  Motivated by the direct method in the calculus of variations in $L^{\infty}$,
our main result identifies the notion of convexity characterizing the
weakly$^*$ lower semicontinuity of nonlocal supremal functionals: Cartesian
level convexity. This new concept coincides with separate level convexity in
the one-dimensional setting and is strictly weaker for higher dimensions. We
discuss relaxation in the vectorial case, showing that the relaxed functional
will not generally maintain the supremal form. Apart from illustrating this
fact with examples of multi-well type, we present precise criteria for
structure-preservation. When the structure is preserved, a representation
formula is given in terms of the Cartesian level convex envelope of the
(diagonalized) original supremand. This work does not only complete the picture
of the analysis initiated in [Kreisbeck \& Zappale, Calc.~Var.~PDE, 2020], but
also establishes a connection with double integrals. We relate the two classes
of functionals via an $L^p$-approximation in the sense of $\Gamma$-convergence
for diverging integrability exponents. The proofs exploit recent results on
nonlocal inclusions and their asymptotic behavior, and use tools from Young
measure theory and convex analysis.
