In my introduction to groupoids, I mentioned that if you have a groupoid, you can find
groups within it. Given a groupoid in categorical form, if you take any object in the
groupoid, and collect up the paths through morphisms from that object back to itself, then
that collection will form a group. Today, I’m going to explore a bit more of the relationship
between groupoids and groups.
Before I get into it, I’d like to do two things. First, a mea culpa: this stuff is out on the edge of what I really understand. My category-theory-foo isn’t great, and I’m definitely
on thin ice here. I think that I’ve worked things out enough to get this right, but I’m
not sure. So category-savvy commenters, please let me know if you see any major problems, and I’ll do my best to fix them quickly; other folks, be warned that I might have blown some of the details.
Second, I’d like to point you at Wikipedia’s page on groupoids as a
reference. That article is quite good. I often look at the articles in Wikipedia and
MathWorld when I’m writing posts, and while wikipedia’s articles are rarely bad, they’re also
often not particularly good. That is, they cover the material, but often in a
somewhat disorganized, hard-to-follow fashion. In the case of groupoids, I think Wikipedia’s
article is the best general explanation of groupoids that I’ve seen – better than most
textbooks, and better than any other web-source that I’ve found. So if you’re interested in
finding out more than I’m going to write about here, that’s a good starting point.
As I’ve explained before, the basic idea of a group is a mathematical structure with an
operation that defines a kind of immunity to transformation. Every transformation through the
group operator preserves some kind of structure. In a groupoid, you don’t have such a strong
symmetry. In a groupoid, you have an operation that includes a symmetric
transformation, but not all applications of it are symmetric: the groupoid operator preserves
structure under some conditions: that is, there are sequences of groupoid
operations that result in symmetric, structure-preserving transformation, and it’s easy to
define just what those sequences of transformations are.
In categorical terms, that idea of the partially symmetric operation of a
groupoid is reflected by the fact that there are many arrows that go from an object (a
structure) to a different object – transforming it into something different –
something distinguishable from the original structure. But there are sequences of operations
that end back at the original object, and those sequences of operations describe symmetric
This leads to the intuition behind the simplest relation between groupoids and groups. If
you have a groupoid with a single object in it, that that groupoid is a group. In a
single-object groupoid, every application of the operation must end back at the
object where it began – so every transformation in a single-object groupoid must be symmetric
– so the single-object groupoid is a group.
That much already shows us some of why the categorical formulation is nice: we’ve gone from something where the relationships and structures are symbolic, to something where we can see what’s going on using a simple diagram: it’s easier to see “all paths from G end at G”
when you draw the structures, than it is to see the corresponding statement in terms of pure algebra.
But there’s more to it. Some of the more subtle things are captured very nicely. For
example, there’s a notion of equivalence in groupoids that is different from isomorphism, which tells us more about the relationship between groupoids and groups. It comes from the categorical idea of a natural transformation. I wrote about
natural transformations here. For a quick reminder, if you start with
a category, you can define structure-preserving transformations between objects using
morphisms; you can define structure-preserving transformations between morphisms using functors; and you can define structure-preserving transformations between functors using
Let’s look at that notion of loose equivalence. Suppose we have a groupoid, O, which is
connected – that is, where in the category for G, given any two objects a and b in O, there
is an arrow from a to b. We can form a group G out of the groupoid, by collapsing
the connected groupoid to a single object. Loosely speaking, you can pick an arbitrary object from the groupoid as a representative, and collapse the arrows – each set of arrows
from the representative to a particular other object becomes an arrow in the collapsed groupoid; those arrows become the objects of the group, and arrow composition is the group operator. The end result is a group formed from the collapsed arrows of any connected groupoid.
What if the groupoid isn’t connected? You take the connected components of the groupoid, and each of them can be collapsed into a group – so you can collapse any groupoid into a collection of disjoint groups.
Here’s where it gets interesting. The collapse of a groupoid isn’t unique. It’s determined by the selection of the representative objects. The collapsed groupoids are all
equivalent. In fact, you can define an isomorphism between them – they’re all ultimately the
same structure; so you can define the isomorphic mappings between objects in the different
But suppose you don’t define the mappings between objects. Suppose all you look at are
the arrows that form the groups. So you’ve discarded the objects from the original groupoid,
and all you have left are single objects corresponding to the different connected components, each of which is a group. What you’ve done, then, is to create a mapping from a groupoid to a collection of groups. It is not an isomorphism – but it is a mapping from the groupoid to a collection of groups that is, in some sense, equivalent to the original groupoid. (Note that I said “collection”, not set. In fact, it’s a multiset; multiple components can collapse down to the same group, but you need to keep an instance of that group for each of the components.) The structure of that collection of groups is determined by the original groupoid.
But it’s not an isomorphism. That transformation has discarded some information about the original structure. There are multiple groupoids that can all be collapsed into the same
collection of groups. Because you didn’t keep the mappings that allow you to identify the arrows to the original groupoid objects, you can’t go back. It’s a one-way transformation.
Category theory allows you to define that loss of information in a very convenient way. Any high-level transformation like that – where you’re basically transforming a category into a different category (in this case, from a groupoid category to a category containing a collection of groups) can be described by categorical transformations. The structure-preserving relations between arrows in the original category – the functors of the original category – in a categorical sense define the structural information contained in the category. When you map from category to category, there
is a mapping between the functors that define the informational relationships between
the two categories. If you can map from one category C to another category D without losing any structural information, that is reflected in category theory by the existence of
a natural transformation from C to D. In the case of the groupoid-to-groups collapse, there is no natural transformation. The non-existence of a natural transformation
means that there is going to be a significant loss of structural information.
What does that mean?
It means that groupoids are richer structures that groups. There is interesting
information about symmetry which we can capture in terms of the algebraic structure of
a groupoid which we cannot capture in terms of the groups that make up the components of a groupoid. Without understanding the groupoids, we are missing something about
the meaning of symmetry!