*September 5 2018 Jimmy Mathews*

These notes are intended to be a minimal account of the material introduced by Andrée Ehresmann and Jean-Paul Vanbremeersch in their 2007 book *Memory Evolutive Systems: Hierarchy, Emergence, Cognition*. I hope that the presentation is convenient for deployment of the ideas in others' own domains of interest. It hardly needs to be mentioned that the original text contains many interesting points which are completely unrepresented here.

- Notions that can reasonably be considered part of standard literature (e.g. in category theory) are not defined.
- Facts and theorems, as well as short comments, are recorded as separate sentences.
- To shorten each definition, intermediate definitions or notation are occasionally given which are not present in the original text.
- Examples and almost all commentary are completely omitted.
- Definitions are presented in the format:
a symbol for a thing **its name**in contextThe definition.

*page 21*

K | configuration category | Category. |

A,B... | component | Object of K. |

f,g... | action, constraint, or information transfer | Morphism of K. |

Per(A) | field or perception field of A | Category of objects over A, object-with-morphisms to A. |

Op(A) | operating field of A | Category of objects under A, object-with-morphisms from A. |

*page 49*

P | pattern | (Directed) graph homomorphism into K. |

sP | sketch of the pattern | Domain graph of the homomorphism. |

implementation of a pattern | Image graph in K. | |

P_{i} | component of the pattern | Value of node i in K. |

x | distinguished links | Value of edges of sP in K. |

analogous patterns | Two patterns with the same sketch. | |

CG(P) | category generated by P | Subcategory of K generated by the implementation of P. |

F=(f_{i}) | collective link from pattern P to object A | Natural transformation between functor CG(P)->K and the A-constant functor (cocone over P). |

Op(P) | operating field of the pattern | Category of collective links from P (and cocone morphisms). |

base or effect functor | Forgetful functor from the operating field of a pattern to targets in K. | |

cP | colimit of the pattern (object) | Colimit object of the diagram generated by P. |

(c_{i}) or (c_{Pi}) | binding link | Morphisms P_{i}->cP of the colimit construction. |

The existence of a colimit constrains the local structure (of P) and the global function (operations of P upon the rest of K).

Metaphors for (colimit) binding include:

- Information, structuring communication between agents
- Function, division of labor
- Morphology, development, or integration of a whole from parts
- Macro- and micro-states for concept of entropy

P | decomposition of object C | Pattern having colimit C. |

*page 73*

g:B->C | link mediated by decomposition P of C=cP | Link to C factoring through a component P_{i} of P. |

g_{i}:B->P_{i} | P-factor of mediated link | Factor of mediation, through given P_{i}. |

zig-zag correlation between two P-factors | P-factors along path in sketch sP comprising with that path a commutative diagram. | |

(g_{i}) | perspective of B for P | Family of P-factors maximal with respect to zigzag correlation among mutually zigzag correlated families. |

g_{i} | aspect of B for P | The P-factors of a perspective. |

If cP exists, a P-mediated link B->cP (with choice of mediation) is equivalent to a perspective of B for P.

If cP exists, Op(P) = Op(cP) = Hom(cP,K).

Per(P) | perception field or field of P | Category of perspectives of objects for P and morphisms of the objects commuting the P-factors. |

base or appearance functor | Forgetful functor from perception field of a pattern to sources in K. |

In general the functor Per(P)->Per(cP) is neither full nor faithful.

cluster from pattern Q to P | Family of perspectives of components of Q for P, closed under composition with distinguished links of Q and P. | |

E | A collection of links from components of Q to components of P, at least one from each Q_{i}, and zigzag correlated if more than one. | |

E' | All compositions of E with distinguished links of Q and P. | |

E'' | cluster generated by E | Saturation of E' with respect to zigzag correlation. |

In(i) | insertion cluster | Cluster generated by distinguished links of P from P_{i}. |

G | cluster from Q to P. | |

G' | cluster from P to Q'. | |

GG' | cluster generated by all compositions of G with G'. |

Patterns in K and clusters between them form a super-category of K, denoted Ind(K).

Every pattern P in K as a pattern in Ind(K) admits a colimit in Ind(K), the pattern P. The binding links are the insertion cluster In(i).

g | (Q,P)-simple link | Morphism cQ->cP binding a cluster G:Q->P; induced by universal definition of cQ from composition links (Q_{i}->cP). |

factors from P to Q | Links of the cluster G binding into g. | |

(Q,P)-complex link | Morphism cQ->cP not binding any cluster G:Q->P. | |

adjacent clusters | Pair of simple links with respect to decompositions in which the target of one is the domain of the other. |

The composition of two simple links having adjacent clusters is simple. It binds the composition of the clusters that the two links bind.

R | sub-pattern of P | The restriction of P to a sub-graph of its sketch sP. |

insertion cluster R->P | Cluster generated by all links of P originating in a component of R (equivalently by the identities of the R_{i}). | |

representative sub-pattern of P | A sub-pattern having the same colimit as P. |

If there is a cluster P->R generated by distinguished links of P, then R is a representative sub-pattern of P.

delegation cluster | A cluster P->R generated by distinguished links of P. | |

(R_{k}) | representatives of P_{i} with respect to a delegation cluster | Components of R admitting a link P_{i}->R_{k} of the cluster. |

P,P^{o} | homologous decompositions | Patterns having the same colimit up to isomorphism. Decompositions of the same object. |

connected decompositions | Homologous patterns whose colomits isomorphism is the binding of a cluster. | |

strongly connected decompositions | Homologous patterns whose colimits isomorphism is the identity and the binding of a cluster. | |

C | multi-fold object | Object admitting non-connected decompositions. |

complex switch | Transfer of attention between the non-connected decompositions of a multi-fold object. | |

homologous patterns | Patterns whose operating fields are isomorphic over their base functors to K. | |

cohomologous patterns | Patterns whose opposites in K^{op} are homologous. | |

connected patterns | Patterns with isomorphic operating fields, the isomorphism induced by a cluster. |

Homologous patterns both admit colimits or neither admit colimits.

Homologous patterns admitting colimits have isomorphic colimits.

Multiplicity principle (degeneracy) | Presence of homologous but not connected patterns. |

The composition of two simple links whose adjacent target and source decompositions are connected is simple.

The composition of two simple links whose adjacent target and source decompositions are homologous may not be simple.

K | hierarchical category | Objects partitioned into finite ordered set. Each object is a colimit of a pattern involving only lower objects. |

P | ramification of length 1 of C | A pattern having colimit C. |

P,(P_{i}) | ramification of length 2 of C | A choice of patterns (P_{i}) having colimit the components P_{i} of P. |

... | ramification of length k of C | ... |

2-iterated colimit of P,(P_{i}) | The object C. | |

ultimate components of a ramification | Components of the ramification in the lowest level. | |

micro-components of C | The ultimate components of a ramification of length 2. |

Every object of level n of a hierarchical category has a ramification of length k whose ultimate components belong to level n-k or less.

Can every object of level n (i.e. an object admitting a ramification of length 2 with microcomponents in n-2 or less) also admit a ramification of length 1 whose **components** are in level n-2 or less? No.

**Reduction Theorem**. If the mid-level distinguished links of a ramification R of C of length 2 are simple with respect to the lower part of the ramification, then the following pattern is a ramification of C of length 1 rooted in levels n-2 or less: The components are the microcomponents of R and the links are the microlinks of R plus the links of the clusters bound by the mid-level links of R.

complexity order of C | The lowest level in which a ramification of length 1 of C can be rooted. | |

C is q-reducible | The complexity order of C is q or less. | |

m-simple link | A link binding a cluster contained in level m or less. | |

K satisfies the multiplicity principle | K is a hierarchical category, and for each non-trivial level n of K, there are multi-fold objects of level n. | |

n-entropy of C | The largest number of mutually non-connected patterns (in levels n or less) with colimit C (of level n+1). |

*page 109*

p | functor K->K' | A functor. |

c | comparison link colim(pP)->pC | The link guaranteed by the universal property of colim(pP) (when it exists) with respect to the p image of the binding links P_{i}->C. |

C is maintained or preserved by p | c is an isomorphism. | |

C is modified by p | c is not an isomorphism. | |

C is disintegrated by p | P is bound into C but pP is not bound. | |

P is integrated by p | P is unbound (C=colim(P) does not exist), but pP is bound. | |

(f_{i}) | distributed link B->P | A system of links f_{i}:B->P_{i} commuting with the distinguished links of P. |

λP | limit or classifier of P | The limit of the diagram of P in K. |

(l_{i}) | projection distributed link λP->P_{i} | Defining distributed link λP->P. |

f | classification B->λP | Link B->λP induced by the universal property of (λP,(l_{i})) applied to (f_{i}):B->P. |

product of (P_{i}) | The limit of P, in case P has no distinguished links. | |

Cl(P) | classifying field of P | The category of distributed links to P. |

If the limit λP exists, there is an equivalence of categories Cl(P)->Per(λP).

P and Q are pro-homologous | The opposites of P and of Q are homologous patterns in the opposite category K^{op}. | |

G | pro-cluster P->Q | Maximal set of links P_{i}->Q_{j} closed under composition with distinguished links and containing at least one link to each Q_{j}. |

Pro(K) | Category of pro-clusters between patterns in K. | |

(P,Q)-pro-simple link | A link λP->λQ binding a pro-cluster P->Q, when P and Q have classifiers λP and λQ. | |

U | elements to be absorbed | Graph. |

V | elements to be eliminated | Set of objects of K. |

W | patterns to be bound | Set of patterns in K (not necessarily unbound). |

forced colimits | Partial function W->Op, where each pattern Q in W in the domain maps to an element of the operating field Op(Q) (to colim(Q), if it already exists). May be undefined only for unbound patterns. Homologous patterns Q and Q' in W must have forced colimits / already-present colimits in Op(Q) and Op(Q') which correspond under a fixed, chosen isomorphism Op(Q)=Op(Q'). A pattern in W not in the domain of this partial function must not be homologous to one that is. | |

X | patterns to be classified | Set of patterns in K (not necessarily classified). |

forced limits | Function X->Cl, where each pattern Q in X maps to an element of the classifying field Cl(Q) (to lim(Q), if it already exists). Co-homologous patterns Q and Q' in X must have forced limits / already-existing limits in Cl(Q) and Cl(Q') which correspond under a fixed, chosen isomorphism Cl(Q)=Cl(Q'). |

*page 117*

Op^{id} | insertion/deletion option on K | Tuple (U,V). |

Op^{sb} | simple binding option on K | An option of the form (W), where the forced colimits function is defined on all of W. |

Op^{sc} | simple classifying option on K | (X). |

Op^{b} | binding option (or just option) on K | Tuple (U,V,W). |

Op^{c} | classifying option on K | Tuple (U,V,X). |

Op^{m} | mixed option on K | Tuple (U,V,W,X). |

Op | Any type of mixed option (includes Op^{id}, Op^{sb}, Op^{sc}, Op^{b}, Op^{c}, Op^{m}). | |

p | Partial functor K->K'. | |

U is absorbed by p | U is identified with a sub-graph of K' in the complement of the image of p. | |

V is eliminated by p | V is identified with a subset of the complement of the domain of p. | |

W is bound by p | The p images of the patterns W have colimits in K' (with isomorphism comparison links from the p images of their colimits, if these colimits already exist in K, or else from the p images of the forced colimit). | |

X is classified by p | The p images of the patterns X have limits in K' (with isomorphism comparison links to the p images of their limits, if these limits already exist in K, or else to the p images of the forced limits). | |

Op/Op by p^{id}/Op^{sb}/Op^{sc}/Op^{b}/Op^{c}/Op^{m} is satisfied | U is absorbed / V is eliminated / W is bound / X is classified by p. If W is non-empty, homologous patterns belonging to W must have the same colimit in K'. | |

p:K->K' | complexification of K achieving or realizing Op | A partial functor satisfying Op through which any other partial functor K->K'' satisfying Op factors as K->K'->K'' for some K'->K''. |

*page 121*

The complexification realizing an insertion/deletion option Op^{id} is unique up to unique isomorphism. It is the full subcategory of the category sum/union of the category K and the graph U regarded as a category, which contains Obj(K)-V and the vertices of U.

Q | Arbitrary element of W, for simple binding option Op^{sb}=(W). | |

(c_{i}) | to-be binding link | Collective link Q->C, equal to the value of the forced colimit function on Q or else the binding collective link if colim(Q) exists. |

(f_{i}) | Arbitrary collective link Q->B in the category K. | |

L_{1} | Category freely generated from K by formally added links f:B->C for every (f_{i}), with formal equation of multiple links B->C binding (f_{i}) if present, subject to fc_{i}=f_{i}. | |

(f_{i}^{2}) | Arbitrary collective link Q->B in the category L_{1}. | |

L_{2} | Category freely generated from L_{1} by formally added links f:B->C for every (f_{i}^{2}), with formal equation of multiple links B->C binding (f_{i}^{2}) if present, subject to fc_{i}=f_{i}^{2}. | |

... | ... | |

K->K' | The category union/colimit of the functors K->L_{1}->L_{2}->... . |

** Complexification Theorem** (simple binding options). The functor K->K' given above is the complexification of K realizing the simple binding option Op^{sb}.

K' contains infinitely many morphisms in general, even if it contains only finitely many objects.

*page 125*

The construction of the complexification realizing Op^{sb} does not apply to the (W) component of a general binding option Op^{b}=(U,V,W), since the construction required the forced colimits function to be defined on all of W.

K_{0} | Target of the complexification p_{0}:K->K_{0} realizing the insertion/deletion component (U,V) of a binding option Op^{b}=(U,V,W). | |

Obj(K_{1}) | The union of objects of K_{0} with the set of homology classes of patterns in W with no bound patterns in the class. | |

P | Object of K_{0} or element of W. | |

P^{0} | A choice of bound pattern belonging to W homologous to P, fixed but arbitrary, in case P is in W and its homology class in W contains such a bound pattern. | |

o(P) | object of P in K_{1} | P, if P is an object of K_{0}, or else colim(P^{o}) in K_{0} if P is an element of W homologous to a bound pattern in W, or else the homology class object [P] in Obj(K_{1}). |

c_{P} | binding link of P in K_{1} | The identity of P=o(P), if P is an object of K_{0}, or else the binding collective link (P_{i}->o(P)) of colim(P^{o}) if P is homologous to a bound pattern in W, or else a formal set of arrows P_{i}->o(P) in Obj(K_{1}). |

G | cluster | A cluster P->P'. |

g | simple link of G | The morphism G:o(P)->o(P') in K_{0}, if o(P) and o(P') are in K_{0}, or else a formal arrow o(P)->o(P') between elements of Obj(K_{1}). |

K_{1} | The category on Obj(K_{1}) freely generated by the set of simple links g subject to the relations of K_{0} and the system of equations c_{P'}G=gc_{P} (that is, c_{P'j}G_{ij}=gc_{Pi} for each i and j). | |

K_{1}->K' | The complexification of K_{1} realizing the simple binding option Op^{sb}=(W), wherein the domain of the forced colimits function is extended to all of W by assigning to each Q the element (o(Q),c_{Q}). | |

K->K' | The composition K->K_{0}->K_{1}->K'. |

**Complexification Theorem** (binding options). The functor K->K' given above is the complexification realizing the binding option Op^{b}.

*page 130*

Op^{m} | A mixed option (U,V,W,X). | |

b(Op^{m}) | binding component | (U,V,W). |

c(Op^{m}) | classifying component | (X). |

K->M_{1} | The complexification of K realizing b(Op^{m}). | |

N_{1} | The opposite category of the complexification of M_{1}^{opposite} realizing the binding option (c(Op^{m}))^{opposite} with forced colimits function equal to the opposite of the forced limits function. | |

N_{1}->M_{2} | The complexification of N_{1} realizing b(Op^{m}). | |

N_{2} | The opposite category of the complexification of M_{2}^{opposite} realizing the binding option (c(Op^{m}))^{opposite} with forced colimits function equal to the opposite of the forced limits function. | |

... | ... | |

K->K' | The category union/colimit of the functors K->M_{1}->N_{1}->M_{2}->N_{2}... |

**Complexification Theorem** (mixed options). The functor K->K' given above is the complexification realizing the mixed option Op^{m}.

**Iterated Complexification Theorem**. It may be the case that the composition of a sequence K->K'->K'' of complexifications is not isomorphic to any single complexification K->K'''.

In particular this may occur because the option realized by K'->K'' requires binding a pattern in K' containing complex links with respect homologies between patterns bound by K->K'.

*page 135*

H | Hierarchical category. | |

H_{n} | The full sub-category of H containing levels n and less. | |

n-simple link | Morphism of H binding a cluster of H_{n}. | |

S_{n} | The smallest set of links containing the links of level n and which is closed under the binding of clusters of links and closed under composition. (Note: The authors write "[containing] any cluster formed of n-simple links" rather than closed under binding of clusters of links, which seems to be a typo.) | |

n-constructible link | An element of S_{n}. | |

n-complex link | An n-constructible link which is not n-simple. | |

H is based (on H_{0}) | Every morphism of H_{n+1} is n-constructible. |

**Lemma**. If H is based, any n-constructible link is also (n-1)-constructible.

A hierarchical category is based if and only if every morphism is 0-constructible, that is either 0-simple or 0-complex.

Based hierarchies are precisely those which are constructed level-by-level by a sequence of complexifications realizing binding options.

*page 140*

material cause | Material or matter contributing to causation. (Aristotelian cause) | |

formal cause | Form or arrangement contributing to causation. (Aristotelian cause). | |

efficient cause | Movement or mover contributing to causation. (Aristotelian cause). | |

final cause | Purpose or reason contributing to causation. (Aristotelian cause). |

For a complexification K->K' with respect to an option Op, the initial configuration K is associated with material cause, the option Op is associated with formal cause, and the implementation or construction of the complexification is associated with efficient cause. Because of the Iterated Complexification Theorem, an iterated complexification K->K'...->K^{(m)} may be said to have indivisible material/formal/efficient causes. This suggests that complexifications may be used to model systems with features which prevent them from being modeled by classical physical dynamical systems, for which the material/formal/efficient causes are typically separately identifiable.