fitting empirically-determined statistical models to real data,
in precisely the sense that one would fit the usual parametric
statistical models to normally distributed data.
The fitting of statistical models does not, of itself, perform
scientific inference. That is done by comparing fitted mod-
els for similar systems under different, or different systems
under similar, conditions, and by examining the structure of
residuals.
One implication of this work, then, is that understanding
complicated processes of gene expression and development -
and their pathologies - will require construction of data analy-
sis tools considerably more sophisticated than now available,
including the present crop of simple models abducted from
neural network studies or stochastic chemical reaction theory.
Most centrally, however, currently popular (and fundable) re-
ductionist approaches to understanding gene expression must
eventually exhaust themselves in the same desert of sand-
grain hyperparticularity that appears to have driven James
Crick from molecular biology into consciousness studies, a
field now mature enough to provide tools for use in the other
direction.
18 Mathematical appendix
18.1 Groupoids
18.1.1 Basic ideas
Following Weinstein (1996) closely, a groupoid, G, is defined
by a base set A upon which some mapping - a morphism - can
be defined. Note that not all possible pairs of states (aj, ak)
in the base set A can be connected by such a morphism.
Those that can define the groupoid element, a morphism
g = (aj, ak) having the natural inverse g-1 = (ak, aj). Given
such a pairing, 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 g1 g2 provided
α(g1g2) = α(g1), β(g1g2) = β(g2), and β(g1) = α(g2). Then
the product is defined, and associative, (g1g2)g3 = g1 (g2 g3).
In addition, there are natural left and right identity ele-
ments λg, ρg such that λgg = g = gρg (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 . Following Cannas da
Silva and Weinstein (1999), we note that a groupoid is called
transitive if it has just one orbit. The transitive groupoids
are the building blocks of groupoids in that there is a natural
decomposition of the base space of a general groupoid into
orbits. Over each orbit there is a transitive groupoid, and
the disjoint union of these transitive groupoids is the original
groupoid. Conversely, the disjoint union of groupoids is itself
a groupoid.
The isotropy group of a ∈ X consists of those g in G with
α(g) = a = β(g). These groups prove fundamental to classi-
fying groupoids.
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.
Groupoids may have additional structure. As Weinstein
(1996) explains, a groupoid G is a topological groupoid over a
base space X if G and X are topological spaces and α, β and
multiplication are continuous maps. A criticism sometimes
applied to groupoid theory is that their classification up to
isomorphism is nothing other than the classification of equiv-
alence relations via the orbit equivalence relation and groups
via the isotropy groups. The imposition of a compatible topo-
logical structure produces a nontrivial interaction between the
two structures. Below we will introduce a metric structure on
manifolds of related information sources, producing such in-
teraction.
In essence, a groupoid is a category in which all morphisms
have an inverse, here defined in terms of connection to a base
point by a meaningful path of an information source dual to
a cognitive process.
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
19