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 inclusionfunctor 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.
A free magma on a set X is the "most general possible" magma generated by the set X (that is there are no relations or axioms imposed on the generators; see free object).
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).
Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation.
A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set.
Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph.
A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
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