# Controlled objects as a symmetric monoidal functor

## TL;DR

This paper constructs a functorial association between symmetric monoidal additive categories with group actions and coarse spaces, enabling a refined equivariant coarse algebraic K-homology.

## Contribution

It introduces a new lax symmetric monoidal functor from $G$-bornological coarse spaces to additive categories, refining equivariant coarse algebraic K-homology.

## Key findings

- Defines a functor $ extbf{V}_{ extbf{A}}^{G}$ for symmetric monoidal additive categories with $G$-action.
- Establishes a lax symmetric monoidal structure on the equivariant coarse algebraic K-homology.
- Provides a framework for functorial and monoidal refinement of equivariant coarse algebraic K-homology.

## Abstract

The goal of this paper is to associate functorially to every symmetric monoidal additive category $\mathbf{A}$ with a strict $G$-action a lax symmetric monoidal functor $\mathbf{V}_{\mathbf{A}}^{G}:G\mathbf{BornCoarse}\to \mathbf{Add}_{\infty}$ from the symmetric monoidal category of $G$-bornological coarse spaces $G\mathbf{BornCoarse}$ to the symmetric monoidal $\infty$-category of additive categories $ \mathbf{Add}_{\infty}$. This allows to refine equivariant coarse algebraic $K$-homology to a lax symmetric monoidal functor.