What's the canonical embedding of the globe category into Top? I'm looking at this characterization of the globe category, and it's not clear to me how to think about it from an intuitive spatial standpoint. If each object $[n]$ corresponds to an $n$-sphere (ball?), then what are $\sigma$ and $\tau$?

They are the maps of the closed $n$-ball to the "northern" and "southern" hemispheres of the surface of the $(n+1)$-ball.