# Affine lines over derivators: Properties

## TL;DR

This paper explores the properties of the affine line over a derivator, including morphisms, monoidal structures, and universal properties, extending to affine spaces and providing foundational insights into derivator geometry.

## Contribution

It introduces a detailed study of the affine line over a derivator, establishing morphisms, monoidal structures, and universal properties, extending to affine spaces.

## Key findings

- Characterization of morphisms between derivators and their affine lines
- Existence of a monoidal structure on the affine line when the derivator is monoidal
- Universal property of the affine line over a derivator, analogous to polynomial rings

## Abstract

In this paper we detail a number of properties of the affine line of a derivator, including a number of morphisms between $\mathbb{D}$ and $\mathbb{A}^1_{\mathbb{D}}$, a monoidal structure on $\mathbb{A}^1_{\mathbb{D}}$ if $\mathbb{D}$ is monoidal, and a universal property of $\mathbb{A}^1_{\mathbb{D}}$ over $\mathbb{D}$ akin to the universal property that $R[t]$ has over $R$. These results also extend naturally to affine space $\mathbb{A}^n$ over a derivator $\mathbb{D}$.