6 Oct
2016
6 Oct
'16
4:57 a.m.
Without naturality we are just asking that Hom(A,C) and Hom(B,C) have the same cardinality, so this is not a very strong restriction. For example the category of non zero finite dimensional vector spaces over an infinite field K satisfies that for any four objects W,X,Y,Z, Hom(W,X) is in (non natural) bijection with Hom(Y,Z) simply because they are both infinite sets of the cardinality of K.
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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