What is exactly the meaning of being isomorphic?

I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or geometry, I often hear that people say that such a structure is complete in the sense that any other set that satisfy their properties is isomorphic. What is the real meaning and implications of being isomorphic? At least to me it is not very clear.

Let's see for example the axioms of geometry. To begin with, there are different systems of axioms, all of them trying to define that object called geometry. So if we are talking about the same object, I suppose the theories generated should be isomorphic each other. Does this make sense? Again what is the real implication of being isomorphic? According to the definition there should be a bijection between the sets, which means that it cannot be larger sets (with a higher cardinality, or even different cardinality) that hold the propierties of the given structure and also the operations defined on each structure should be mantained in the bijection. Is this all? Please, if I'm saying bullshit forgive me I just want to really understand this concept. I will really appreciate any comment.

Edit: In my example I meant Euclidian Geometry.

Solution 1:

When we say that two things are isomorphic, we are saying that they are essentially the same. To make this into something with rigorous meaning we, of course, must say what is it we mean by 'essentially' here. Different interpretations will lead to different notions of isomorphisms.

For instance, suppose we really like triangles in the plane and we want to study triangles. We quickly realize that if we take one triangle, apply a rotation and/or a translation and/or a reflection we obtain another triangle that is different since it is located in a different area of the plane, and perhaps it's rotated now, but it's very much like the original triangle. We may then choose to regard any two triangles $T_1,T_2$ such that $T_1$ may be obtained from $T_2$ by applying a rotation and/or a translation and/or a reflection. To make this precise, we define two such triangles to be congruent and agree that for all intents and purposes we treat them as essentially identical. So, we are actually studying equivalence classes of congruent triangles.

Now, the same principle applies to other mathematical structures. For instance, it is a known fact that any two models of Peano arithmetic + Induction are isomorphic. One says that the theory is categorical (though this has nothing to do with category theory). Here the meaning of isomorphism is that there exists a bijection between the modeling sets which respects all of the structure. In particular, it implies that anything true about the first model translated via the bijection to something true about the second model. Thus, there is essentially only one model of the natural numbers, where here 'essentially' means 'up to isomorphism', and the implication is that there is basically just one model of the natural numbers, so that they are all the same except for possibly having different names for the numbers. A similar categoricity is well known for the reals when axiomatized as a complete ordered field.

Category theory is a very general setting for talking about structure, and thus also for talking about essential sameness, and thus about isomorphisms. A category can be a very abstract thing where the objects need not be sets at all. Nonetheless, we say that two objects $x,y$ in a category are isomorphic if there is an isomorphism between them. An isomorphism in a category is an arrow $fx\to y$ such that there is an arrow $g:y\to x$ such that $f\circ g =id_y$ and $g\circ f =id_x$. Two isomorphic objects in a category are interchangeable as far as categorical properties are concerned. That is, whatever categorical property that $x$ satisfies, $y$ satisfies too. So, if you are interested in a mathematical problem that can be formulated inside some category, then you can't possibly care if the answer to your question is $x$ or $y$.

As for geometry in general, there are plenty of different notions of geometry and it can get a bit tricky, so I'll leave it unanswered.

It should also be noted that often a weaker notion than isomorphism is important. For instance, if two category are isomorphic, then they are certainly essentially the same, but this turns out to be rather strong a condition. A much weaker condition is that of equivalence of categories. All of this is strongly related to the recent book Voevodsky's Univalent Axiom, a new foundations for mathematics.

Solution 2:

The concept of isomorphism is used to describe situations in which two objects are indistinguishable. More specifically, when one is working with a category, one takes the point of view that any object is characterized by the way it interacts with the other objects of the category, i.e. which maps it receives from other objects and which maps it has to other objects. This can be made precise - see Yoneda Lemma. Two objects are then isomorphic if they interact in precisely the same way with any other object of the category. Note that this notion of "indistinguishability" depends on the category you're working in. In many situations, the objects one is considering will fit in many different categories and whether two objects will be isomorphic depends on the choice of of the category you're working with.

Here is an example: in the category of real vector spaces with linear maps, $\mathbb{R}^2$ and $\mathbb{C}$ will be isomorphic (every complex number is determined by a pair of real numbers, its real and imaginary part). However, both $\mathbb{R}^2$ and $\mathbb{C}$ can also be considered as objects in the category of commutative rings with ring homomorphisms (define multiplication on $\mathbb{R}^2$ componentwise) and in this category they are not isomorphic. This means, that as real vectorspaces, $\mathbb{R}$ and $\mathbb{C}$ are indistinguishable, but when we allow ourselves to take into account their structure of commutative rings we can tell a a difference between them.

Now, when your hear a statement about something being determined up to isomorphism, there is usually an implicit category that one has in mind in which this statement holds: In a given category $C$ all objects that have a certain property must be isomorphic, in that category.