The Complementary Blog: Diamond Strategies

Have a look at the complementary Blog to the Chinese Challenge Blog:

Rudy's Diamond Strategies.

The new Blog is presenting, step by step, new insights into the mathematical theory of Diamonds.


Chiasm (Categorification, Diamondization)

Diamond: 2-graphs with 2-structures

i) Data: 2-diagram C1–s,t––>Co/Co<–diff–C1 in 2-Set Objects in diamonds are involved into 2 operations: coincidence and difference. Coincidence is producing composition and therefore commutativity.
Differences are producing hetero-morphisms and therefore jumpoids.

ii) Structure: composition, identities + complement, differences

iii) Properties: unit, associativity + diversity, jump law

iv) Interaction: Chiasm between category and saltatory.

"In ordinary category theory we have 1-dimensional arrows ––>; in higher-dimension category theory we have higher-dimension arrows."

"...n-categories are studying morphisms between morphisms." Tom Leinster

Hence, Diamond theory is neither studying linear ordered arrows nor morphisms between linear ordered arrows but the complementarity of morphisms and hetero-morphisms, acceptional and rejectional morphisms, i.e., the relations between the operation of composition and its complementary morphisms.

In this sense, diamond theory is studying 2-dimensional, i.e., tabular categories, independent from the questions of n-categories or others.

Thus, diamond theory is the study of tabular categories as an interaction of categories and saltatories. Saltatories are the complementary diamond structures of categories.

The term interaction is correct because the interplay between categories and saltatories happens inside the diamond definition and is not only a meta-theoretical fact like the duality of categories in category theory.

Compositions as operations are not thematized in Category theory but only their result, which are new morphisms.

Diamond theory is thematizing the activity of the composition operator not as a morphogram but as a complementarity to the operator, implemented as a hetero-morphism.

Diamonds are thematizing the basic operation of category theory as such: the operation of composition. The thematization is modeled into the hetero-morphisms.

In a general setting of graphematic analysis of composition the morphogrammatics of the operator "composition" has to be taken into account, too. That is, the neither-nor gesture of categorical object and morphism has a double face: hetero-morphism and morphogram of composition.

As Categories can be generalized to n-Categories, Diamonds can be generalized to n-Diamonds.

Topics
Category Theory: object/morphism
n-Category: morphism/morphism
Diamond Theory: categories/saltatories.