Question

If one views a group as a one object category with the elements of the group as morphisms then a natural transformation between functors of such categories is an inner automorphism, i.e. if we have two group homomorphisms $f,g: A\to B$ then a natural transformation $\eta :f\to g$ is just an element $b\in B$ such that $f(a)\cdot b = b \cdot g(a)$ which can be rewritten as $f(a)=b \cdot g(a)\cdot b^{-1}$. This isn't the only way to turn groups into categories. Another way is to take the elements of the group as objects and to have a morphism $h_a:a\to b$ if $h\cdot a=b$. If we view groups in this way then are the natural transformations again something nice like inner automorphisms?

Answer 1

The comments thread is getting a bit long, so here's an answer. The category $C(G)$ that David associates to a group $G$ (by his second recipe) has the elements of $G$ as its objects, and exactly one morphism between any given pair of objects. It's what category theorists call an indiscrete or codiscrete category, and graph theorists call a complete graph or clique. You can form the indiscrete category on any set: it doesn't need a group structure.

A functor from one indiscrete category to another is simply a function between their sets of underlying objects. In particular, given groups $G$ and $H$, a functor from $C(G)$ to $C(H)$ is simply a function from $G$ to $H$. That's any function (map of sets) whatsoever -- it completely ignores the group structure.

Given indiscrete categories $C$ and $D$ and functors $P, Q: C \to D$, there is always exactly one natural transformation from $P$ to $Q$. In particular, given groups $G$ and $H$ and functors $P, Q: C(G) \to C(H)$, there is always exactly one natural transformation from $P$ to $Q$.

Answer 2

I like the notation $\mathcal{B}G$ and $\mathcal{E}G$ for the two constructions of a category out of a group $G$ in David's question. $\mathcal{E}G$ is what Tom calls the codiscrete category $C(G)$.

Of course there is a third construction: it has $G$ as the objects, and only identity morphisms. Let's denote this category again by $G$.

The notation is nice because you can take the nerve of any category $\mathcal{C}$, and then geometrically realize. If we denote the resulting space by $|\mathcal{C}|$,

1. $|\mathcal{B}G|$ is a classifying space for $G$.
2. $|\mathcal{E}G|$ is a universal principal $G$-bundle
3. $|G|$ is just $G$.