# A universal rigid abelian tensor category

@inproceedings{BarbieriViale2021AUR, title={A universal rigid abelian tensor category}, author={Luca Barbieri-Viale and Bruno Kahn}, year={2021} }

We prove that any rigid additive symmetric monoidal category can be mapped to a rigid abelian symmetric monoidal category in a universal way. This yields a novel approach to Grothendieck’s standard conjecture D and Voevodsky’s smash nilpotence conjecture.

## References

SHOWING 1-10 OF 49 REFERENCES

Universal rigid abelian tensor categories and Schur finiteness

- Mathematics
- 2022

We study the construction of [4] in more detail, especially in the case of Schur-finite rigid ⊗-categories. This leads to some groundwork on the ideal structure of rigid additive and abelian…

Monoidal abelian envelopes

- MathematicsCompositio Mathematica
- 2021

We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new…

Tensor structure for Nori motives

- MathematicsPacific Journal of Mathematics
- 2020

We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative…

Algebraic cycles on an abelian variety

- Mathematics
- 2009

Abstract It is shown that to every ℚ-linear cycle ᾱ modulo numerical equivalence on an abelian variety A there is canonically associated a ℚ-linear cycle α modulo rational equivalence on A lying…

Exactness and faithfulness of monoidal functors

- Mathematics
- 2021

Inspired by recent work of Peter O’Sullivan, we give a condition under which a faithful monoidal functor between abelian ⊗-categories is exact.

Representations in Abelian Categories

- Mathematics, Philosophy
- 1966

Consider the general but imprecise question: given a category how nicely can it be represented in an abelian category?

Motives , numerical equivalence , and semi-simplicity

- Mathematics
- 2005

In this note we show that the category of motives, as defined via algebraic correspondences modulo an adequate equivalence relation, is a semi-simple abelian category i f and in fact, only i f the…

Autonomization of Monoidal Categories

- MathematicsACT
- 2019

It is shown that contrary to common belief in the DisCoCat community, a monoidal category is all that is needed to define a categorical compositional model of natural language, and the applications of this principle to various distributional models of meaning are illustrated.

A nilpotence theorem for cycles algebraically equivalent to zero

- Mathematics
- 1995

In this paper we prove that a correspondence from a smooth projective variety over a field to itself which is algebraically equivalent to zero is a nilpotent in the ring of correspondences modulo…

Definable Additive Categories: Purity and Model Theory

- Mathematics
- 2011

Definable additive categories and their model theory are the topic of this paper. We begin with background and preliminary results on additive categories. Then definable subcategories, their…