Magma category
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In mathematics, the category of magmas (see category, magma for definitions), denoted by Mag, has as objects sets with a binary operation, and morphisms given by homomorphisms of operations (in the universal algebra sense).
The category Mag has direct products, so the concept of a magma object (internal binary operation) makes sense. (As in any category with direct products).
There is an inclusion functor from Set to Med to (inclusion) Mag as trivial magmas, with operations: right, say, projections (bad references, we need projection maps) : x T y = y.
An 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.es:Magma (categorķa)