Review summary: Memory Evolutive Systems

Part B (Memory Evolutive Systems, pages 147-173)

(Chapter 5 only)

page 150

T time scale An ordered set; finite subset or interval of the positive real numbers.
Kt configuration at time t A configuration category.
k(t,t') transition from t to t' A partial functor Kt->Kt', for t less than t' (t,t' in T).
At Object of Kt.
At' Image k(t,t')(At), when this exists (notation convention).

page 153

K evolutive system A system of configuration categories Kt for t in a time scale T and transitions k(t,t') for t
L evolutive subsystem of K Evolutive system consisting of subcategories Lt in Kt 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 Kt->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 At for each t, in Kt, containing all k(t,t') images of At for later t'.
TA time span of A The times t for which At is defined.

page 157

(gt) link A->B of K A system of links gt:At->Bt, at most most for each t, containing all k(t,t') images of gt for later t'.
pattern in K A system of patterns in the constituent categories Kt 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 t0 be the meeting time, so At0=Bt0:

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.