Barycentric Subdivision and Isomorphisms of Groupoids

Abstract

Given groupoids $\mathscr G$ and $\mathscr H$ as well as an isomorphism $\Psi:\text{Sd}\,\mathscr G\cong\text{Sd}\,\mathscr H$ between subdivisions, we construct an isomorphism $P:\mathscr G\cong\mathscr H$. If $\Psi$ equals $\text{Sd} F$ for some functor $F$, then the constructed isomorphism $P$ is equal to $F$. It follows that the restriction of $\text{Sd}$ to the category of groupoids is conservative. These results do not hold for arbitrary categories.