Difference between revisions of "Composition"
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Compsition can be thought of as a data type in [[Category Theory]]. It is often made of [[object]]s and [[map]]s. In [[Category Theory]], compositions must obey the rule of [[associativity]], which preserves sequential invariance at the [[map]] level. According to [[Jeremy Mann|Mann]]<ref>{{:Video/Intro to Category Theory I: the Data of a Category}}</ref>, the compositional structure of maps must string together in a '''compatible''' fashion. This compatibility is realized by having [[map]]s connected to the same [[object]]s in the pairing structure. This '''compatibility''' condition can be satified using the notion of [[associativity]]. This property of being [[associativity|associative]] allows for systems to be packaged in different combinatorial options. | Compsition can be thought of as a data type in [[Category Theory]]. It is often made of [[object]]s and [[map]]s. In [[Category Theory]], compositions must obey the rule of [[associativity]], which preserves sequential invariance at the [[map]] level. According to [[Jeremy Mann|Mann]]<ref>{{:Video/Intro to Category Theory I: the Data of a Category}}</ref>, the compositional structure of maps must string together in a '''compatible''' fashion. This compatibility is realized by having [[map]]s connected to the same [[object]]s in the pairing structure. This '''compatibility''' condition can be satified using the notion of [[associativity]]. This property of being [[associativity|associative]] allows for systems to be packaged in different combinatorial options. | ||
<math>A \tensor B \tensor C<> = <math>A \tensor (B \tensor C)<> = <math>(A \tensor B) \tensor C<> | <math>A \tensor B \tensor C</math> = <math>A \tensor (B \tensor C)<></math> = <math>(A \tensor B) \tensor C</math> | ||
[[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]]. | [[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]]. | ||
Revision as of 03:04, 27 March 2022
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[1], 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.
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= Failed to parse (unknown function "\tensor"): {\displaystyle (A \tensor B) \tensor C}
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.
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:
- Closure: Symmetrical operations on symmetries always create symmetries
- Associativity: Symmetries composition with symmetries are symmetries Associative
- Identity/Unit: Doing nothing is a symmetrical operation
- 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[2].
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 3[3]on Lambda Calculus.
References
Related Pages
- ↑ Mann, Jeremy (Jun 25, 2020). Intro to Category Theory I: the Data of a Category. local page: Jeremy Mann.
- ↑ Wildberger, Norman J. (Nov 24, 2021). A (somewhat) new paradigm for mathematics and physics. local page: Insights into Mathematics.
- ↑ Recursively Enumerable Sets