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I am starting to learn category theory. Because an isomorphism between groups A and B is just a function that is simultaneously a homomorphism from A to B and a homomorphism from B to A, it seems like that two groups being isomorphic doesn't mean that they are actually the same object in Grp. In general, it seems like having a morphism from A to B and a morphism from B to A is the same as having the natural equivalence in that category. So when we talk about Grp, are we talking about the collection of groups up to isomorphism, or are we using a different fundamental notion of "group"?

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    $\begingroup$ The objects of Grp, as typically defined, are groups, full stop (as in sets together with binary operations satisfying blah). There are other categories you might write down which are equivalent to Grp, and some of them are skeletons of Grp in the sense that they contain exactly one object for each isomorphism class of groups; in those categories you can think of an object as being an isomorphism class of groups if you want. Everything in category theory is invariant under equivalence of categories so in some sense it doesn't matter which of these categories you use. $\endgroup$ – Qiaochu Yuan Dec 3 '16 at 6:58
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    $\begingroup$ An isomorphism is not "just a function that is simultaneously a homomorphism from A to B and a homomorphism from B to A". And I'm not sure in what sense you mean by "having a morphism from A to B and a morphism from B to A is the same as having the natural equivalence in that category"; what categories is this somehow a natural equivalence between? $\endgroup$ – Malice Vidrine Dec 3 '16 at 7:04
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The definition I know for Grp has as objects groups and as morphisms homomorphisms of groups. This means two distinct but isomorphic groups are considered as two distinct objects.

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