Category of monoids

In mathematics, the category of magmas, denoted 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: Set → Med ↪ Mag as trivial magmas, with operations given by projection: 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.

Because the singleton ({*}, *) is the zero object of Mag, and because Mag is algebraic, Mag is pointed and complete.

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