Non-associative Categories of Octonionic Bimodules

Category is put to work in the non-associative realm in the article. We focus on a typical example of non-associative category. Its objects are octonionic bimodules, morphisms are octonionic para-linear maps, and compositions are non-associative in general. The octonionic para-linear map is the main object of octonionic Hilbert theory because of the octonionic Riesz representation theorem. An octonionic para-linear map f is in general not octonionic linear since it subjects to the rule Re (f (px)-pf (x))= 0. The composition should be modified so that it preserves the octonionic para-linearity. In this non-associative category, we introduce the Hom and Tensor functors which constitute an adjoint pair. We establish the Yoneda lemma in terms of the new notion of weak functor. To define the exactness in a non-associative category, we introduce the notion of the enveloping category via a universal property. This allows us to establish the exactness of the Hom functor and Tensor functor.