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Mathematics has many different notions of equivalence and, correspondingly, many different notions of uniqueness. Sometimes, it is sufficient to show that two things are equivalent (resp. something is unique) up to isomorphism in a given category: for instance, we usually speak of "the" symmetric group $S_4$ or "the" complete graph $K_7$ or what-have-you. On the other hand, sometimes we find it desirable that two things are equivalent (resp. something is unique) up to unique isomorphism, for instance when we speak of the real numbers $\mathbb{R}$.

My question is: in what sorts of contexts do we need uniqueness up to unique isomorphism, and in what sorts of contexts does uniqueness up to isomorphism suffice? Is there a formal principle we can appeal to before hand to know if we will need one or the other in a certain proof? Are there ever times when we in fact need some stronger notion of uniqueness short of actual equality? What might it be?

M. Sperling
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I'd say that "up to isomorphism" and "up to unique isomorphism" play a slightly different role. One slogan is that isomorphic objects ought to be indiscernable within that category. For instance, any purely group-theoretic property satisfied by a group $G$ should be satisfied by any group isomorphic to $G$. If a property fails this, it is not appropriate to call it a group-theoretic property. In this sense, "up to isomorphism" is the appropriate notion of sameness in a given context (to be precise, "in a given context" means "in a given category" already). In fact, any appropriate notion of sameness ought to be a notion of isomorphism in some category. If this fails in your context, you might not be considering your objects in the correct ambient category (for me, this is also a guiding principle in the design of new categorical structures: for instance, higher categories allow us to see weaker notions like weak homotopy equivalence as a notion of isomorphism in a more convenient way that homotopy $1$-categories).

However, you generally do not want to identify isomorphic objects: you do not want to pretend they are "literally" equal, unless you have some single preferred isomorphism between them. The reason is that, in any proof that isomorphic objects have the same properties, you use a particular choice of isomorphism to "translate" from one object to the other. You have to use the inverse of this isomorphism when translating back, because you will introduce nontrivial automorphisms otherwise. So at least, you have to be consistent in your choice of isomorphism. But once I want to identify more than two objects, I have to make sure my choices are still consistent once I start composing isomorphisms to form new ones. Without some very clear notion of preferred isomorphism between objects, this is essentially not possible. This means that you do not want to pretend that isomorphic objects are actually equal (whatever that means), but you can do that when your objects are unique up to unique isomorphism in a way. In other situations, like in $\pi_1(S^1)\cong\mathbb{Z}$, we sort of make a choice of isomorphism at the beginning of the universe and always work with this one. The choice in this case does not matter in this case since it is about whether positive rotations are in clockwise or counterclockwise direction, so the only thing that matters is that you stick to your choice once you make it. (You sometimes can get away with identifyng two isomorphic objects if you stick to your choice of isomorphism, but you have to be careful, especially if there are "genuinely different" isomorphisms without a preferred choice, for instance between a finite-dimensional vector space and its dual: in this case, the choice of an isomorphism is the choice of an inner product, and this is not something you want to fix for the rest of time on your vector spaces.)

Maybe this answer of mine here is also an interesting read for you.

Daniël Apol
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