5 Oct
2016
5 Oct
'16
8:33 p.m.
Dear Michael, Consider (Z×Z×Z)-modules A = Z×Z×1, B = Z×1×Z where Z denotes the ring of integers. The full subcategory {A, B} is a (counter)example. Indeed, all the homsets Hom(A,A)=Z×Z×1, Hom(A,B)=Z×1×1, Hom(B,A)=Z×1×1, and Hom(B,B)=Z×1×Z are countable. There is no pair (f: A->B, g: B->A) s.t. g f = id. -- Toshiki Kataoka 2016-10-05 3:21 GMT+09:00 Michael Barr <barr@math.mcgill.ca>:
We all know that if Hom(A,-) is naturally equivalent to Hom(B,-), then A is isomorphic to B. But can you find a category in which for each object C, Hom(A,C) is isomorphic to Hom(B,C) but no naturality of the isomorphism without A being isomorphic to B?
Michael
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