Classification of Concrete Geometrical Categories

A precise concept of concrete geometrical category is introduced in an axiomatic way. To any algebra L for an many-sorted infinitary algebraic theory T is associated a concrete geometrical category Geo(L), the so-called classifying concrete geometrical category of L, satisfying a universal property. The terminology "geometrical" is justified firstly for Geo(L) and secondly for any concrete geometrical category by proving that they are all classifying ones. The legitimate category CGC of concrete geometrical categories is build up and proved to be the dual of the legitimate category TGC of topological geometrical categories.