Auto magma object

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In mathematics, a magma in a category, or magma object, can be defined in a category with a cartesian product. This is the 'internal' form of definition of a binary operation in a category.

As Mag the magma category has direct products, the concept of an (internal) magma (or internal binary operation) in Mag is defined, say

T′: (X,T) × (X,T) → (X,T).

Since T′ is a morphism we must have

(x T′ y) T (u T′ z) = (x T u) T′ (y T z).

If we want to take the original operation, this will be allowed only if the medial identity

(x T y) T (u T z) = (x T u) T (y T z)

is valid.

This operation, which gives a medial magma, can have a two-sided identity only if it is a commutative monoidal operation. The if direction is obvious.

As a result Med, the medial category, has all its objects as medial objects; and this characterizes it.