Abstract. We give a survey with some explanations but no proofs of the new notion of b-minimality by the author and F. Loeser [b-minimality, Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 195–227, math.LO/0610183]. We compare this notion with other notions like o-minimality, C-minimality, p-minimality, and so on.
Introduction. As van den Dries notes in his book , Grothendieck's dream of tame geometries found a certain realization in model theory, at first by the study of the geometric properties of definable sets for some nice structure like the field of real numbers, and then by axiomatizing these properties by notions of o-minimality, minimality, C-minimality, p-minimality, v-minimality, t-minimality, b-minimality, and so on. Although there is a joke speaking of x-minimality with x = a, b, c, d, …, these notions are useful and needed in different contexts for different kinds of structures, for example, o-minimality is for ordered structures, and v-minimality is for algebraically closed valued fields.
In recent work with F. Loeser , we tried to unify some of the notions of x-minimality for different x, for certain x only under extra conditions, to a very basic notion of b-minimality. At the same time, we tried to keep this notion very flexible, very tame with many nice properties, and able to describe complicated behavior.
An observation of Grothendieck's is that instead of looking at objects, it is often better to look at morphisms and study the fibers of the morphisms.