On the Relative Projective Space

Let $(\mathcal{C},\otimes ,\mathbb{1})$ be an abelian symmetric monoidal category satisfying certain exactness conditions. In this paper we define a presheaf ${\mathbb{P}}_{\mathcal{C}}^n$ on the category of commutative algebras in $\mathcal{C}$ and we prove that this functor is a $\mathcal{C}$-scheme in the sense of Toen and Vaqui'e. This construction gives us a context of non-associative relative algebraic geometry. The most important example of the construction is the octonionic projective space.