  We develop a limit theory for controlled path-dependent mean field stochastic
partial differential equations (SPDEs) within the semigroup approach of Da
Prato and Zabczyk. More precisely, we prove existence results for mean field
limits and particle approximations, and we establish set-valued propagation of
chaos in the sense that we show convergence of sets of empirical distributions
to sets of mean field limits in the Hausdorff metric topology. Furthermore, we
discuss consequences of our results to stochastic optimal control. As another
application, we deduce a propagation of chaos result for Peng's $G$-Brownian
motion with drift interaction.
