In mathematics, the category of medial magmas, also known as the medial category, and denoted Med, is the category whose objects are medial magmas (that is, sets with a medial binary operation), and whose morphisms are magma homomorphisms (which are equivalent to homomorphisms in the sense of universal algebra).
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