Abstract: The set $M(A)$ of multiplicative seminorms on a commutative ring $A$ carries a natural topology. By considering completions of $A$ with respect to these seminorms, Berkovich obtained a structure sheaf of $\mathcal O_A$ of "analytic" functions on $M(A)$.

We consider (the rudiments of) a categorification of Berkovich’s theory, associating to every symmetric monoidal stable infinity category $\mathcal A$ a topological space $M(\mathcal A)$ and sheaf of categories $\mathcal O_{\mathcal A}$. We emphasize the concrete nature of these objects. For example, for $k$ a field and any $r > 0$ the function $$N_{r,k}: \mathsf{Ob}(\mathsf{Sp}^\mathrm{fin}) \to \mathbb{R}_{\geq 0}$$ $$ X \mapsto \sum_i \mathrm{dim} H_i(X; k) r^i$$ is a point in $\mathcal M(\mathsf{Sp}^\mathrm{fin})$. Moreover, the skeletal filtration of a spectrum is often a "Cauchy sequence."

We carry through the theory far enough to compute $M(\mathsf{Sp}^\mathrm{fin})$ (which the reader familiar with chromatic homotopy theory may now guess).