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Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing mathematicians: there's a difference between equality upto isomorphism and equality upto unique isomorphism, and that's it (not to forget about isomorphism and natural isomorphism).

But personally, I'm not totally satisfied with this stance when looking at truly abstract structures like unlabelled graphs (finite or infinite, countable or uncountable), conceived as nothing-but-dots-and-arrows.

Are two abstract structures not to be considered equal in the strongest sense - being one and the same - as soon as there is an isomorphism between them, regardless of being unique, natural, and/or not?

The existence of isomorphisms in turn tells us something about the symmetries of the abstract structure, but of one and only one.

Maybe there are no such abstract structures per se, but only concrete structures (models) and/or concrete presentations of them (like adjacency matrices). Then the question misses a subject. But if so?

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No. A major lesson of category theory is that in practical situations it is not enough to know that two things are isomorphic; one needs to keep track of how they are isomorphic because in practice one has to work with multiple such isomorphisms at the same time. (Hans, more generally might I suggest that you spend less time thinking about mathematics and more time doing mathematics? Then you would get to learn these lessons the hard way.) – Qiaochu Yuan Feb 17 '11 at 11:19
I also disagree with the premise of the question; higher category theorists care a lot about different notions of isomorphism. – Qiaochu Yuan Feb 17 '11 at 12:03
Honestly, I can but agree with Qiaochu... Doing math will make it evident to you why $V$ and $V^*$ are felt to be different. – Mariano Suárez-Alvarez Feb 17 '11 at 16:41
@Mariano: ... or at least, not "as equal" as $V$ and $V^{\ast\ast}$. To paraphrase George Orwell, all $n$-dimensional vector spaces are isomorphic, but some are more isomorphic than others. Hans: It wasn't a hint, it's a fact that becomes apparent with use. Trying to work with $V^\ast$ as if it were "exactly the same" as $V$ leads to a lot of problems (especially when dealing with inner products and representability of functionals), whereas this is simply not the case with $V$ and $V^{\ast\ast}$; the coordinate-free nature of the latter isomorphism is key to many important properties. – Arturo Magidin Feb 17 '11 at 16:43
@Hans (...cont) So it's not "just do the hard work", but rather "I know it's hard to see why we distinguish between these, but that's because you are standing a little too far off and you can't see the finer points. Come, stand a bit closer so you can take a closer look and see what we're pointing to" – Arturo Magidin Feb 17 '11 at 16:56
up vote 2 down vote accepted

In practice, God does not hand you abstract structures: you construct them from other structures, and the point of caring about isomorphisms is that you want to keep track of this construction process. For example, you are almost never handed a vector space $V$ and its dual $V^{\ast}$. Usually you are performing some construction (e.g. on the tangent space $T_p(M)$ of a manifold $M$ at a point $p$) which naturally involves elements of $T_p(M)$ as well as elements of the cotangent space $T_p(M)^{\ast}$, and if you are foolish enough to think that they are the same space then you will literally not be capable of doing calculations in this setting (e.g. changing coordinates).

One way to say this is that abstract structures often arise functorially, and even if $F(c), G(c)$ are isomorphic where $F, G : C \to D$ are functors and $c \in C$ is an object, the functors $F, G$ need not be naturally isomorphic, and usually we actually care about the functors, not the objects $F(c), G(c)$ in isolation. In the above example taking duals is a contravariant functor $\text{Vect} \to \text{Vect}$, and since it is contravariant it does not behave at all like the identity functor.

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Thanks, Qiaochu. The phrase "the point of caring about isomorphisms is that you want to keep track of this construction" is really enlightening. That gives me a lot of thinking. – Hans Stricker Feb 17 '11 at 16:55
@Hans: perhaps the following analogy might help you see what people have been trying to tell you. You are going up to a group of carpenters and asking them "why do you use so many different kinds of wood? It all comes from trees, right?" All they are trying to tell you in response is "maybe you should make some chairs and then get back to us." Certainly this is a much better way to learn something about carpentry than thinking about carpenting all the time. – Qiaochu Yuan Feb 17 '11 at 17:16

Let me draw your attention to the newborn founding endeavor of Voevodsky, which attempts, among other things, to capture exactly that isomorphic structures are indeed identical; this is supposed to be the content of his so-called "univalence axiom" (as explained by Awodey in relevant lectures).

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May I ask you to explain a bit more what Voevodsky's work entails, and how it relates to the question Hans posed? – Willie Wong Jul 25 '12 at 15:24
Sorry for the late response! I'm afraid I can't explain Voevodsky's work---I've hardly studied it myself. I just happen to know a couple of slogans like the above. For info of varying degree of depth, you might want to look at the "HoTT" site---ah, this western bent on clever acronyms... :-) --- – Basil Oct 1 '12 at 12:49

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