This post started out as a response to a question in the comments of my last post on groupoids. Answering those questions, and thinking more about the answers while sitting on the train during my commute, I realized that I left out some important things that were clear to me from thinking about this stuff as I did the research to write the article, but which I never made clear in my explanations. I’ll try to remedy that with this post.

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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.

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Today’s entry is short, but sweet. I wanted to write something longer, but I’m very busy at work, so this is what you get. I think it’s worth posting despite its brevity.

When we look at groups, one of the problems that we can notice is that there are things
that seem to be symmetric, but which don’t work as groups. What that means is that despite the
claim that group theory defines symmetry, that’s not really entirely true. My favorite example of this is the fifteen puzzle.

The fifteen puzzle is a four-by-four grid filled with 15 tiles, numbered from 1 to 15, and one empty space. You can make a move in the puzzle by sliding a tile adjacent to the empty
space into the empty. In the puzzle, you scramble up the tiles, and then try to move them back so that they’re in numerical order. The puzzle, in its initial configuration, is shown to the right.

If you look at the 15 puzzle in terms of configurations – that is, assignments of the pieces to different positions in the grid – so that each member of the group describes a single tile-move in a configuration, you can see some very clear symmetries. For example, the moves that are possible when the empty is in any corner are equivalent to the moves that are possible when the empty is in any other corner. The possible moves when the space is in any given position are the same except for the labeling of the tiles around them. There’s definitely a kind of symmetry there. There are also loops – sequences of moves which end in exactly the same state as the one in which they began. Those are clearly symmetries.

But it’s not a group. In a group, the group operation most be total – given any pair of values x and y in the group, it must be possible to combine x and y via x+y. But with the 15 puzzle, there moves that can’t be combined with other moves. If x = “move the ‘3’ tile from square 2 to square 6”, and y = “move the ‘7’ tile from square 10 to square 11”, then there’s no meaningful value for “x+y”; the two moves can’t be combined.

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So far, I’ve spent some time talking about groups and what they mean. I’ve also given a
brief look at the structures that can be built by adding properties and operations to groups –
specifically rings and fields.

Now, I’m going to start over, looking at things using category theory. Today, I’ll start
with a very quick refresher on category theory, and then I’ll give you a category theoretic
presentation of group theory. I did a whole series of articles about category theory right after I moved GM/BM to ScienceBlogs; if you want to read more about category theory than this brief introduction, you can look at the category theory archives.

Like set theory, category theory is another one of those attempts to form a fundamental
abstraction with which you can build essentially any mathematical abstraction. But where sets
treat the idea of grouping things together as the fundamental abstraction, category
theory makes the idea of mappings between things as the fundamental abstraction.

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