Group Theory via Category Theory

I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for someone who already knows category theory but doesn't know any group theory. I have been given the liberty to decide how I build the theory. I already have a vague idea of what is to be done.

1. However, I would like to hear ideas about what should be done. So I solicit advice on things I should emphasize, the ways I can exploit the given familiarity with category theory for a more economic presentation and/or a exposition through "the path of least resistance". And please feel free to also mention any tips or precautions.
2. Please refer me to material along this line.

Thanks!

Solution 1:

Here's some food for thought (although its nowhere near enough material):

1. A groupoid is a category in which every morphism is an isomorphism (i.e. all morphisms are invertible). Groupoids are an oidification of the concept "group"; in other words, a group is just a one-object groupoid.

2. The inclusion $\mathbf{Grp} \hookrightarrow \mathbf{Grpd}$ does not preserve coproducts. Hence the coproduct of groups is part of what gives group-theory its own special flavour, distinct from the flavour of groupoid theory. (Perhaps use the wedge sum of nice topological spaces to motivate the group-theoretic coproduct in its own right, since the fundamental group of a wedge sum is the coproduct of the fundamental groups under certain general conditions.)

3. A group is just a connected groupoid, up to equivalence. Hence every (small) groupoid is the (groupoid-theoretic) coproduct of a set-indexed family of groups, up to equivalence. Looking at it another way, groups are kind of like the "atoms" from which we can build up any groupoid. (Note that the same relationship does not hold between monoids and categories.)

4. Let $\mathbf{C}$ denote a category. Then the isomorphisms of $\mathbf{C}$ form a groupoid called the core of $\mathbf{C}$, typically denoted $\mathrm{core}(\mathbf{C}).$ Now consider the automorphism function $\mathrm{Aut}_\mathbf{C} : \mathrm{Obj}(\mathbf{C}) \rightarrow \mathbf{Grp}$ that takes any object $X$ of $\mathbf{C}$ to its group of automorphisms $\mathrm{Aut}_\mathbf{C}(X).$ This cannot be regarded as a functor out of $\mathbf{C}$ in any sensible way. However, it can be regarded as a functor out of $\mathrm{core}(\mathbf{C})$.

5. The action of a group $G$ on a set (a "$G$-set") is precisely an object of the functor category $[G,\mathbf{Set}].$ More generally, the action of a group $G$ on a "foo" is just an object of $[G,\mathbf{Foo}]$, where $\mathbf{Foo}$ is the category of foos. This makes sense even if $G$ is allowed to be a general groupoid, or even a general category.

   This explains why every homomorphism $\varphi : G \rightarrow H$ of groups gives rise to a functor $[H,\mathbf{Set}] \rightarrow [G,\mathbf{Set}]$ "going the other way"; its because $\mathrm{Hom}$ is contravariant in its first argument.

6. We may consider group objects in any category with finite products (note that we don't even need finite limits; finite products will do). For example, the group objects in $\mathbf{Top}$ (the category of topological spaces) are precisely the topological groups.

7. Given a monoid $M$, every $M$-set $X$ is associated with a category $\tilde{X}$ called its "translation category," defined as follows. The object set of $\tilde{X}$ is just the underlying set of $X$. Given objects $x,y \in X$, we define that an arrow $x \rightarrow y$ is just a pair $(x,m)$ such that $m \in M$ and $mx = y$. Composition of arrows is by multiplication. Explicitly:

   $$(x,m_y)(y,m_z) = (x,m_z m_y).$$

   Furthermore, if $G$ is a group and $X$ is a $G$-set, then the translation category $\tilde{X}$ of $X$ is always a groupoid, called the translation groupoid of $X.$ Note also that every monoid acts on itself by left-multiplication. So given a monoid $M$, we may think of $M$ as an $M$-set, and hence we may write $\tilde{M}$ and speak of the translation category of $M$. As a special case, if $G$ is a group, we may write $\tilde{G}$ and speak of the translation groupoid of $G$.

8. For something a little more advanced, George Bergman has recently shown how to define the concepts of "inner automorphism" and "inner endomorphism" for objects of an arbitrary category. This could be an interesting viewpoint to take in your own report.

Solution 2:

You may want to look at Paolo Aluffi's book called Algebra Chapter zero. He begins talking about categories first and then introduces groups. He says a group is a groupoid (category in which every morphism is invertible) with only one object.

Solution 3:

On the [sub]topic of bibliography, there is a reasonable chapter (#4) in Steve Awodey's Category Theory book; the chapter introduces group theory with the help of category theory. Awodey is a philosophy prof at CMU, which probably explains why he put such a chapter in his book; the target audience isn't assumed have first learned group theory as the math undergrads normally do. A version of that chapter (probalby the one that appeared in the first edition of the book) is freely available on a CMU class webpage.