Universal Weil cohomology
L. Barbieri-Viale, B. Kahn
TL;DR
This work constructs a universal Weil cohomology over any field $k$, universal among cohomologies valued in rigid tensor categories and, dually, among abelian targets; it develops a 2-categorical framework to encode cohomologies as functors on Chow correspondences and motives, and proves a representability theorem yielding a universal object $W_{\mathcal{V}}^{*}$ (with a Tate twist $\mathbb{L}$) and its abelian version. By varying the admissible class $\mathcal{V}$ and allowing extensions to the base and to graded settings, the authors obtain initial and abelian initial universals, and relate them to André’s category of motivated cycles, traditional cohomology theories, and Chow–Künneth decompositions under standard conjectures. The paper also integrates Lefschetz theory, Albanese invariance, and conjectural conditions (C, D, V) into the universal framework, proving that tight cohomologies admit refined universals (and tight abelian variants) and enabling a structured comparison across cohomology theories. Collectively, the results yield a flexible, universal motive-theoretic landscape in which classical and new cohomologies interrelate, with potential applications to base change, abelian-type motives, and the broader program of generalized motivic theory. The synthesis advances the understanding of how cohomological realizations can be universal, functorial, and controllable via axioms, while clarifying the role of conjectures in shaping the universality and the resulting motivic categories.
Abstract
We construct a new Weil cohomology for smooth projective varieties over a field, universal among Weil cohomologies with values in rigid additive tensor categories. A similar universal problem for Weil cohomologies with values in rigid abelian tensor categories also has a solution. We give a variant for Weil cohomologies satisfying more axioms, like Weak and Hard Lefschetz. As a consequence, we get a different construction of André's category of motives for motivated correspondences and show that it has a universal property. This theory extends over suitable bases.