Category of magmas

In mathematics, the category of magmas, denoted Mag, is the category whose objects are magmas (that is, sets equipped with a binary operation), and whose morphisms are magma homomorphisms.

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.[1]

References

Borceux, Francis; Bourn, Dominique (2004). Mal'cev, protomodular, homological and semi-abelian categories. Springer. pp. 7, 19. ISBN 1-4020-1961-0.

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[1] Borceux, Francis; Bourn, Dominique (2004). Mal'cev, protomodular, homological and semi-abelian categories. Springer. pp. 7, 19. ISBN 1-4020-1961-0.