Toward a descent theoretic formulation for organization and emergence - An initial object sheaf α hypothesis and its consequence

Abstract

Let αα be an initial object (let ω be a terminal object) of a temporal topos S^; by definition for an arbitrary object mm of S^ there exists a uniquely determined morphism δ_m:α→m (σ_m:m→ω). Sheaf α, an initial object of S^, plays a crucial and functional role for our cognitive study in this paper, where a terminal object ω plays an essential role for the temporal topos theoretic methods for the fundamental aspects of quantum physics and quantum gravity as in [5], [6], [7], [8], [9]. Via the methods of descent theory e.g., as in [1], sheaf α can give explicit categorical formulations for cognitive concepts and phenomena in what will follow. Under the hypothesis on the existence of sheaf α as an initial object of tt-topos, the functionality of the role among fundamental phenomena will be exposed. As an example, a sheaf (stack)- categorical formulation is given for the mechanism for the emergence from locally trivial individual ant's behavior of a leaf-collecting ant colony to the global and organized consequence as a colony. A fundamental principle of this paper is that a prestack is a stack if a system is self-organized or local objects may be patched together to become a global object.