Could not include topos theory - contents
Totally connectedness is a stronger form of local connectedness that arose in the work of M. Bunge and J. Funk on topos distributions.
The properties of totally connected geometric morphisms are largely dual to those of local geometric morphisms.
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.
Totally connected geometric morphisms are closed under composition and stable under pullback.
$f$ is totally connected iff $f$ is connected and has a right adjoint in $\mathfrak{Top}$, the 2-category of toposes and geometric morphisms.
Totally connected geometric morphisms are orthogonal to grouplike morphisms (cf. Johnstone (2002)).
A connected, locally connected $p:\mathcal{E}\to\mathcal{S}$ that satisfies the Lawvere Nullstellensatz (i.e. the canonical $\theta:p_*\to p_!$ is epic) is totally connected precisely iff it is a quality type over $\mathcal{S}$ (i.e. $\theta$ is iso; cf. Johnstone (2011)).
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
Marta Bunge, Jonathon Funk, Spreads and the Symmetric Topos , JPAA 113 (1996) pp.1-38.
Marta Bunge, Jonathon Funk, Spreads and the Symmetric Topos II , JPAA 130 (1998) pp.49-84.
Peter Johnstone, Sketches of an Elephant , Oxford UP 2002. (vol. 2, Section C.3.6, pp.707-710)
P. Johnstone, Remarks on Punctual Local Connectedness , TAC 25 no.3 (2011) pp.51-63. (pdf)