Generic Mathematical Structures, Wiesław Kubiś

A mathematical object can be called “generic” if it appears, up to isomorphism, with probability one as the result of a natural stochastic process. Instead of probability, one may its topological counterpart, using the Baire category theorem. Yet another option is using a natural infinite game for two players, declaring an object U ”generic” if one of the players has a suitable winning strategy leading to the isomorphic copy of U. The story of generic mathematical structures goes back to Cantor, who was the first to identify the set of rational numbers as the generic countable linearly ordered set. About half a century later, Fraïssé developed an abstract theory of universal homogeneous structures (nowadays called ”Fraïssé limits”) which until recent years was viewed as a part of model theory. As it happens, Fraïssé limits are particular cases of generic mathematical objects which can be found in several branches of mathematics, starting from model theory, algebra, functional analysis, and geometric topology. We will try to explain why pure and enriched category theory is the suitable language and framework for studying these objects. Krakow Methodological Conference: 2019 is financed from the funds of the Minister of Science and Higher Education allocated to the dissemination of science [761/P-DUN/2019]. Krakow Methodological Conference: 2019 - zadanie finansowane w ramach umowy 761/P-DUN/2019 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę. #CategoryTheory #KrakowMethodologicalConference https://23kmc.copernicuscenter.edu.pl https://www.adamwalanus.pl/2019/konfmet/index.html