in a canonical fashion between two limits, defined by the min-
imum and maximum of f on M . Let the part of M below a
be defined as
Ma = f-1 (-∞, a] = {x ∈ M : f(x) ≤ a}.
These sets describe the whole manifold as a varies between
the minimum and maximum of f.
Morse functions are defined as a particular set of smooth
functions f : M → R as follows. Suppose a function f has
a critical point xc , so that the derivative df (xc) = 0, with
critical value f(xc). Then f is a Morse function if its critical
points are nondegenerate in the sense that the Hessian matrix
of second derivatives at xc , whose elements, in terms of local
coordinates are
Hi,j = ∂2f /∂xi∂xj ,
has rank n, which means that it has only nonzero eigen-
values, so that there are no lines or surfaces of critical points
and, ultimately, critical points are isolated.
The index of the critical point is the number of negative
eigenvalues of H at xc .
A level set f-1 (a) of f is called a critical level if a is a
critical value of f , that is, if there is at least one critical point
xc ∈ f-1(a).
Again following Pettini (2007), the essential results of
Morse theory are:
[1] If an interval [a, b] contains no critical values of f, then
the topology of f-1 [a, v] does not change for any v ∈ (a, b].
Importantly, the result is valid even if f is not a Morse func-
tion, but only a smooth function.
[2] If the interval [a, b] contains critical values, the topology
of f -1 [a, v] changes in a manner determined by the properties
of the matrix H at the critical points.
[3] If f : M → R is a Morse function, the set of all the
critical points of f is a discrete subset of M, i.e., critical
points are isolated. This is Sard’s Theorem.
[4] If f : M → R is a Morse function, with M compact, then
on a finite interval [a, b] ⊂ R, there is only a finite number of
critical points p of f such that f (p) ∈ [a, b]. The set of critical
values of f is a discrete set of R.
[5] For any differentiable manifold M , the set of Morse func-
tions on M is an open dense set in the set of real functions of
M of differentiability class r for 0 ≤ r ≤ ∞.
[6] Some topological invariants of M , that is, quantities that
are the same for all the manifolds that have the same topology
as M, can be estimated and sometimes computed exactly once
all the critical points of f are known: Let the Morse numbers
μi(i = 0,..., m) of a function f on M be the number of critical
points of f of index i, (the number of negative eigenvalues of
H). The Euler characteristic of the complicated manifold M
can be expressed as the alternating sum of the Morse numbers
of any Morse function on M ,
m
χ = X(-1)iμi.
i=1
The Euler characteristic reduces, in the case of a simple
polyhedron, to
χ=V -E+F
where V, E , and F are the numbers of vertices, edges, and
faces in the polyhedron.
[7] Another important theorem states that, if the interval
[a, b] contains a critical value of f with a single critical point
xc , then the topology of the set Mb defined above differs from
that of Ma in a way which is determined by the index, i, of
the critical point. Then Mb is homeomorphic to the manifold
obtained from attaching to Ma an i-handle, i.e., the direct
product of an i-disk and an (m - i)-disk.
Again, see Pettini (2007) or Matusmoto (2002) for details.
18.3 Generalized Onsager Theory
Understanding the time dynamics of groupoid-driven infor-
mation systems away from the kind of phase transition criti-
cal points described above requires a phenomenology similar
to the Onsager relations of nonequilibrium thermodynamics.
This also leads to a general theory involving large-scale topo-
logical changes in the sense of Morse theory.
If the Groupoid Free Energy of a biological process is
parametized by some vector of quantities K ≡ (K1 , ..., Km),
then, in analogy with nonequilibrium thermodynamics, gra-
dients in the Kj of the disorder, defined as
Sg ≡ Fg(K) - X Kj∂FG∣∂Kj
j=1
(22)
become of central interest.
Equation (22) is similar to the definition of entropy in terms
of the free energy of a physical system.
Pursuing the homology further, the generalized Onsager re-
lations defining temporal dynamics of systems having a GFE
become
dKj/dt = ^LjTidSG/dKi
i
(23)
where the Lj,i are, in first order, constants reflecting the
nature of the underlying cognitive phenomena. The L-matrix
is to be viewed empirically, in the same spirit as the slope and
intercept of a regression model, and may have structure far
21