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We introduce the concept of an abstract evolution system, which provides aconvenient framework for studying generic mathematical structures and theirproperties. Roughly speaking, an evolution system is a category endowed with aselected class of morphisms called transitions, and with a selected objectcalled the origin. We illustrate it by a series of examples from several areasof mathematics. We formulate sufficient conditions for the existence of theunique "most complicated" evolution. In case the evolution system "lives" inmodel theory and nontrivial transitions are one-point extensions, the limit ofthe most complicated evolution is known under the name Fraisse limit, a uniquecountable universal homogeneous model determined by a fixed class of finitelygenerated models satisfying some obvious axioms.  Evolution systems can also be viewed as a generalization of abstractrewriting systems, where the partially ordered set is replaced by a category.In our setting, the process of rewriting plays a nontrivial role, whereas inrewriting systems only the result of a rewriting procedure is relevant. Ananalogue of Newman's Lemma holds in our setting, although the proof is a bitmore delicate, nevertheless, still based on Huet's idea using well foundedinduction.

Evolution systems: A framework for studying generic mathematical structures