Syntactic categories for Nori motives

Abstract

We give a new construction, based on categorical logic, of Nori's Qlinear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over Q. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a Q-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, ℓ-adic, p-adic or motivic cohomology. We prove that the Q-linear abelian categories of mixed motives associated to different (co)homology functors are equivalent if and only a family (of logical nature) of explicit properties is shared by these different functors. The problem of the existence of a universal cohomology theory and of the equivalence of the information encoded by the different classical cohomology functors thus reduces to that of checking these explicit conditions.

Multisorted modules and their model theory

“…In fact, it turns out that Nori's category of motives is exactly the abelian category of functors/pp-pairs associated to the representation given by singular homology. In essence this first appeared in [7], though it is not said this way. There Caramello used the methods of categorical model theory, in particular classifying toposes for regular logic, and showed that Nori's category is the effectivisation of the regular syntactic category for the regular theory associated to Nori's diagram D. This is a much simpler construction than Nori's original one, in particular there is no need to approximate the final result through finite subdiagrams of D or to go via coalgebra representations.…”

“…In [7] additivity appears at a relatively late stage of the construction. If we build that in from the beginning then, as we show in [9], we are able to apply the existing model theory of additive structures and, in particular, realise Nori's category of motives as a category of pp-pairs (equivalently as a localisation of the free abelian category on the preadditive category Z − → D generated by Nori's diagram D).…”

“…As mentioned there, an important additional feature is that there should be a tensor product structure on motives. Here I briefly report on work [8] with Barbieri-Viale and Huber which extends the constructions of [7] and [9] to an induced tensor product and I present an example, in considerable detail, illustrating a key part of that process, namely the extension of a tensor product on the category of modules to a tensor product on the category of pp-pairs. For the general construction see [8].…”

Tensor product of motives via Künneth formula

“…In the following, we shall denote by T op a cohomology theory as above and we shall say that H is a T op -model is synonymous with H is a cohomology (see the next section and also [2, §3] for details). For any such cohomology H we then get the regular theory T op H of that model, obtained by adding to T op all regular axioms which are valid in H, and, therefore, we obtain a universal category (it is abelian in the additive case), which is the (Barr) exact completion A[T op H ] of the regular syntactic category associated to T op H (see [2, §4], [5, §2.2] and [4]). For example, we get that ECM = A[T op H sing ] is the category of (effective, cohomological) Nori motives if H = H sing is singular cohomology and C is the category of (affine) schemes over a subfield of the complex numbers (e.g., see [11] for details on Nori's original construction and [4] for its reconstruction via syntactic categories).…”

“…For any such cohomology H we then get the regular theory T op H of that model, obtained by adding to T op all regular axioms which are valid in H, and, therefore, we obtain a universal category (it is abelian in the additive case), which is the (Barr) exact completion A[T op H ] of the regular syntactic category associated to T op H (see [2, §4], [5, §2.2] and [4]). For example, we get that ECM = A[T op H sing ] is the category of (effective, cohomological) Nori motives if H = H sing is singular cohomology and C is the category of (affine) schemes over a subfield of the complex numbers (e.g., see [11] for details on Nori's original construction and [4] for its reconstruction via syntactic categories). Moreover, in general, for any cohomology H but in the additive case only, there is a canonical equivalence (*) A[T op H ] ∼ = A(H) with the universal abelian category A(H) given by H regarded as a representation of Nori's diagram D Nori associated to C (see [5,Cor.…”