turkmath.org

The theory of motives was conceived as a universal cohomology theory for algebraic varieties. Today it is a vast subject systematically developed in many directions spanning algebraic geometry, arithmetic geometry, higher category theory, and abstract homotopy theory. After briefly reviewing the construction of the stable homotopy category of schemes following Voevodsky, we will pivot to a discussion of the so-called enhanced models of the classical derived category of a scheme. Starting with a foray into enriched and higher category theory, we will then proceed to touch on the notion of a universal additive invariant and introduce some "categorical" cohomology theories such as Hochshild Homology, Topological Cyclic Homology and Algebraic K-theory. We will then discuss the construction of so-called noncommutative motives of dg-categories following Kontsevich, Tabuada, and Robalo, which generalize and encompass pure and mixed motives of schemes and help one assemble the aforementioned cohomology theories associated with derived categories (and much more) into a motivic formalism in a very precise sense of the word. Since there is a great deal of abstract machinery involved, we will try to keep the discussion conceptual.