*page 150*

T | time scale | An ordered set; finite subset or interval of the positive real numbers. |

K_{t} | configuration at time t | A configuration category. |

k(t,t') | transition from t to t' | A partial functor K_{t}->K_{t'}, for t less than t' (t,t' in T). |

A_{t} | Object of K_{t}. | |

A_{t'} | Image k(t,t')(A_{t}), when this exists (notation convention). |

*page 153*

K | evolutive system | A system of configuration categories K_{t} for t in a time scale T and transitions k(t,t') for t |

L | evolutive subsystem of K | Evolutive system consisting of subcategories L_{t} in K_{t} and restrictions of the k(t,t') for t and t' in a sub time scale S in T. |

p | evolutive functor K->K' | For sub time scale T in T', a system of functors K_{t}->K'_{t} compatible (commuting) with the k(t,s) and k'(t,s) (t,s in T). |

A | component of K | A collection of constituents, at most one A_{t} for each t, in K_{t}, containing all k(t,t') images of A_{t} for later t'. |

TA | time span of A | The times t for which A_{t} is defined. |

*page 157*

(g_{t}) | link A->B of K | A system of links g_{t}:A_{t}->B_{t}, at most most for each t, containing all k(t,t') images of g_{t} for later t'. |

pattern in K | A system of patterns in the constituent categories K_{t} of K, consisting of an initial pattern and all its later configurations. | |

colimit of a pattern in K | The minimal component of K containing the colimits of the patterns in the constituent categories. | |

birth/death time of component A | Infimum/supremum of the time span of A. |

*page 159*

The trajectories of two components A and B in K either meet or do not meet. If they meet, they may meet once or more than once.

There are 3 cases concerning the trajectories of components A and B that meet once. Let t_{0} be the meeting time, so A_{t0}=B_{t0}:

- Mixture. For all t>t
_{0}, t is in TA if and only if t is in TB, and A_{t}=B_{t}. - Absorption. t
_{0}is the death time of B. - Fusion. t
_{0}is the death time of A and of B, but there is a new component C with birth time t_{0}such that C_{t0}=A_{t0}=B_{t0}.

In a 4th case, not technically part of the foregoing list, **unison**, neither i, ii, nor iii takes place at time t0, but rather a new component is born, M, which is something like a complexification binding A and B.

*page 161*

hierarchical evolutive system | An evolutive system whose configuration categories are hierarchical and whose transition functors preserve levels. |

*page 163*

e(t) | (forward) stability span of A at t | The largest number such that there exists a ramification Qt of At which remains so for all t' between t and t+e(t) (non-inclusive), in the sense that Qt' is a ramification of At'. |

complex identity of A | A sequence Pm of patterns ramifying A, supported on a respective covering of the time scale of A, agreeing on overlaps in the sense that Pm and Pm+1 have a common representative subpattern at a distinguished time (all times?) belonging to the overlap between the time scale of Pm and Pm+1. |

*page 167*

Complex identity of A should probably be considered up to some equivalence, which we do not consider here yet.

r(t) | renewal span of A at t | The smallest number such that there exists a ramification of At+r(t) made of components which were not part of (a chosen ramification of) A at time t but which were introduced (born?) between t and t+r(t). |

c(t) | continuity span of A at t | The greatest number such that there exists a pattern P between times t and t+c(t) with the property that Pt ramifies At, and for each t' in this time span (non-inclusive of endpoints), At' is the colimit of a pattern consisting either of components of Pt' or components replacing these components (in the sense of (i) mixture, (ii) absorption, or (iii) fusion?). |

*page 170*

propagation delay | A system of functors from the configuration categories Kt to the groupoid of additive or multiplicative real numbers. |

*page 171*

FK | fibration associated to K | A quasi-category (not all compositions are defined) consisting of all formal composites of vertical links vt':At'->Bt' (belonging to one of the Kt) with horizontal links htt':At->At' (formally inserted morphisms from objects At to some forward image At'); horizontal links first. The composite of vt'htt' with wt''kt't'' is defined when wt'':Bt''->Ct'', kt't'' is the canonical morphism Bt'->Bt'', and vt' has a configuration vt'':At''->Bt'' at time t'', in which case the composite is (wt''vt'')(lt't''htt'), where lt't'':At'->At'' is the canonical morphism. |

transverse link | A non-trivial morphism of FK (neither the vertical nor the horizontal factor is the identity). |

The authors' notation omits the horizontal link, since it is canonical once the source and target are specified. Thus one would write: the composite of vt' with wt'' is wt''vt'', where vt'' is the configuration of vt' at time t''.

base functor of FK | The functor FK->T, where T is regarded as a category with respect to the ordering of the reals. | |

α | local section of the base functor | A partial functor T->FK such that the image αt of any t belongs to Kt, and for t->t', αt' is a the configuration of αt at t' (the image of t->t' must be the canonical morphism αt->αt'). |

ordering of local sections | Partial ordering defined by restriction of mappings. |

A component of K as an evolutive system is the same as a maximal local section of FK->T.

time-constrained subgraph of FK (In case propagation delays are defined) | Subgraph containing all horizontal and vertical links, and transverse links vt'htt' such that the delay of vt' is less than t'-t (greater than?). | |

memory of K | Evolutive sub-system of K. | |

records | Components of the memory of K. |