Morphisms in the category of finite-dimensional absolute valued algebras
This is a study of morphisms in the category of finite-dimensional absolute valued algebras whose codomains have dimension four. We begin by citing and transferring a classification of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this cross-section.