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.
So - in the last post I explained a bit about the categorical viewpoint on why
we should care about groupoids. Every groupoid contains groups. The groups
capture the symmetries that exist within the groupoid. So why should we care about
The symmetries of the groups that exist within the groupoid are built using the
groupoid operator. That should be pretty obvious. But the groupoid operator is more
than just a stepping stone to the symmetries. The groupoid also relates the symmetries of the group - and that's one of the really important things about groupoids. They don't just define the symmetries, but they show you how they're built and how they're related.
Look back at the 15 puzzle for a moment. Given a configuration, there's a group consisting of the set of sequences of moves that both start and end at that configuration. The immunity to transformation is a kind of permutation symmetry - you're performing actions that shuffle things around, and end up with something that's indistinguishable from what you started with.
If you look more at the 15 puzzle, you can see another interesting feature. There's a group for each possible configuration of the puzzle. But if you look at that collection of groups, many of them are isomorphic. For example, the groups for every configuration with a corner square unoccupied are isomorphic. That's an interesting property - and an important one for really understanding the nature of the symmetries of the puzzle.
If you look at it from a different perspective, you can find configurations which can be transformed into other configurations - non-symmetric transformations, formed by the same steps that are used to build the symmetric transformations, and which also define the relationships between the configurations that are the roots of symmetries. You can
also find particular configurations which cannot be transformed into
specific other configurations - once again, defined by the nature of the elements of
the symmetries of the puzzle. For example, if you have the tiles 1 through 13 in their
"home" positions, and 14 and 15 swapped, you can't define any sequence
of steps that will end with every tile in its "home" position.
Those things are part of the structure of the puzzle, and part of the structure that defines the symmetries. But the groups themselves don't talk about it. In terms of the categorical formulation, they're part of the information that is lost in the non-natural transformation from the groupoid to the set of groups.
Some readers were confused that the interesting groupoid properties were related to the
discrete nature of the 15 puzzle. That's my fault. My work revolves pretty much entirely around discrete math. Unless I make a specific effort to think about continuous math, I tend to automatically think about things in discrete terms - so the examples that come to mind for me tend to be discrete examples. But the importance of groupoids doesn't just apply to continuous math.
In fact, the specific example of the groupoid over the 15 puzzle has a very nice
continuous counterpart, in the land of topological spaces. In a topological space, you can
define something called the fundamental groups of a topological space based on the closed loops from a point in the space to itself. When we examine those groups, we find that in a topological space, the set of groups for closed loops on points in the space reduces to a small number of isomorphic groups. Those groups are close counterparts of the configuration groups of the 15 puzzle.
The full structure of the topological space isn't captured by the groups on that space. The essential symmetries of the space are captured by the groups - but there's more to the structure of those symmetries and the relationships between them than you can get from just
the collection of groups. The groupoid defines the symmetries of the groups, and also the
structure of, and relationships between, those symmetries. If you're interested, I wrote about the fundamental groupoids of topological spaces, and some of their interesting properties here back when I was writing about topology.