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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.

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  • $\begingroup$ Two objects $a$ and $b$ in category called isomorphic, if there exist two arrows $f\colon a\to b$ and $g\colon b\to a$, such that $g\circ f=id(a)$ and $f\circ g=id(b)$. It is the general definition. As corollary, we get that two sets(objects in $\mathbf{Set}$) are isomorphic iff there exists a bijection between them. $\endgroup$ – Oskar Jun 16 '13 at 8:51
  • $\begingroup$ In arbitrary category, it may happen, that objects are not sets, even not sets with some additional structure. So in such cases you can use only general definition of isomorphism. $\endgroup$ – Oskar Jun 16 '13 at 8:54
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    $\begingroup$ Intuitively speaking, if two objects in category are isomorphic, then they have the same categorical properties(e.g. if one object is terminal, so it is true for another too). But, in some sense, all mathematical properties are categorical... $\endgroup$ – Oskar Jun 16 '13 at 8:57
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    $\begingroup$ Saying that two mathematical objects are isomorphic means that they behave the same way in some relevant sense. The example you're trying to understand is relatively sophisticated, so you should start with much simpler examples. For example, learn some group theory and get a feel for what it means to say that two groups are isomorphic. $\endgroup$ – Qiaochu Yuan Jun 16 '13 at 9:06
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    $\begingroup$ The example of geometry is badly chosen. There are many different forms of geometry (Euclidean, projective, Riemannian, differential, to name a few) which are totally incomparable; not only they are not isomorphic, they don't even have the same language, so talking about isomorphism between then is meaningless. In general isomorphism applies to objects (instances, models) of one theory, not between different theories altogether. $\endgroup$ – Marc van Leeuwen Jun 16 '13 at 9:57
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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.

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  • $\begingroup$ Nice answer +1. Only the last paragraph is hard to follow. $\endgroup$ – Marc van Leeuwen Jun 16 '13 at 10:02
  • $\begingroup$ @Ittay Weiss Could you give an example of a structure that have two models? This would be very helpful. $\endgroup$ – Daniela Diaz Jun 16 '13 at 10:03
  • $\begingroup$ @MarcvanLeeuwen thanks! As for the last paragraph, it's meant as a teaser to something quite exciting in current research which I wish I knew all about. $\endgroup$ – Ittay Weiss Jun 16 '13 at 10:09
  • $\begingroup$ @DanielaDiaz a structure with two different models (and more than two) is, e.g., a group, i.e., There are plenty of non-isomorphic groups. But I'm not sure that is what you meant. Can you elaborate? $\endgroup$ – Ittay Weiss Jun 16 '13 at 10:11
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    $\begingroup$ @IttayWeiss: OK I understand for the teaser. But you could at least explain what you mean for two categories to be isomorphic (as opposed to equivalent). In fact, now that I looked it up, just providing the obvious links en.wikipedia.org/wiki/Isomorphism_of_categories and en.wikipedia.org/wiki/Equivalence_of_categories would do nicely. $\endgroup$ – Marc van Leeuwen Jun 16 '13 at 10:25
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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.

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Maybe the following analogy with comparing arabic and roman numerals is helpful: Among these systems, we have a bijection $1\leftrightarrow I$, $2\leftrightarrow II$, $3\leftrightarrow III$, $4\leftrightarrow IV$ and so on. These bijections are compatible with addition and multiplication of numbers. Therefore, two people using these different systems will necessarily agree on all essential (i.e. based on arithmetic) aspects of the natural numbers carried by these notations. For example, they one will say that $1729$ is the smallest natural number that can be written in $2$ ways as sum of $3$rd powers and the other will say that $MDCCXXIX$ is the smallest natural number that can be written in $II$ ways as sum of $III$rd powers. Even though showing $XII^{III}+I^{III}=X^{III}+{IX}^{III}$ may be harder to verify with pen and paper than $12^3+1^3=10^3+9^3$. They will also agree that there are $21$ primes between $10$ and $100$ (or $XXI$ primes between $X$ and $C$) and among these is $17$ (or $XVII$), but not $499$ (or $ID$). But will they agree what a two-digit prime is and that there are $21$ two-digit primes? (This shows the problem with this ananlogy as it actually is about representaions of natural numbers more than about natural numbers).


Or, in graph theory consider the two graphs pictured here: enter image description here

They are isomorphic, which means there is a bijection between vertices and between edges that is compatible with the vertex-edge-incidence. If I had given you a description "Start with two vertices and an edge between them; add a path via a third vertex between them; ad another poth via yet another vertex between them; finally add a path of length three via two more vertices", you might have produced either ofthese drawings from that description. But wait! Only one of these has two triangular faces, and only the other has two quadrangular faces? How can they be isomorphic? Via the isomorphism, these graphs agree on every aspect in the theory of graphs. So either both or none of them has a Hamiltonian path. Both or none of them can be three-coloured. But if we want to talk about faces, we need plane graphs, i.e. graphs that are embedded into the plane. The embedding belongs to the data, and as seen, the same graph can have non-isomorphic embeddings. That is: Some aspects of an object may be unambiguously determined by an axiom system (or a textual description of a graph), but one has to be careful which aspects may not be covered.

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  • $\begingroup$ There is no bijection between arabic and roman numerals. There is no 0 for roman numerals and some numbers can be written in more than one way using roman numerals. $\endgroup$ – Bakuriu Jun 16 '13 at 13:06
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    $\begingroup$ As excuse for the first, I mention (too hidden maybe) that I talk about natural numbers and take the freedom to let $\mathbb N=\{1,2,3\ldots\}$ for the sake of this analogy. For an excuse for the second, we may assume one of the writing variants as normative- For an overall excuse, this was intended rather as an analogy than as a perfect example. :) $\endgroup$ – Hagen von Eitzen Jun 16 '13 at 21:05
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It's not about structural completeness, but structural indistinguishability. Say you're to place the integers on the real number line. If you place 0 and 1 at the same point, can you still add them like they're the original integers? No. So it's not a homomorphism. Morphisms define changing one thing into another successfully. Isomorphisms show the change didn't affect it - it can be changed right back. So, it defines for every definition of change a definition of being the same. Take a look at the algebra formed by $\{1,-1\}$ under multiplication. It works the same way as addition on a two-hour clock, and as adding 6 and 12 hours on the 12-hour clock, and even+even=even, odd+odd=even, etc.. By taking homomorphisms to be the rule for saying whether an operation turns one thing into another, we find we can turn each of these algebras into one another.

However, your latter situation is different, depending on what you mean by a theory. What if the theory of an object involves deriving a new object from another? Take, for example, the dual vector space. It turns out in finite-dimensional situations this is isomorphic to the original vector space. But when you flatten the original space to a lower-dimensional one, there isn't a corresponding flattening of the dual space. Instead, the lower-dimensional dual space gets placed into the higher-dimensional one. So we say that the isomorphism between a vector space and its dual space is unnatural - the identity functor is not doing the same thing as the dual space functor - and therefore the vector space and its dual space aren't the same thing, even if they're isomorphic.

On the other hand, if a constructed object were to change when an isomorphism of the original object were taken, we would certainly consider the idea unstable, not well-defined. Imagine if the Cayley graph of the 2-hour clock depended on the representation chosen from the list above. You would have to think the construction wasn't about algebras at all, but maybe about the structure of clocks.

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Each category defines its own definition of isomorphic. It may be helpful to consider similar categories with the same objects but different definitions of isomorphic.

Example with shapes

Triangles in the Euclidean plane are a nice collection of objects. We'll define three categories, each with their own idea of what isomorphic means, and give three different definitions of a triangle. The completeness of the definition will depend on which category we work in.

(Arbitrary) definitions of isomorphic

The first category contains all Euclidean triangles, and calls two triangles isomorphic if one can be taken to the other using reflections.

The second category contains all Euclidean triangles, and calls two triangles isomorphic if one can be taken to the other using reflections and dilations.

The third category contains all Euclidean triangles, and calls two triangles isomorphic if one can be taken to the other using “shadows” (as in, draw one triangle on the plane z=1, the other on the plane z=0; can you shine a light from somewhere so that the shadow of the z=1 triangle is exactly the z=0 triangle?)

Summaries of isomorphic

In the first category, two triangles are isomorphic iff they are congruent.

In the second category, two triangles are isomorphic iff they are similar.

In the third category, all triangles are isomorphic.

Complete definitions

A definition of a triangle is “complete” means that if two triangles satisfy it, then they are isomorphic.

In the first category, the "SSS congruence theorem" says that giving the three side lengths is a complete definition of a triangle.

In the second category, the "AAA similarity theorem" says that giving the three angle measures is a complete definition of a triangle. This is not complete in the first category.

In the third category, the "projective equivalence theorem" says that "it has three corners" is a complete definition of a triangle. This is not complete in the first or second category.

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