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I've cleaned up the comment thread of Are there two non-isomorphic modules such that all the Hom-sets are isomorphic?. Here's a snapshot:
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This follows from Yoneda lemma if you assume this isomorphisms are natural. – Piotr Pstrągowski 21 hours ago
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Among the worst titles in my memory. – Igor Rivin 20 hours ago
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And if you do not assume the isomorphisms between Hom-sets to be natural, then for example over a field the question boils down to whether it is possible for two non-isomorphic vector spaces to have isomorphic duals. Over the field with two elements this is simply a question about the cardinality of power sets, which might very well be independant of ZFC. – Piotr Pstrągowski 20 hours ago
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@Igor: I guess it would be nice to edit the title and then inform the OP, so that next time he poses a question he would be more precise. – Chandrasekhar 12 hours ago
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Flagged Igor's comment as offensive and hate speech. – Guntram 11 hours ago
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@Guntram: You keep using that word. I do not think it means what you think it means. – Harry Gindi 9 hours ago
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I suggest "Are modules isomorphic if their Hom-sets are all isomorphic?" (or something like that). – Mark Grant 5 hours ago
@Piotr: could you please explain the "boils down" a bit further? The implication "isomorphic duals" ⟹ "all hom spaces isomorphic" seems to require some implication of the sort 2κ=2λ⟹ακ=αλ for all cardinals α. Is this true? – a-fortiori 5 hours ago
For Igor's comment to make sense, it should be noticed that the original title of the question was "If the homomorphisms are their isomorphic then they are isomorphic".
Oops, my attention span needs some work.
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