Summary "How to Compose?"

Composing the answers of

"How to compose?"

This is a systematic summary of the paper "How to Compose?"
It may be used as an introduction into the topics of a general theory of composition.

Categorical composition

Category theory is defining the rules of composition. It answers the question: How does composition work? What to do to compose morphisms?

Answer: Category Theory.


It is focused on the surface-structures of the process of composing morphism, realized by the triple DPS of

Data (source, target),
Structure (composition, identity) and
Properties (unity, associativity)

by fulfilling the matching conditions for morphisms.

The properties (axioms) of categories are the global conditions for the final realization of the local rules of composition, i.e., the matching conditions for morphisms to be composed.
Categories are based on their global Properties of "unit" and "associativity", understood as the axioms of categorical composition of morphisms.

Proemial composition

Proemiality answers the question: What enables categorical composition? What is the deep-structure of categorical composition?

Answer: proemial relationship.
Proemial relationship is understood as a cascade of order- and exchange-relations, as such it is conceived as a pre-face (pro-oimion) of any composition.

Parts of the categorial Structure are moved into the proemial Data domain. Or inverse: Parts of the Data (source, target) are moved into the Structure as exchange relation.

Thus,
Data (order relation=morphism),
Structure (exchange relation, position; identity, composition).
Properties (diversity; unit, associativity)

That is, categorical Structure is distributed by "positions" over different levels of the proemial relationship.

Proemiality is based on order- and exchange relations.
That is, order relations are based on a cascade of exchange relations and exchange relations are founded in cascade of order relations.

But this interlocking mechanism is not inscribed into the definition of proemiality, it occurs as an interpretation, only.

Hence, proemiality as a pre-face may face the essentials of composition but not its Janus-faced movements.

Chiastic composition

Chiastic approach to composition answers the question: How is proemiality working? What enables proemiality to work?

Answer: Chiasm of the proemial constituents, i.e., order- and exchange relation.

The chiasm of composition is the inscription of the reading of the proemial relationship. It is mediating the upwards and downwards reading of proemiality, which in the proemial approach is separated.

Hence, it is realizing the Janus-faced movements of double exchange relations.

To avoid empty phantasms and eternal dizziness of the Janus-faced double movements of exchange relations, iterative and accretive, up- and downwards, the coincidence relations of chiasms have to enter the stage.

That is, the matching conditions have to be applied to the exchange relations as well as to the coincidence relations to perform properly the game of chiasms on trusted arenas.

Thus, proemiality, with its single exchange relation and lack of coincidence, is still depending on logo-centric thematizations even if its result are surpassing radically its limits by the introduction of polycontexturality.

Hence, proemiality is depending on a specific reading, i.e., a mental mapping of chiasms. This proemial reading has to imagine the double movements of the way up and the way down. And the coherence of the different levels, formalized in chiasms by the coincidence relations.

The DSP-transfer is:
Data (morphisms),
Structure (exchange, coincidence, position; identity, composition),
Properties (diversity; unity, associativity)

Diamond of composition

The diamond approach answers the question: What is the deep-structure of composition per se, i.e., independent from the definition or view-point of morphisms and its chiasms?

Answer: the interplay of acceptional and rejectional process/structures as complementary movements of diamonds. Without such an interplay there is no chiasm, and hence, no proemiality nor categorial composition.
The acceptional parts are defining categories, the rejectional saltatories, both together are defining diamonds.

The DSP-transfer is:
Data (morphisms, hetero-morphism),
Structure (double-exchange, coincidence, position; identity, difference, composition, de-composition),
Properties (unity, diversity, associativity, complementarity).

In fact, diamonds don’t have Data and Structure, everything is in the Properties as an interplay of global and local parts.

Hence, diamonds are playing the

Properties (global/local, surface/deep-structure),

which is realized by the interplay of categories and saltatories, hence, again,

Properties (categories, saltatories).

Saltatories are founded in categories and categories are founded in saltatories; both together in their interplay are realizing the diamond structure of composition.

Descriptive Definition of Diamond

Interplay of the 4 approaches

How are the 4 approaches related? What’s their interplay? What is the deep-structure of "interplay"?

Answer: Diamonds as the interplay of interplays, i.e., the play of global/local and surface-/deep-structures are realizing the autonomous process/structure "diamond".

Diamond (categories, saltatories)

Kenogrammatics of Diamonds

Diamonds are taking place, they are positioned, hence their positionality is their deep-structure.

The positionality of diamonds, marked by their place-designator, is the kenomic grid with its tectonics of proto-, deutero- and trito-structure of kenogrammatics.
(Don't ask, where the kenomic grid is located?!)

Kenogrammatics answers the question: How to get rid of diamonds (without loosing them)?

In other words, kenogrammatics is inscribing diamonds without the necessity to relate them to the drama of composition.

Therefore, the kenogrammatics of diamonds is opening up a composition-free calculus of "composition".

Polycontexturality of Diamonds

Because of the iterability of diamonds based in the fact that diamonds are placed and situated in a kenomic grid they can be repeated in an iterative and a accretive way.

Iteration is application inside the framework of a diamond system, hence iteration remains mono-contextural.

Polycontexturality of diamonds is an accretive repetition, i.e., a dissemination of frameworks of diamonds.