In mathematics, the medial category Med, that is, the category of medial magmas has as objects sets with a medial binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense).

The category Med has direct products, so the concept of a medial magma object (internal binary operation) makes sense. As a result, Med has all its objects as medial objects, and this characterizes it.

There is an inclusion functor from Set to Med as trivial magmas, with operations being the right projections

(x, y) → y.

An injective endomorphism can be extended to an automorphism of a magma extension — the colimit of the constant sequence of the endomorphism.

See also

• Eckmann-Hilton argument