Cohomology and homotopy
In higher category theory
A locally connected site is a site satisfying sufficient conditions to make its topos of sheaves into a locally connected topos.
Locally connected toposes from sites
We discuss that the sheaf toposes over locally connected sites are locally connected toposes.
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.
This means that the inverse image functor has a left adjoint .
By remark 1 it follows that the constant presheaf functor has a left adjoint given by taking colimits along (this is one of the equivalent definitions of the colimit operatiion.) Since constant presheaves on are sheaves, is just a factorization of through , 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 any sheaf, we may write it, using the co-Yoneda lemma as a coend over representables
The left adjoint functor 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 .
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 is generated by a single arrow , then is a weakly terminal object? of the sieve (qua full subcategory of ), so the sieve is connected.