A Comparison of Categories of Nori Motivic Sheaves

We show that two different possible theories of Nori motivic sheaves, introduced by Ivorra–Morel and by Ayoub, respectively, are canonically equivalent. The proof of this result, which exploits the six functor formalism systematically, is based on the Tannakian theory of motivic local systems. As a consequence, we obtain a system of realization functors of Voevodsky motivic sheaves into Nori motivic sheaves compatible with the six operations, previously constructed by Tubach using different methods.

💡 Research Summary

The paper establishes a canonical equivalence between the two existing constructions of Nori motivic sheaves: the Ivorra–Morel approach, which emphasizes a universal abelian category factoring the homological functor from Voevodsky motives to perverse sheaves of geometric origin, and the Ayoub approach, which builds on the action of the motivic Galois group (G^{\mathrm{mot}}_{A}(k)) on the same perverse‑geometric categories. Both constructions aim to extend Nori’s original abelian category of mixed motives (M(k)) to a six‑functor formalism over arbitrary (k)-varieties.

The authors first recall the necessary background on Voevodsky motives, the Betti realization, and the construction of the triangulated categories (\mathrm{D}^{\mathrm{geo}}{\mathrm{B}}(X)) of constructible complexes of geometric origin. They equip these categories with the perverse (t)-structure whose heart is (\mathrm{Perv}^{\mathrm{geo}}(X)). Ayoub’s motivic Galois group acts on (\mathrm{D}^{\mathrm{geo}}{\mathrm{B}}(X)) in a (t)-exact way, allowing one to form the homotopy‑fixed‑point category (\mathrm{D}^{\mathrm{geo}}{\mathrm{B}}(X)^{G^{\mathrm{mot}}{A}(k)}). By a G‑equivariant version of Beilinson’s equivalence, the heart of this fixed‑point category is identified with (\mathrm{Perv}^{\mathrm{geo}}(X)^{G^{\mathrm{mot}}_{A}(k)}).

On the Ivorra–Morel side, the abelian category (\mathcal{M}(X)) is defined as the universal abelian category through which the composite functor
\