In this blog post, Terence Tao gives the following definition of a group.
Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of $d$-simplices are bijective for $d > 1$ (and merely surjective for $d \leq 1$).
I understand the benefit of this definition is that it is easier to generalize to higher dimensions, but I'm not able to crack the geometric intuition behind it. I'm hoping some examples will help. So, what are some examples of (relatively) basic facts in group theory which become obviously geometric in terms of this definition? What are some palpable "applications" of this approach?
