Medial category
In mathematics, the category of medial magmas (see category, magma, medial, magma object for definitions), denoted by Med, 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 productss, so the concept of a medial magma object (internal binary operation) makes sense. (As in any category with direct products). But now, 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 magmass, with operationss: right, say, projections (bad references, we need projection maps) : x T y = y.
A very important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.