Non-associative Categories of Octonionic Bimodules

TL;DR

This paper develops a non-associative categorical framework for octonionic bimodules, introducing new functors, a Yoneda lemma, and exactness concepts tailored to the non-associative setting, advancing octonionic Hilbert theory.

Contribution

It introduces a novel non-associative category for octonionic bimodules, defining Hom and Tensor functors, and establishes foundational categorical results like the Yoneda lemma and exactness.

Findings

Defined octonionic para-linear maps and their composition rules.

Established the Yoneda lemma in a non-associative context.

Proposed the enveloping category to define exactness.

Abstract

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.