Syntactic categories for Nori motives

TL;DR

This paper introduces a categorical logic-based construction of Nori's mixed motives, extending it to various cohomology theories and establishing criteria for their equivalence based on explicit logical properties.

Contribution

It generalizes Nori's construction to infinite-dimensional vector spaces and different cohomology theories, providing a logical framework for their comparison.

Findings

Constructs a new categorical logic-based approach to mixed motives.

Shows equivalence of motives from different cohomology theories under explicit conditions.

Extends the scope of Nori's motives to broader cohomology functors.

Abstract

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, $\ell$-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.