A 6-functor formalism for Z_- and Q_-sheaves on diamonds

For every nuclear Z_ Z ℓ -algebra Λ Λ and every small v-stack X on perfectoid spaces, we construct an ∞ ∞ -category Dnuc(X,Λ ) D nuc ( X , Λ ) of nuclear (i.e., “ind-Banach”) Λ Λ -modules on X . We then construct a full 6-functor formalism for these sheaves, generalizing the étale 6-functor formalism for Λ = F_ Λ = F ℓ . Prominent choices for Λ Λ are Z_ Z ℓ , Q_ Q ℓ and Q_ ̄ Q ℓ ¯ . We also provide and study an abstract notion of ULA sheaves in this setting, whose definition and basic properties can be carried over to any 6-functor formalism. Applied to classifying stacks, we obtain a robust theory of nuclear representations, i.e., continuous representations on filtered colimits of Banach spaces.