Abstract We offer a groupoid‐theoretic approach to computing invariants. We illustrate this approach by describing the Gel'fand‐MacPherson correspondence and the Gale transform. We also provide Zariski‐local descriptions of the moduli space of ordered points in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {P}^1$\end{document} . We give an explicit description of the moduli space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M_0(\mathbb {P}^1,2)$\end{document} over \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mbox{Spec}\mathbb {Z}$\end{document} . In characteristic 2, the singularity at the totally ramified cover is isomorphic to the affine cone over the Veronese embedding \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {P}^1 \rightarrow \mathbb {P}^4$\end{document} .