The vertex fundamental groupoid of a sphere is thin

Put $S=(0,0,-1)$, $U_N=S^2\setminus \{N\}$ and $U_S=S^2\setminus \{S\}$. The chart in [F3] identifies $U_N$ with $\mathbb{C}\cong {\mathbb{R}}^2$. The reflection $r(x,y,t)=(x,y,-t)$ is a continuous self-inverse map of the sphere and interchanges $N$ and $S$, so $\mathrm{\Pi }\circ r$ identifies $U_S$ with the same plane. Both sets are open, path-connected and have trivial fundamental group: in either chart, the homotopy $(t,u)\mapsto (1-u)\alpha (t)+u\alpha (0)$ contracts any based loop while fixing its basepoint. Their overlap is identified by $\mathrm{\Pi }$ with $\mathbb{C}\setminus \{0\}$, since $\mathrm{\Pi }(S)=0$; this is path-connected, as any two nonzero points can be joined by two line segments through a point outside their two real lines through zero. The two charts cover the sphere and have nonempty overlap, so the sphere is path-connected. Choose $x_0\in U_N\cap U_S$. Van Kampen gives ${\pi }_1(S^2,x_0)$ as the pushout of two trivial groups over the overlap group. This pushout is trivial: any compatible pair of homomorphisms from the two trivial factors to a group has a unique factorization through the trivial group.

Fix a homeomorphism $h:K\to S^2$ from [F1]. Composing paths and endpoint-fixed homotopies with $h$ and $h^{-1}$ gives inverse functors on the fundamental groupoids in [F2]. Thus $K$ is path-connected and its loop classes at $a_0=h^{-1}(x_0)$ are trivial by step 1.1. For any $A\in K$, choose a path $p:a_0\to A$. If $\lambda$ is a loop at $A$, then $p\ast \lambda \ast \bar{p}$ is a loop at $a_0$ and has the identity class. Cancellation using the groupoid laws in [F2] gives $[\lambda ]=[c_A]$. Consequently all loop classes at every point of $K$ are trivial.

For any $A,B\in V$ and paths $\alpha ,\beta :A\to B$, the loop $\alpha \ast \bar{\beta }$ is null-homotopic by step 2.1. In the composition convention of [F2], this says $[\beta {]}^{-1}\circ [\alpha ]=1_A$, hence $[\alpha ]=[\beta ]$. A path exists by path-connectedness, so every ambient hom-set between these vertices is a singleton. The full subgroupoid retains exactly these hom-sets. Each directed crossing edge is such a path, so it represents the unique morphism between its endpoints. The opposite crossing is a distinct one-cell, but its class equals the inverse class because the hom-set in the reverse direction is also a singleton. This proves the Statement.