Category Jord1 of unital Jordan algebras and unital maps

For a field $k$ of characteristic different from $2$ , $\mathbf{Jord}_{1,k}$ has unital Jordan algebras as objects. Morphisms $f:J\to K$ are $k$ -linear, preserve the Jordan product, and satisfy

f(1_J)=1_K.

Identity maps and composites obey all three requirements.

Relationship to UJord

This is a subcategory of $\mathbf{UJord}_k$ with the same objects and possibly fewer morphisms. The zero map to a nonzero algebra and a proper matrix-corner inclusion are excluded. The scalar map $k\to J$ , $t\mapsto t1_J$ , is the unique unit-preserving $k$ -linear algebra map from the scalar Jordan algebra.

Endomorphisms versus automorphisms

Restricting to unital maps can shrink the endomorphism monoid. It does not shrink the automorphism group of a unital object, because every bijective Jordan homomorphism preserves the unit automatically.