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I was browsing through the questions and read one about whether defining a group as $G$ a set with certain features instead of an ordered pair $\langle G, \circ \rangle$, was abuse of language.

Someone mentioned that one could also define a group as an object of the Category of Groups. My question is: is that all one needs to say? are the group axioms implied from the category?

I haven't taken Category Theory so I apologize if the question is "stupid".

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    $\begingroup$ AFAIK, the "category of groups" is just the category the objects of which are those pairs and the morphisms of which are the usual functions. $\endgroup$
    – user228113
    Commented Feb 8, 2017 at 6:11
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    $\begingroup$ If you're talking to someone who have never in their life seen a dog, you could try to explain what a dog is... Or you could just say that it is a member of the species "dog". That certainly explains everything. Your question is not stupid, but that "definition" is. $\endgroup$
    – Arthur
    Commented Feb 8, 2017 at 6:27
  • $\begingroup$ An aside: One can define a semigroup as a category with one object. (Semigroup elements are morphisms, multiplication is the composition of morphisms.) Of course, I would not advise to use this as a definition in an undergraduate abstract algebra class. $\endgroup$
    – Moishe Kohan
    Commented Feb 8, 2017 at 11:21

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No, that definition would not work for, say, a set theorist; what might work, though, is to define a group as a category with one object, all of whose morphisms are iso.

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