Paulina Radecka
supervisor: Agata Pilitowska Wiesław Kubiś
An evolution system is a category with a distinguished class of arrows - called transitions - and a fixed object called the origin. This simple yet powerful concept turns out to be a convenient framework for studying both infinite and finite Fraïssé limits – the unique (up to an isomorphism), most complicated, generic objects in a suitable category.
First we state the definition of an evolution system and show some relevant examples of such systems to develop a bit of intuition. Next, we define the most important, in our setting, version of the amalgamation property, which is a notion well-known in model theory. Then, we will state necessary conditions for existance of the unique (up to an isomorphism) evolution with the absorption property. The colimit of such an evolution is precisely the Fraïssé limit of our category. Turns out that the crucial notion for the existence of such an evolution is our amalgamation property and some kind of ‘smallness’ of the system.
We shall explain these ideas, showing a few interesting properties of evolution systems - regularity, termination and determination. Moreover we present the notion of confluence and local confluence in evolution systems and how is it connected to the amalgamation property. Lastly, we show that a variant of famous Newman's lemma (also known as the diamond lemma) holds in our setting.
References:
Kubiś, W., Radecka, P., Abstract evolution systems, link to preprint: https://arxiv.org/abs/2109.12600