What is an isomorphism?

Question

I'm familiar with the concepts of group isomorphism, ring isomorphism, and graph isomorphism, but it's never been presented to me what an isomorphism is in general: given any X, what is an X isomorphism?

Informally, I understand isomorphism as "preservation of structure", where "preservation" is domain specific. Is there a formal definition?

Answer 1

As the comment suggests, looking up the word "morphism" in the context of "Categories" and Objects will give you a more rigorous and general idea about isomorphisms. But I always like to think of isomorphism as something that allows you to make copies of a given "object", be it rings or groups, or Topological spaces or manifolds,etc.

Conveniently you can also look at it as a transformation that allows you to preserve the algebraic / geometric / topological structure or atleast the most basic and desired properties pertaining to these.

Edit: The word "copies" might be very misleading actually. Its more prevalent in the context of "Product spaces" than here. But I could not find a better word.Actually I think it might be quite wrong to say copies. Instead I would go along with saying that it allows you to collect spaces which though not essentially copies of the original space still basically carry the same "bloodline" if you will. I realise that this is not a technically sound answer, but was trying to see how best I could put this without resorting too many technical terms.I apologise if this isn't what you need

Answer 2

Suppose you have two structures $A$ and $B$, and suppose you call a function $f:A\to B$ "good" if it satisfies certain properties (for $A,B$ topological spaces, we might want $f$ continuous, for $A,B$ vector spaces we might $f$ to be a linear transformation, &c). Suppose that $f$ is invertible, and $f^{-1}$ is also "good". Then we say $f$ is an isomorphism. In the setting say of vector spaces, rings or groups, whenever $f$ is invertible, it is automatically "good" too, so sometimes people define an isomorphism to be an "invertible homomorphism" say when talking about groups or rings. However, one can give counterexamples when $A,B$ are topological spaces: there exists invertible continuous $f:A\to B$ such that $f^{-1}:B\to A$ is discontinuous.