Dimension and partial groups

A partial group with $n+1$ elements is, when regarded as a symmetric simplicial set, of dimension at most $n$. This dimension is $n$ if and only if the partial group is a group. As a consequence of the first statement, finite partial groups are genuinely finite, despite being seemingly specified by infinitely much data. In particular, finite partial groups have only finitely many im-partial subgroups. We also consider dimension of partial groupoids.

💡 Research Summary

The paper “Dimension and Partial Groups” investigates partial groups from the viewpoint of symmetric simplicial sets, establishing a robust notion of dimension that sharply distinguishes genuine groups from more general partial groups. The authors work in the category Υ whose objects are the finite ordinals