Cone and join in the asymptotic categories

Анотація

We will prove that the join $\mathrm{ℝ}*{\mathrm{ℝ}}_{+}$ is isomorphic to the half-space ${\mathrm{ℝ}}_{+}^{n+1}$ and extend this result onto the class of $\gamma$-weakly convex and $\delta$-weakly concave geodesic spaces. We will also show that the cone over divergent sequence is not isomorphic to the join of this sequence and  ${\mathrm{ℝ}}_{+}$ in the asymptotic category $\mathcal{A}$.