Quantales, persistence, and magnitude homology

We construct a nerve functor parametrized by a choice of quantale, exhibiting both the Vietoris-Rips complex and the magnitude nerve as instances of this nerve for different choices of monoidal structure on $\mathbb{R}$. Furthermore, the difference between how persistent homology processes the Vietoris-Rips complex and how magnitude homology processes the magnitude nerve is cast as a choice of whether or not to "localize" the corresponding nerves along $\mathbb{R}$ in a precise sense. Lastly, we mention some application-oriented observations naturally suggested by the perspective mentioned above.