Could not include topos theory - contents
A geometric morphism $f\colon E\to S$ is totally connected if
It is locally connected, i.e. its inverse image functor $f^*$ has a left adjoint $f_!$ which is $S$-indexed, and
The functor $f_!$ is left exact, i.e. preserves finite limits.
When thinking of $E$ as a topos over $S$ via $f$, we say that it is a totally connected $S$-topos. In particular, when $S=Set$ and $f = (L Const, \Gamma)$ is the unique global sections geometric morphism, we call $E$ a totally connected topos.
Of course, any totally connected geometric morphism is connected, since the terminal object is a particular finite limit. It is also strongly connected, since finite products are also finite limits.
A topos $Sh(X)$ of sheaves on a topological space is totally connected iff $X$ has a dense point (a single point whose closure is all of $X$).
A presheaf topos $Psh(C)$ is totally connected iff $C$ is cofiltered.
A small site $C$ is called totally connected if
$C$ is cofiltered, and
Every covering sieve in $C$ is connected, when regarded as a subcategory of a slice category.
The second condition implies that all constant presheaves are sheaves, and hence that the left adjoint $Colim\colon Psh(C) \to Set$ of $Const\colon Set\to Psh(C)$ restricts to $Sh(C)$ to give a left adjoint of $L Const$. Cofilteredness of $C$ is exactly what is needed for left exactness of $Colim\colon Psh(C) \to Set$, essentially by definition. Hence the topos of sheaves on any totally connected site is totally connected.
Conversely, one can show that any totally connected topos can be (but need not be) presented by some totally connected site.
and
locally connected site / locally ∞-connected site
totally connected site / totally ∞-connected site
Chapter C3.6 in