Syntactic categories for Nori motives

We give a new construction, based on categorical logic, of Nori’s $\mathbb Q$-linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over $\mathbb Q$. This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a $\mathbb{Q}$-linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, $\mathrm{\ell }$-adic, $p$-adic or motivic cohomology. We prove that the $\mathbb{Q}$-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.