What's the significance of defining group as a group object in category $\mathcal{Set}$?

Question

At first sight, redefining group as a group object in the category of sets $\mathcal{Set}$ seems just like a meaningless restatement, but when we apply this definition to other categories, interesting things happens. For example, abelian group is the group object in the category of groups $\mathcal{Grp}$, and free group is the cogroup object in $\mathcal{Grp}$, etc..

Any reason this happens? It just seems too good to be an accident...

Answer 1

As you noted redefining groups in categorical terms as a group object in $\mathbf{Set}$ is nothing but just rephrasing the classical definition of groups in terms of morphism... so basically is not redefining since the two definition are the same.

Nonetheless groups objects are a generalization of groups (or should we say $\mathbf{Set}$-groups) to groups in other product categories. The reason to study them in this way is that we can generalize many proof of facts about $\mathbf{Set}$-groups to groups in arbitrary product categories in a straightforward and automatic (and of course categorical) way.

Note: that restating the definition of groups in $\mathbf{Set}$ as group objects in $\mathbf{Set}$ is to prove that the notion of internal group is really a generalization of the notion of group.

About the nice proprieties that arise when we consider groups in $\mathbf {Grp}$ those properties arise as consequences of the property $\mathbf{Grp}$. If you really like to see why ... well the group axioms imply that the moltiplication homomorphism is in fact the multiplication of the group and the fact that multiplication is an homomorphism implies that it's a commutative operation, but that's a fact that depends strictly by the properties of groups, I'm not aware if there's a generalization of this property, although I suspect this property could be deduced in some way by the Eckemann-Hilton.

Edit: internal groups are just one example of the bigger pehnomena: internalization. The idea of internalization is to generalize in some way a construction usually carried out in some category to other categories of a given type. The reason to do so are various and may depend on the object of study.

Anyway I hope this not so long answer address your doubts.

Answer 2

The significance of the title question is that we can apply the group object definition to other categories, and it gives us a systematic way to define groups in other categories rather than having ad-hoc definitions (e.g. "continuous group") for every situation of interest.