Barycentric Subdivision and Isomorphisms of Groupoids

Abstract

Given groupoids $\mathcal{G}$ and $\mathcal{H}$ as well as an isomorphism $\Psi :\text{Sd}\mathcal{G}\cong \text{Sd}\mathcal{H}$ between subdivisions, we construct an isomorphism $P:\mathcal{G}\cong \mathcal{H}$. 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.

55 pages, written for the 2015 UChicago REU