Composition

Compsition can be thought of as a data type in Category Theory. It is often made of objects and maps. In Category Theory, compositions must obey the rule of associativity, which preserves sequential invariance at the map level. According to Mann, the compositional structure of maps must string together in a compatible fashion. This compatibility is realized by having maps connected to the same objects in the pairing structure. This compatibility condition can be satified using the notion of associativity. This property of being associative allows for systems to be packaged in different combinatorial options.

$A\otimes B\otimes C$ = $A\otimes (B\otimes C)$ = $(A\otimes B)\otimes C$

The different arrangements of parenthesis packaging of expressions indicate that a system can be packaged in different sub systems. In other words, this condition:associativity, enables system abstraction and packaging in ways that are more convenient for mental absorption or complexity reduction. Composition is closely related to Monad and Symmetry. It is also important to know that if something is composable from some basic building blocks, that means almost everything can be decomposed into certain universal component.

An Example of Composition

A concrete example of composition can be formulated this way:

Let objects be elemets ($x_{i}$) of a set $X$, let maps, say $f$ and $g$ be functions between these objects. Then, composition is simply a composition of different functions.

In this example, $f$ and $g$ can be composed in the following way:

$h$ =$f\circ g$ = $f(g(x))$

In the above example, $h$ is said to be a new function that is composed of $f\circ g$ in the specified sequence.

Symmetries as the first Meta-Rule

According to Mathemaniac, symmetries can be thought of as mathematical operands that gets to be manipulated through some operations that preserves the properties of being symmetrical. These four most general properties are:

1. Closure: Symmetrical operations on symmetries always create symmetries
2. Associativity: Symmetries composition with symmetries are symmetries Associative
3. Identity/Unit: Doing nothing is a symmetrical operation
4. Inverse Exists: Symmetrical operations can be undone, and returns to the original symmetry.

A mathematical treatment of this subject was explained by Norm Wilberger in a video.

Talks about Combinators and Enumerable Sets here

Particularly talks about SK Combinators, and showing that these ideas, and enumerability, determines whether certain kinds of building blocks can be recursively composed or not. Watch Dana Scott's lecture 3on Lambda Calculus.