Could not include topos theory - contents
A locally connected site is a site satisfying sufficient conditions to make its topos of sheaves into a locally connected topos.
Let $C$ be a small site; we say it is a locally connected site if all covering sieves of any object $U\in C$ are connected, as full subcategories of the slice category $C_{/U}$.
(In particular, this means that all covering families are inhabited.)
We discuss that the sheaf toposes over locally connected sites are locally connected toposes.
If $C$ is locally connected, then every constant presheaf on $C$ is a sheaf.
The fact that all covering families are inhabited makes the constant presheaves be separated presheaves (see this example) and then the connectedness condition further makes them be sheaves.
If $C$ is a locally connected site, then the sheaf topos $Sh(C)$ is a locally connected topos.
This means that the inverse image functor $L Const\colon Set \to Sh(C)$ has a left adjoint $\Pi_0$.
By remark 1 it follows that the constant presheaf functor $Const \colon Set \to Psh(C)$ has a left adjoint given by taking colimits along $C^{op}$ (this is one of the equivalent definitions of the colimit operatiion.) Since constant presheaves on $C$ are sheaves, $L Const$ is just a factorization of $Const$ through $Sh(C)$, and thus it also has a left adjoint given by the colimit operation.
The colimit over a representable functor is always the singleton set.
So for $X \in Sh(C)$ any sheaf, we may write it, using the co-Yoneda lemma as a coend over representables
The left adjoint functor $\Pi_0$ commutes with the coend and the tensoring in the integrand to produce
We may think of this as computing the set of plot-connected components of $X$.
If $C$ furthermore has a terminal object, then colimits along $C^{op}$ preserve the terminal object, so that $Sh(C)$ is moreover a connected topos.
Note that a non-locally-connected site can still give rise to a locally connected topos of sheaves, but every locally connected topos can be defined by some locally connected site.
any small subcategory of Top on connected topological spaces (with the standard open cover coverage).
Any site whose topology is generated by a singleton pretopology, i.e. a Grothendieck pretopology in which all covering families consist of single arrows. For if a covering sieve on $U$ is generated by a single arrow $p:V\to U$, then $p$ is a weakly terminal object? of the sieve (qua full subcategory of $C/U$), so the sieve is connected.
and
locally connected site / locally ∞-connected site