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

Definition

A ∞-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)-limits.

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

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

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

• local site

• cohesive site