  In categorical realizability, it is common to construct categories of
assemblies and categories of modest sets from applicative structures. These
categories have structures corresponding to the structures of applicative
structures. In the literature, classes of applicative structures inducing
categorical structures such as Cartesian closed categories and symmetric
monoidal closed categories have been widely studied. In this paper, we expand
these correspondences between categories with structure and applicative
structures by identifying the classes of applicative structures giving rise to
closed multicategories, closed categories, monoidal bi-closed categories as
well as (non-symmetric) monoidal closed categories. These applicative
structures are planar in that they correspond to appropriate planar lambda
calculi by combinatory completeness. These new correspondences are tight: we
show that, when a category of assemblies has one of the structures listed
above, the based applicative structure is in the corresponding class. In
addition, we introduce planar linear combinatory algebras by adopting linear
combinatory algebras of Abramsky, Hagjverdi and Scott to our planar setting,
that give rise to categorical models of the linear exponential modality and the
exchange modality on the non-symmetric multiplicative intuitionistic linear
logic.
