Is there such thing as a functor from the Category of all Groups to that of all binary operations?

Question

I know that it is possible to define "forgetful" functors that maps the Category of Groups in that of Sets. Is it possibile to do the the same with binary operations? In other words, is it possible to "forget" the underlying set of a group?

Answer 1

I'm not sure if a forgetful functor is the correct way to approach the problem of studying groups without regard for their underlying sets (but maybe such a one does exist in the context of what I'm about to suggest). Instead, a natural way to study groups as a collection of operations instead of as sets with extra structure is using the concept of an algebra over a Lawvere theory. You can look here for an introductory survey. You could also look into algebraic theories in general as a solution to the problem of encoding the binary, and in general $n$-ary, operations of familiar algebraic objects without concern for their underlying sets.