| The universal construction of the Grothendieck group K associated to to an abelian monoid is extremely simple yet very powerful. We now know K explicitly in apparently different contexts such as that of symmetric monoidal categories and pretriangulated categories in geometry, Waldhausen and DG categories in homotopy theory and much more. The similarity between all of these constructions is highlighted by the notion of "linearization", the most classical example of linearization being cell decomposition in complex geometry and elementary topology.
n this survey talk, I review several constructions of the Grothendieck group in algebraic, geometric and topological categories and show how they are related to the homotopy theory of schemes and to the still-mysterious theory of "motives" in algebraic geometry by making the philosophy of linearization precise in terms of Chow groups. Time permitting, we will hint at the generalization of this philosophy to higher dimensional cases using Quillen K-theory. |