  Given an algebra in a monoidal 2-category, one can construct a 2-category of
right modules. Given a braided algebra in a braided monoidal 2-category, it is
possible to refine the notion of right module to that of a local module. Under
mild assumptions, we prove that the 2-category of local modules admits a
braided monoidal structure. In addition, if the braided monoidal 2-category has
duals, we go on to show that the 2-category of local modules also has duals.
Furthermore, if it is a braided fusion 2-category, we establish that the
2-category of local modules is a braided multifusion 2-category. We examine
various examples. For instance, working within the 2-category of 2-vector
spaces, we find that the notion of local module recovers that of braided module
1-category. Finally, we examine the concept of a Lagrangian algebra, that is a
braided algebra with trivial 2-category of local modules. In particular, we
completely describe Lagrangian algebras in the Drinfeld centers of fusion
2-categories, and we discuss how this result is related to the classifications
of topological boundaries of (3+1)d topological phases of matter.
