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Many people say that an object satisfying a property is unique up to isomorphism if every such object belongs to a unique isomorphism equivalence class. Is there a term for an object that is both unique up to isomorphism and also satisfies the converse: if an object is isomorphic to an object with this property, then this object also has the property?

For most of the objects we study in category theory, this if and only if holds. One can, however, create artificial examples where the converse does not hold.

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I'm not sure I understand the question: any time two objects are isomorphic, they must satisfy this property, whatever it is? And no matter what category they're in? – Eric Auld Aug 10 '13 at 16:30
"if any" is at best an ambiguous phrase and I wouldn't use it in this way. "if any such object belongs to..." could be construed to mean "if there is any such object that belongs to..." But I don't think that's what you meant. Simply changing it to "if every such object belongs to..." would be entirely unambiguous. – Michael Hardy Aug 10 '13 at 16:32
Thanks Michael for catching that. – Aaron McBride Aug 10 '13 at 16:34
This question is very unclear. "If an object is isomorphic to an object with this property, then this object also has the property" is not the converse of anything, but obvious. – Christian Blatter Aug 10 '13 at 19:54
Are you saying that if a property defines an object uniquely up to isomorphism, then this property is automatically invariant under isomorphism? – Aaron McBride Aug 10 '13 at 20:18
up vote 5 down vote accepted

"Invariant under isomorphism."

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Perfect! Thanks. – Aaron McBride Aug 10 '13 at 16:53
This is a term for "the converse" (answering the question in the title). It is not "a term for an object that is both unique up to isomorphism and also satisfies the converse"...but I am not aware of the existence of a single term for that. – Pete L. Clark Aug 10 '13 at 17:00
@PeteL.Clark, I think it's different in either direction. A property is invariant under isomorphism if and only if it is satisfied on a union of isomorphism classes, but the question is about a property holding on exactly one full isomorphism class. – zyx Aug 11 '13 at 19:17
Note that almost every sensible notion is invariant unter (higher) isomorphism. – Martin Brandenburg Aug 12 '13 at 11:55

There is the term "topological invariant": Any two topological spaces homeomorphic to this one specified space that have property $X$. That is expressed by saying $X$ is a topological invariant. One reason this term gets used is that things initially defined in terms of metric properties of a space unexpectedly turn out to be topological invariants. Thus the integral of Gaussian curvature with respect to area of a compact surface without boundary requires more than topological ideas for its definition, but every pair of such manifolds that are homeomorphic to each other have the same integral of Gaussian curvature with respect to area.

I don't know of any examples for other sorts of isomorphism that seem as good as things like that.

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Isn't a topological invariant the converse of what you said? Two spaces can have the same fundamental without being homeomorphic ($\Bbb R^2$ and $\Bbb R^3$) , but any two with different fundamental groups must not be homeomorphic. At least this is how I have heard topological invariant used. – PVAL Aug 10 '13 at 16:52
Sorry --- that was a typo. (When I wrote "Edward Snowden is a heretic who should be burned at the stake", that was a typo. What I actually meant was "Edward Snowden is a saint who should be venerated.") – Michael Hardy Aug 10 '13 at 16:56

The object is "unique up to arbitrary isomorphism". Which is to say that

The property "defines the object up to arbitrary isomorphisms". Which is to say that

The property defines [identifies, specifies, cuts out, assigns, etc] a unique isomorphism class of objects.

The last formulation seems like the most standard use of language, and makes it clear that isomorphism could be replaced by any equivalence relation.

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Thank you! Jeez, what an obvious word to use: "arbitrary". Boy, terminology can be a pain sometimes! – Aaron McBride Aug 12 '13 at 21:26

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