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For every nuclear $\mathbb Z_\ell$-algebra $\Lambda$ and every small v-stack$X$ we construct an $\infty$-category $\mathcal D_{\mathrm{nuc}}(X,\Lambda)$ ofnuclear $\Lambda$-modules on $X$. We then construct a full 6-functor formalismfor these sheaves, generalizing the \'etale 6-functor formalism for $\Lambda =\mathbb F_\ell$. Prominent choices for $\Lambda$ are $\mathbb Z_\ell$, $\mathbbQ_\ell$ and $\bar{\mathbb Q_\ell}$ and especially in the latter two cases, nosatisfying 6-functor formalism has been found before. Applied to classifyingstacks we obtain a theory of nuclear representations, i.e. continuousrepresentations on filtered colimits of Banach spaces.

The 6-Functor Formalism for $\mathbb Z_\ell$- and $\mathbb Q_\ell$-Sheaves on Diamonds