Affine lines over derivators: Properties

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 $\mathbb{D}$.