Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras $A$ and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of $A$-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras $A$ the full subcategory of symmetric $A$-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.