In [[mathematics]], the '''category of magmas''' (see [[category theory|category]], [[magma (algebra)|magma]] for definitions), denoted by '''Mag''', has as objects sets with a [[binary operation]], and [[morphism]]s given by [[homomorphism]]s of operations (in the [[universal algebra]] sense).
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The category '''Mag''' has [[Direct_product#Categorical_Product|direct product]]s, so the concept of a [[magma object]] (internal binary operation) makes sense. (As in any category with direct products).
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There is an ''inclusion'' [[functor]] from '''Set''' to '''[[Medial category|Med]]''' to (inclusion) '''Mag''' as trivial [[magma (algebra)|magma]]s, with [[binary operation|operation]]s: right, say, [[projection]]s ('''bad references, we need projection maps''') : x T y = y.
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A very important property is that an [[injective]] [[endomorphism]] can be extended to an [[automorphism]] of a magma [[extension (algebra)|extension]], just the [[colimit]] of the ([[constant]] [[sequence]] of the) [[endomorphism]].
