an object in the category Ghtp of groupoids and ho-
motopy classes of morphisms.
The groupoid approach has become quite popular in the
study of networks of coupled dynamical systems which can
be defined by differential equation models, (e.g., Golubitsky
and Stewart 2006).
18.1.2 Global and local symmetry groupoids
Here we follow Weinstein (1996) fairly closely, using his ex-
ample of a finite tiling.
Consider a tiling of the euclidean plane R2 by identical 2 by
1 rectangles, specified by the set X (one dimensional) where
the grout between tiles is X = H ∪ V , having H = R × Z and
V = 2Z × R, where R is the set of real numbers and Z the
integers. Call each connected component of R2\X, that is, the
complement of the two dimensional real plane intersecting X ,
a tile.
Let Γ be the group of those rigid motions of R2 which leave
X invariant, i.e., the normal subgroup of translations by ele-
ments of the lattice Λ = H ∩ V = 2Z × Z (corresponding to
corner points of the tiles), together with reflections through
each of the points 1∕2Λ = Z × 1/2Z, and across the horizontal
and vertical lines through those points. As noted by Weinstein
(1996), much is lost in this coarse-graining, in particular the
same symmetry group would arise if we replaced X entirely
by the lattice Λ of corner points. Γ retains no information
about the local structure of the tiled plane. In the case of
a real tiling, restricted to the finite set B = [0, 2m] × [0, n]
the symmetry group shrinks drastically: The subgroup leav-
ing X ∩ B invariant contains just four elements even though
a repetitive pattern is clearly visible. A two-stage groupoid
approach recovers the lost structure.
We define the transformation groupoid of the action of Γ
on R2 to be the set
G(Γ, R2) = {(x, γ, y|x ∈ R2,y ∈ R2,γ ∈ Γ,x = γy},
with the partially defined binary operation
(x, γ, y)(y, ν, z) = (x, γν, z).
Here α(x, γ, y) = x, and β(x, γ, y) = y, and the inverses are
natural.
We can form the restriction of G to B (or any other subset
of R2) by defining
G(Γ,R2)∣b = {g ∈ G(Γ,R2)∣α(g),β(g) ∈ B}
[1]. An orbit of the groupoid G over B is an equivalence
class for the relation
x ^G y if and only if there is a groupoid element g with
α(g) = x and β(g) = y.
Two points are in the same orbit if they are similarly placed
within their tiles or within the grout pattern.
[2]. The isotropy group of x ∈ B consists of those g in G
with α(g) = x = β(g). It is trivial for every point except
those in 1∕2Λ ∩ B, for which it is Z2 × Z2 , the direct product
of integers modulo two with itself.
By contrast, embedding the tiled structure within a larger
context permits definition of a much richer structure, i.e., the
identification of local symmetries.
We construct a second groupoid as follows. Consider the
plane R2 as being decomposed as the disjoint union of P1 =
B ∩ X (the grout), P2 = B\P1 (the complement of P1 in B,
which is the tiles), and P3 = R2\B (the exterior of the tiled
room). Let E be the group of all euclidean motions of the
plane, and define the local symmetry groupoid Gloc as the set
of triples (x, γ, y) in B × E × B for which x = γy , and for
which y has a neighborhood U in R2 such that γ(U ∩Pi) ⊆ Pi
for i = 1, 2, 3. The composition is given by the same formula
as for G(Γ, R2 ).
For this groupoid-in-context there are only a finite number
of orbits:
O1 = interior points of the tiles.
O2 = interior edges of the tiles.
O3 = interior crossing points of the grout.
O4 = exterior boundary edge points of the tile grout.
O5 = boundary ‘T’ points.
O6 = boundary corner points.
The isotropy group structure is, however, now very rich
indeed:
The isotropy group of a point in O1 is now isomorphic to
the entire rotation group O2 .
It is Z2 × Z2 for O2 .
For O3 it is the eight-element dihedral group D4 .
For O4, O5 and O6 it is simply Z2 .
These are the ‘local symmetries’ of the tile-in-context.
18.2 Morse Theory
Morse theory examines relations between analytic behavior of
a function - the location and character of its critical points
- and the underlying topology of the manifold on which the
function is defined. We are interested in a number of such
functions, for example information source uncertainty on a
parameter space and ‘second order’ iterations involving pa-
rameter manifolds determining critical behavior, for example
sudden onset of a giant component in the mean number model
(Wallace and Wallace, 2008), and universality class tuning in
the mean field model of the next section. These can be re-
formulated from a Morse theory perspective. Here we follow
closely the elegant treatments of Pettini (2007) and Kastner
(2006).
The essential idea of Morse theory is to examine an n-
dimensional manifold M as decomposed into level sets of some
function f : M → R where R is the set of real numbers. The
a-level set of f is defined as
f-1 (a) = {x ∈ M : f(x) = a},
the set of all points in M with f(x) = a. If M is compact,
then the whole manifold can be decomposed into such slices
20