Guarded Fraïssé Banach spaces

We characterize separable Banach spaces having $G_δ$ isometry classes in the Polish codings $\mathcal{P}$, $\mathcal{P}_\infty$ and $\mathcal{B}$ introduced by Cúth-Doležal-Doucha-Kurka [13] as those being guarded Fraïssé, a weakening of the notion of Fraïssé Banach spaces defined by Ferenczi-Lopez-Abad-Mbombo-Todorcevic [18]. We prove a Fraïssé correspondence for those spaces and make links with the notion of $ω$-categoricity from continuous logic, showing that $ω$-categorical Banach spaces are a natural source of guarded Fraïssé Banach spaces. Using those results, we prove that for many values of $(p, q)$, the Banach space $L_p(L_q)$ has a $G_δ$ isometry class; we precisely characterize those values.