Barycentric Subdivision and Isomorphisms of Groupoids

TL;DR

This paper demonstrates that the subdivision functor on groupoids is conservative, meaning isomorphisms between subdivisions uniquely determine isomorphisms between the original groupoids, highlighting a special property not shared by all categories.

Contribution

The paper proves that the subdivision functor on groupoids is conservative and provides a method to recover original groupoid isomorphisms from subdivision isomorphisms.

Findings

Subdivision functor on groupoids is conservative.

Isomorphisms between subdivisions induce isomorphisms between original groupoids.

Results do not extend to arbitrary categories.

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.