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Whats the meaning of "unique up to isomorphism" in the context of initial algebras.

It seems that up to means "to ignore" (sometimes said as "modulo"). Isomorphism means that the objects are the same in some way (with a bidirectional mapping). However, "unique ignoring that they are the same" still perplexes me.

Please try to keep your answer(s) as simple as possible. I'm a professional programmer, not a mathematician.

Steven Shaw
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    It means that any two objects that satisfy whatever it is being talked about, happen to be isomorphic e.g. given a field $F$ its algebraic closure is unique upto isomorphism, meaning if $K_1,K_2$ are algebraic closures of $F$ then $K_1$ and $K_2$ are isomorphic fields. – James May 08 '16 at 00:53
  • When mathematicians say "up to" they're generally referring to an equivalence relation in some way. In this case, the equivalence relation is isomorphism. "Up to X" certainly does not mean "ignore X"; it means X is an equivalence relation and you want to talk about its equivalence classes. – Qiaochu Yuan May 08 '16 at 00:55
  • it means for example that (the algebra of complex polynomials) $\mathbb{C}[X]$ is clearly the same as $\mathbb{C}[Y]$, even if $P(X) - P(Y) \ne 0$ when putting them in the same formula – reuns May 08 '16 at 01:07

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My algebra professor used to equate this expression with unique "up to a renaming of the elements;" which makes sense to me as it is unique in terms of all of the structures we care about staying the same, we just sent all the elements to some other analogous element with a different symbol.

(I tried to keep this nontechnical, as OP requested).

operatorerror
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