I finally started writing down some of my thoughts on the consequences of approaching group theory from the groups-as-groupoids perspective. The first interesting thing I noticed is that, since Grpd inherits its cartesian closed structure from Cat, we can define an exponentiation operator on groups (which returns groupoids that are not, in general, themselves groups).
Since Grp is a full subcategory of Grpd, this exponentiation pretty much has all the properties you’d expect it to have. The trivial group raised to any power is itself, and any group raised to the trivial-group-th power is itself. A natural transformation between group homomorphisms A,B: G -> H is an element of H which, acting by conjugation, interchanges the images of A,B (and natural transformations compose according to the group multiplication); we can use this fact to construct exponentials explicitly. And we can use the definition of exponentiation as right adjoint to the product to interpret them in terms of group homomorphisms.
Observation: Consider the simplest nontrivial group exponential, X = (C_2)^(C_2). This should be a groupoid such that groupoid homomorphisms G -> X are in bijection with group homomorphisms G × (C_2) -> (C_2). Since (C_2) has just two endomorphisms, and since neither of its inner automorphisms do anything (just like in any abelian group), we can see that X is two disjoint copies of (C_2), i.e. X = (C_2) + (C_2), where “+” denotes the coproduct in Grpd. A homomorphism G -> X thus consists of a single bit of information that determines which copy of (C_2) is in its image, plus an arbitrary homomorphism G -> (C_2). We can see that any homomorphism f: G × (C_2) -> (C_2) is determined by the same data: the bit is b = f(e,1), where “e” is the identity of G, and the homomorphism f`: G -> (C_2) is f` = f(g,0).
Observation: Since any abelian group has no nontrivial inner automorphisms, A^G for any abelian group A consists of a disjoint copy of A for each homomorphism G -> A. The above observation is an example of this, and similarly we can see that, for example, T^(C_n) = n*T, where T is the circle group and n*T is the coproduct of n copies of T.
Observation: For any group G, the groupoid G^G has a connected component for each outer automorphism of G.
(this research is supported by my patrons)
