Category Jord of Jordan algebras

For a field $k$ of characteristic different from $2$ , $\mathbf{Jord}_k$ is the category of Jordan $k$ -algebras and all Jordan homomorphisms. Objects need not be unital. A morphism is a $k$ -linear map satisfying

f(x\circ y)=f(x)\circ f(y).

Identities and composition are the corresponding linear functions.

The catalogue convention

The real and complex categories are separate scalar choices. The UJord category restricts the objects to unital algebras but keeps all these maps. The Jord1 category additionally requires maps to preserve units. These names are explicit conventions of this catalogue.

Zero maps

The zero map always belongs to a Jord Hom-set, so such a Hom-set is never empty. A statement $\operatorname{Hom}(J,K)=\{0\}$ means there is exactly one map, not no maps.