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.
I sent an email to Mark CC, which he might not have read, as it was titled "Paper on Peristalsis" and related to my current computational biomathematical research, which email included the PDF and other links to a smashingly wonderful paper on Groupoid applications, and a few of my groupoid sequences on the OEIS...
Ian Stewart and Martin Golubitsky
NONLINEAR DYNAMICS OF NETWORKS: THE GROUPOID FORMALISM
AMERICAN MATHEMATICAL SOCIETY, 2006
May 3, 2006
A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos.
Symmetry is a rather restrictive assumption, and a general theory of
networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the 'input sets'. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of 'robust' patterns of synchrony in terms of the combinatorial structure of the network.
Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables--the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology--which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.
=== end abstract ===