  In this article, the theory of hierarchical evolutive systems of Ehresmann
and Vandremeersch [Bull. Math. Bio. 49, 13-50 (1987)] is improved by
considering the categories of the theory as fuzzy sets whose elements are the
composite objects formed by the arrows and corresponding vertices of their
embedded graphs. This way each category can be represented as a point in the
states space [0.1]**N. The introduction of a diffeomorphism, that acts in this
context as a functor between categories, allows to define a measure preserving
dynamical system. In particular, we apply this formalism to describe a living
single cell. We propose for its state at a given time a hirerchical category
with three levels (molecular, coarse-grained and cellular levels) related by
adequate colimits. Each level involves the main functional and structural
modules in which the cell can be partitioned. The time evolution of the cell is
drived by a transformation which is a N-dimensional generalization of the
Ricker map whose parameters we propose to be determined by requiring that, as
hallmark of its behavior, the living cell to evolve at the edge of chaos. From
the dynamical point of view this property manifests in the fact that the
largest Lyapunov exponent is equal to zero. Since in a rather complete model of
the living cell the huge number of involved parameters can make of the
calculations a hard task, we also propose a toy model, with fewer parameters to
be determined, which emphasizes the cellular fission.
