nLab totally infinity-connected site

Idea

A totally $\infty$-connected site is a site satisfying sufficient conditions to make the (∞,1)-sheaf (∞,1)-topos over it a totally ∞-connected (∞,1)-topos.

Definition

Proposition

Let $C$ be a locally and globally ∞-connected site; we say it is a strongly $\infty$-connected site if it is also a cofiltered (∞,1)-category.

Properties

Proposition

If $C$ is a totally $\infty$-connected site, then the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over it is a totally ∞-connected (∞,1)-topos.

Proof

We need to check that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite (∞,1)-limits.

By the discussion at ∞-connected site we have that $\Pi$ is given by the (∞,1)-colimit (∞,1)-functor $\lim_\to : Func(C^{op}, \infty Grpd) \to \infty Grpd$. On the opposite and therefore filtered (∞,1)-category $C^{op}$ these preserve finite (∞,1)-limits.

Examples

• 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