Lax colimits of posets with structure sheaves: applications to descent

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 étale 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.