A geometry of information, I: Nerves, posets and differential forms

TL;DR

This paper explores a unified differential geometric framework for spatial representations, using nerve constructions and refinement, aiming to handle dynamic changes in spaces and data, with potential applications to complex spaces like generalized Cantor sets.

Contribution

It introduces a novel approach to the differential geometry of spatial representations, combining nerve constructions, posets, and differential forms to model dynamic spatial data.

Findings

Proposes a differential geometry framework for spatial representations.

Develops an algebra of differential forms for dynamic spaces.

Suggests generalized Cantor sets as test spaces for the theory.

Abstract

The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial Representation: Continuous vs. Discrete'. Spatial representation has two contrasting but interacting aspects (i) representation of spaces' and (ii) representation by spaces. In this paper, we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations change, data changes, spaces change. We will examine the possibility of a `differential geometry' of spatial representations of both types, and in the sequel give an algebra of differential forms that has the potential to handle the dynamical aspect of such a geometry. We will discuss briefly a conjectured class of spaces, generalising the Cantor set which would seem ideal as a test-bed for the set of tools we are developing.