Evolution systems: A framework for studying generic mathematical structures

TL;DR

This paper introduces evolution systems as a general framework for studying mathematical structures, encompassing categories with morphisms called transitions, and explores their properties, existence conditions, and connections to other systems like rewriting systems.

Contribution

It formalizes evolution systems, provides conditions for their most complex evolution, and links them to concepts like Fraisse limits and rewriting systems.

Findings

Existence of a unique 'most complicated' evolution under certain conditions

Evolution systems generalize abstract rewriting systems

An analogue of Newman's Lemma is established in this setting

Abstract

We introduce the concept of an abstract evolution system, which provides a convenient framework for studying generic mathematical structures and their properties. Roughly speaking, an evolution system is a category endowed with a selected class of morphisms called transitions, and with a selected object called the origin. We illustrate it by a series of examples from several areas of mathematics. We formulate sufficient conditions for the existence of the unique "most complicated" evolution. In case the evolution system "lives" in model theory and nontrivial transitions are one-point extensions, the limit of the most complicated evolution is known under the name Fraisse limit, a unique countable universal homogeneous model determined by a fixed class of finitely generated models satisfying some obvious axioms. Evolution systems can also be viewed as a generalization of abstract rewriting systems, where the partially ordered set is replaced by a category. In our setting, the process of rewriting plays a nontrivial role, whereas in rewriting systems only the result of a rewriting procedure is relevant. An analogue of Newman's Lemma holds in our setting, although the proof is a bit more delicate, nevertheless, still based on Huet's idea using well founded induction.