Syntactic categories for Nori motives
Luca Barbieri-Viale, Olivia Caramello, Laurent Lafforgue

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 -linear abelian category of mixed motives associated to a cohomology or homology functor with values in finite-dimensional vector spaces over . This new construction makes sense for infinite-dimensional vector spaces as well, so that it associates a -linear abelian category of mixed motives to any (co)homology functor, not only Betti homology (as Nori had done) but also, for instance, -adic, -adic or motivic cohomology. We prove that the -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…
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