given base point, but those that can define the groupoid el-
ement, a morphism g = (aj , ak) having the natural inverse
g-1 = (ak , aj). Given such a pairing, connection by a mean-
ingful path, it is possible to define ‘natural’ end-point maps
α(g) = aj , β(g) = ak from the set of morphisms G into A,
and a formally associative product in the groupoid g1g2 pro-
vided α(g1g2) = α(g1), β(g1g2) = β(g2), and β(g1) = α(g2).
Then the product is defined, and associative, i.e. (g1g2)g3 =
g1 (g2g3).
In addition there are natural left and right identity elements
λg, ρg such that λgg = g = gρg whose characterization is left
as an exercise (Weinstein, 1996).
An orbit of the groupoid G over A is an equivalence class
for the relation αj∙ ~ Gak if and only if there is a groupoid
element g with α(g) = aj and β(g) = ak.
The isotopy group of a ∈ X consists of those g in G with
α(g) = a = β(g).
In essence a groupoid is a category in which all morphisms
have an inverse, here defined in terms of connection by a
meaningful path of an information source dual to a cognitive
process.
If G is any groupoid over A, the map (α, β) : G → A × A is
a morphism from G to the pair groupoid of A. The image of
(α, β) is the orbit equivalence relation ~ G, and the functional
kernel is the union of the isotropy groups. If f : X → Y is a
function, then the kernel of f, ker(f) = [(x1, x2) ∈ X × X :
f(x1) = f(x2)] defines an equivalence relation.
As Weinstein (1996) points out, the morphism (α, β) sug-
gests another way of looking at groupoids. A groupoid over
A identifies not only which elements of A are equivalent to
one another (isomorphic), but it also parametizes the different
ways (isomorphisms) in which two elements can be equivalent,
i.e. all possible information sources dual to some cognitive
process. Given the information theoretic characterization of
cognition presented above, this produces a full modular cog-
nitive network in a highly natural manner.
Brown (1987) describes the fundamental structure as fol-
lows:
“A groupoid should be thought of as a group with
many objects, or with many identities... A groupoid
with one object is essentially just a group. So the no-
tion of groupoid is an extension of that of groups. It
gives an additional convenience, flexibility and range
of applications...
EXAMPLE 1. A disjoint union [of groups] G =
∪λGλ, λ ∈ Λ, is a groupoid: the product ab is defined
if and only if a, b belong to the same Gλ, and ab is
then just the product in the group Gλ. There is an
identity 1λ for each λ ∈ Λ. The maps α, β coincide
and map Gλ to λ, λ ∈ Λ.
EXAMPLE 2. An equivalence relation R on [a
set] X becomes a groupoid with α, β : R → X the
two projections, and product (x, y)(y, z) = (x, z)
whenever (x, y), (y, z ) ∈ R. There is an identity,
namely (x, x), for each x ∈ X...”
Weinstein (1996) makes the following fundamental point:
“Almost every interesting equivalence relation on
a space B arises in a natural way as the orbit equiv-
alence relation of some groupoid G over B . Instead
of dealing directly with the orbit space B/G as an
object in the category Smap of sets and mappings,
one should consider instead the groupoid G itself as
an object in the category Ghtp of groupoids and ho-
motopy classes of morphisms.”
Later we will explore homotopy in paths generated by in-
formation sources.
It is interesting to note that the equivalence class structure
which Asanovic et al. (2006) identify for problem types can
be recast in groupoid language, and may serve as a model
for projecting structure onto potential hyperparallel architec-
tures.
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; Stewart et al. (2003), Stewart (2004)). Here
we have outlined how to extend the technique to networks of
interacting information sources which, in a dual sense, char-
acterize cognitive processes, and cannot at all be described by
the usual differential equation models. These latter, it seems,
are much the spiritual offspring of 18th Century mechanical
clock models. Cognitive and conscious processes in humans
involve neither computers nor clocks, but remain constrained
by the limit theorems of information theory, and these permit
scientific inference on necessary conditions.
4. Internal forces breaking the symmetry groupoid
The symmetry groupoid, as we have constructed it for cog-
nitive modules, in a kind of information space, is parame-
tized across that space by the possible ways in which states
aj , ak can be equivalent, i.e. connected to some origin by a
meaningful path of an information source dual to a cognitive
process. These are different, and in this approximation, non-
interacting cognitive processes. But symmetry groupoids, like
symmetry groups, are made to be broken: by internal cross-
talk akin to spin-orbit interactions within a symmetric atom,
and by cross-talk with slower, external, information sources,
akin to putting a symmetric atom in a powerful magnetic or
electric field.
As to the first process, suppose that linkages can fleet-
ingly occur between the ordinarily disjoint cognitive mod-
ules defined by the network groupoid. In the spirit of Wal-
lace (2005a), this is represented by establishment of a non-
zero mutual information measure between them: a cross-talk
which breaks the strict groupoid symmetry developed above.
Wallace (2005a) describes this structure in terms of fixed
magnitude disjunctive strong ties which give the equivalence
class partitioning of modules, and nondisjunctive weak ties
which link modules across the partition, and parametizes the
overall structure by the average strength of the weak ties, to
use Granovetter’s (1973) term. By contrast the approach of
Wallace (2005b), which we outline here, is to simply look at
the average number of fixed-strength nondisjunctive links in
a random topology. These are obviously the two analytically