A groupoid is defined to be a (small)category where every morphism is an isomorphism. As we know every group can be viewed as as groupoid with a single object. I wonder if the converse of this is also true.
Let $\mathscr C$ be a groupoid containing at least two objects $A,B$. I have a feeling that such a groupoid can never have a group structure(no matter how morphisms in this category are built) since there will be at least two "identity-like" elements $1_A, 1_B$ making trouble in here.
Am I right? If so, how to prove this rigorously(note that rigorously speaking, we haven't even defined the terminology "viewing a category as a group", which is obvious in the case for the groupoid with one object)? If not, what a counterexample will serve the purpose?
The problem is that the multiplication is only partially defined: you can't multiply any two elements of a groupoid. If your groupoid has two objects $A$ and $B$ then you must have identity morphisms $1_A$ and $1_B$. The morphisms $1_A$ and $1_B$ cannot be composed, so the multiplication is not a group multiplication.