site is a site satisfying sufficient conditions to make its topos of sheaves into a connected topos.

Definition

Proposition

Let $C$ be a locally connected site; we say it is a connected locally connected site if it also has a terminal object.

Properties

Proposition

If $C$ is connected locally connected site, then the sheaf topos $Sh(C)$ is a locally connected topos and connected topos.

Proof

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

The terminal object in the site represents the terminal presheaf on $C$, 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 $C$, hence the terminal object of $Sh(C)$ 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 $\Pi_0$ in the sheaf topos over a locally connected site is given by the colimit functor $\lim_\to : [C^{op}, Set]\to Set$. The colimit over a representable functor is always the point (this is the (co)-Yoneda lemma in slight disguise), hence indeed $\Pi_0 * = *$.

Examples

• The category of connected open subsets (as a full subcategory of the usual category of open subsets) of a connected topological space with the standard open cover-coverage is a connected site, with the terminal object being $X$ itself. If the space is also locally connected it is a dense sub-site of the category of all open subsets, so that even though that site is never connected the topos of sheaves on it can then at least be represented by a connected site.

• CartSp, ThCartSp.

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site