We verify that the Künneth and Mayer–Vietoris formulae for magnitude homology of graphs, proven by Hepworth and Willerton, generalise naturally to the metric setting. Similarly, we extend the notion of diagonality of graphs to metric spaces, and verify its stability under products, retracts, and filtrations. As an application, we show that median spaces are diagonal; in particular any Menger convex median space has vanishing magnitude homology.

中文翻译：

---

幅度同源性，对角线和中位数空间

我们验证了由Hepworth和Willerton证明的图的幅值同源性的Künneth和Mayer–Vietoris公式自然可以推广到度量设置。同样，我们将图的对角线概念扩展到度量空间，并验证其在乘积，缩回和过滤条件下的稳定性。作为一个应用，我们表明中位数空间是对角线；特别是任何Menger凸中值空间都具有消失的幅度同源性。