 Memory Evolutive Systems:
 Categorical modelisation of hierarchical, evolving and selfregulating systems (≃ 30 years of research already!).
A. Ehresmann  Mathematician (Université de Picardie):


J.P. Vanbremeersch  Physician (Université de Picardie): specialty in geriatry 
Flexibility
Adaptability
Structures preserved by morphisms$\qquad \overset{\text{+ associativity, identity}}{\rightsquigarrow} \qquad \underbrace{\text{Category Theory}}_{\text{focus on relations rather than objects}}$
Examples:
Objects preserved by morphisms:
Structures forming one category:
Terminal objects
Products
Equalizers (Kernels), Pullbacks, …
Initial objects
Coproducts
Coequalizers (Quotients), Pushouts, …
⟹ Colimits are more than sums
Hierarchical category $H_t$: layers of complexity: objects at level $n$ are colimits of patterns of level $<n$
Multiplicity Principle: for each time and each level, there is at least one multifaceted component, i.e. two patterns with the same colimit that are not isomorphic in the category of patterns and clusters.
“Standard changes” of living systems (cf. René Thom): birth: (addition of new components), death (addition of certain components), collision (formation of new pattern bindings), scission (destruction of certain bindings).
Complexification preserves the Multiplicity Principle.
In a HES, there may be:
to store patterns $\textbf{p}_1, ⋯, \textbf{p}_n$ in the Hopfield network, the weight matrix is set to:
cf. Jupyter Notebook for simulations
Hopfield networks: