nLab stably locally connected topos

Definition

A sheaf topos $\mathcal{E}$ is called stably locally connected if it is a locally connected topos

$$(\Pi_0 \dashv \Delta \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{\Delta}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set$$

such that the extra left adjoint $\Pi_0$ in addition preserves finite products (the terminal object and binary products).

This means it is in particular also a connected topos.

If $\Pi_0$ preserves even all finite limits then $\mathcal{E}$ is called a totally connected topos.

If a stably locally connected topos is also a local topos, then it is a cohesive topos.

Related concepts

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

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

  • stably locally 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

References

• Peter Johnstone. Remarks on punctual local connectedness. Theory and Application of Categories 25.3 (2011): 51-63. (TAC)