"Grothendieck rings, Euler characteristics, and Schanuel dimensions of models"

Abstract: A distributive category is a category C with finite limits and colimits, an initial object and a final object satisfying the distributivity condition that for any three objects A, B, and C, the canonical map from (A x B) + (A x C) to A x (B + C) is an isomorphism. If C is a small distributive category, then the set of isomorphism types of objects in C carries a natural L_ring = L(+, *, 0, 1) structure, called the rig of C. I will discuss this construction in the case that C is the category of (parametrically) definable sets in some structure (with definable functions as morphisms). These rigs impinge on many areas of mathematics including o-minimal structures, field arithmetic, proof theory, motivic integration, pseudofinite structures, p-adic analysis, and stability theory. I will explain these connections and how in some cases the formalism of rigs of definable sets can be used to solve outstanding problems.
 

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