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We systematically study noncommutative and nonassociative algebras A andtheir bimodules as algebras and bimodules internal to the representationcategory of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms ofthe monoidal category of A-bimodules by internal homomorphisms, and describeexplicitly their evaluation and composition morphisms. For braided commutativealgebras A the full subcategory of symmetric A-bimodule objects is a braidedclosed monoidal category, from which we obtain an internal tensor productoperation on internal homomorphisms. We describe how these structures deformunder cochain twisting of the quasi-Hopf algebra, and apply the formalism tothe example of deformation quantization of equivariant vector bundles over asmooth manifold. Our constructions set up the basic ingredients for thesystematic development of differential geometry internal to the quasi-Hopfrepresentation category, which will be tackled in the sequels to this paper,together with applications to models of noncommutative and nonassociativegravity such as those anticipated from non-geometric string theory.

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