Introduction aux dynamiques cat\'egoriques connectives

TL;DR

This paper explores the categorical framework for connective dynamics, linking connectivity spaces, representations, and dynamical systems to understand their inherent phenomena and structures.

Contribution

It introduces a categorical approach to connective dynamics, integrating connectivity spaces with dynamical systems and representations in a unified framework.

Findings

Connective representations often originate from dynamical systems.

Categorical perspective enables unified treatment of temporalities and dynamics.

Connectivity order can be applied to categorical connective dynamics.

Abstract

This text is a continuation to my former article "On Connectivity Spaces". It takes into account that connectivity spaces gives rise to phenomena which are essentially dynamic. In a first stage, the representation of finite connectivity spaces by links (Brunn-Debrunner-Kanenobu's theorem) leads to the notion of connective representation. But examples of connective representations often come from dynamical systems. And this is even more obvious when we study the adjoint notion of connective foliation. To apply those notions to dynamics, we first need to consider dynamical systems in an unified way. This is done with a categorical point of view on temporalities and dynamics. It is then possible to define categorical connective dynamics, and to apply to them the various connective notions, specially the connectivity order of a connectivity space.