  We consider categories of posets with $\mathfrak{C}$-valued structure sheaves
for any category $\mathfrak{C}$ and see how they possess poset-indexed lax
colimits that are both easy to describe and "weakly equivalent" to their
ordinary colimits in a certain sense. We employ this construction to study
descent problems on schematic spaces -- a particular scheme-like kind of ringed
poset -- , proving a general Seifert-Van Kampen Theorem for their \'etale
fundamental group that recovers and generalizes the homonym result for schemes
to the topology of flat monomorphisms. The techniques are general enough to
consider their applications in many other frameworks.
