Motivic Topos

In this talk, I will provide an introduction to motivic toposes together with theoretical motives. Considering a regular (co)homology theory T on a base category C as a fragment of a first-order theory whose models are certain functors to (families of internal) groups satisfying some exactness conditions, we obtain a motivic topos E[T] given by sheaves on a motivic site. Inside E[T] there is a canonical Grothendieck abelian category of effective T-motives and a subcategory of constructible objects A[T], which can also be regarded as the Barr exact completion of the regular syntactic category. In particular, if C is the category of algebraic schemes over a subfield of the complex numbers, we get an exact functor from constructible effective T-motives to Nori effective motives which lifts to T-motivic complexes. Finally, I explain a way to construct a functor from the category of T-motivic complexes to the category of effective (unbounded) Voevodsky motivic complexes and provide some evidence for the latter being obtained as a (Bousfield) localization of the former.