(Co)homology theories for structured spaces arising from their corresponding poset

TL;DR

This paper explores how cohomology theories for posets can be applied to structured spaces, extending previous work on cohomologies related to these spaces and their associated posets.

Contribution

It demonstrates that various (co)homologies for posets can be adapted to structured spaces, broadening the applicability of these theories.

Findings

Poset (co)homologies can be applied to structured spaces under certain conditions.

The paper extends previous cohomology frameworks to new classes of structured spaces.

It establishes connections between poset-based (co)homologies and structured spaces.

Abstract

In [1] we introduced the notion of 'structured space', i.e. a space which locally resembles various algebraic structures. In [2] and [3] we studied some cohomology theories related to these space. In this paper we continue in this direction: while in [2] we mainly focused on cohomologies arising from $f_s$ and $h$, and in [3] we considered cohomologies for generalisations of objects which involved structured spaces, here we deal with (co)homologies coming from the poset associated to a structured space via an equivalence relation defined at the end of Section 4 in [1]. More precisely, we will show that various (co)homologies for posets can also be applied (under some assumptions) to structured spaces.