nLab
strongly infinity-connected (infinity,1)-topos

Definition

A locally ∞-connected (∞,1)-topos

$$(\Pi ⊣\Delta ⊣\Gamma ):H\to \mathrm{\infty }\mathrm{Grpd}$$(\Pi \dashv \Delta \dashv \Gamma) : \mathbf{H} \to \infty Grpd

is called strongly connected if $\Pi$ preserves finite (∞,1)-products (hence in particular the terminal object, which makes it also an ∞-connected (∞,1)-topos).

Similarly for an $n$-connected $(n,1)$-topos.

For $n=1$ this yields the notion of strongly connected topos.

If in addition $H$ is a local (∞,1)-topos then it is a cohesive (∞,1)-topos.

Related concepts

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

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

  • strongly connected topos / strongly $\mathrm{\infty }$-connected $(\mathrm{\infty },1)$-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

• local site

• cohesive site