Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1_k$ by the action induced by $G\to\mathrm{Aut}_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space $\mathbb{A}^1_k/\mathbb{Z}$ is just $\mathrm{Spec}(k)$. At least to me this is surprising at the moment since dividing out the analogue action of $\mathbb{Z}$ on the topological spaces $\mathbb{R}$ resp. $\mathbb{C}$ gives the sphere $S^1$ (up to homotopy).

**Q.:** Is that simply a pathological example of a categorical quotient or is there a geometric explanation why this is the right orbit space?