I read in Awodey's Category Theory book that the definition of isomorphism in category theory is more general than the one in abstract algebra. For example, he says, the definition of isomorphism from abstract algebra doesn't make sense for monoids. Why not?
[Math] Why is ‘isomorphism’ defined more generally in Category theory than in Abstract Algebra
abstract-algebracategory-theory
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Best Answer
Notice that Awodey writes "Moreover, in some cases only the abstract definition makes sense, for example, in the case of a monoid", not "in the case of monoids".
For Awodey, a monoid is a category with one object, and the isomorphisms are the invertible arrows, i.e. the invertible elements of the monoid. Since the arrows aren't actually functions, it doesn't make sense to ask if they are bijective.