# The weight complex functor is symmetric monoidal

## TL;DR

This paper proves that the weight complex functor is symmetric monoidal within stable symmetric monoidal $$-categories, extending its applicability and developing new $$-categorical tools.

## Contribution

It establishes the symmetric monoidal property of the weight complex functor in the setting of stable symmetric monoidal $$-categories under natural compatibility conditions.

## Key findings

- The weight complex functor is symmetric monoidal under certain conditions.
- Development of additive and stable symmetric monoidal $$-categorical Yoneda embeddings.
- Potential for broader applications in motivic homotopy theory.

## Abstract

Bondarko's (strong) weight complex functor is a triangulated functor from Voevodsky's triangulated category of motives to the homotopy category of chain complexes of classical Chow motives. Its construction is valid for any dg enhanced triangulated category equipped with a weight structure. In this paper we consider weight complex functors in the setting of stable symmetric monoidal $\infty$-categories. We prove that the weight complex functor is symmetric monoidal under a natural compatibility assumption. To prove this result, we develop additive and stable symmetric monoidal variants of the $\infty$-categorical Yoneda embedding, which may be of independent interest.