Appears in collection : Topos à l'IHES

Key results due to O. Caramello show us that there is a regular theory such that the Barr exact completion of its regular syntactic category is equivalent to the category of Nori effective motives. In this talk, I will explain and consider a (co)homology theory T on any base category C as a fragment of a first-order theory whose models are certain functors to (families of internal abelian) groups satisfying some exactness conditions. Denote A[T] the Barr exact completion of the regular syntactic category: this is an abelian category whose objects may be called constructible effective T-motives. Furthermore, under mild conditions on the base category C we get a T-motivic functor from C to D(Ind-A[T]) the (unbounded) derived category of the Ind category of A[T]: we may call T-motivic complexes the objects of (a suitable localization of) the category D(Ind-A[T]). 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, if C is the category of algebraic schemes,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.