higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
A sheaf topos $\mathcal{E}$ is called strongly connected if it is a locally connected topos
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 strongly connected topos is also a local topos, then it is a cohesive topos.
The “strong” in “strongly connected” may be read as referring to $f_! \dashv f^*$ being a strong adjunction in that we have a natural isomorphism for the internal homs in the sense that
This follows already for $f$ connected and essential if $f_!$ preserves products, because this already implies the equivalent Frobenius reciprocity isomorphism. See here for more.
locally connected topos / locally ∞-connected (∞,1)-topos
strongly connected topos / strongly ∞-connected (∞,1)-topos
and