Lax colimits of posets with structure sheaves: applications to descent

TL;DR

This paper develops a framework for lax colimits of posets with structure sheaves, applying it to descent problems and proving a generalized Seifert-Van Kampen theorem for schematic spaces, extending classical results to new topologies.

Contribution

It introduces a novel approach to lax colimits in categories of posets with structure sheaves and applies it to prove a generalized descent theorem for schematic spaces.

Findings

Established a description of poset-indexed lax colimits with structure sheaves.

Proved a Seifert-Van Kampen theorem for the étale fundamental group of schematic spaces.

Extended classical scheme results to the topology of flat monomorphisms.

Abstract

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.