Difference between revisions of "Monoidal category"
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{{WikiEntry|key=Monoidal category|qCode=1945014}} is a category that admits [[tensor product]]s | {{WikiEntry|key=Monoidal category|qCode=1945014}} ([[單項範疇]]/[[么半範疇]]/[[么乘範疇]]/[[張量範疇]]) is a category that admits [[tensor product]]s, in other words, it is the formal mathematical data structure to represent [[hyperlink|hyperlinked entities]]. [[Bob Coecke]] claims that [[Monoidal Category]] is the [[universal construct]] that can be used to construct everything, physical or informational<ref>{{:Book/Picturing Quantum Processes}}</ref>. It can be used as the building block for all languages, including natural languages, see [[Quantum Natural Language Processing]]<ref>{{:Video/Bob Coecke, From Quantum Linguistics to Spacetime Linguistics, and Cognition}}</ref>. | ||
[[Bob Coecke]]'s argument about [[Monoidal Category]] is closely related to the concept of [[monad]] as illustrated in [[Leibniz]]'s [[Monadology]]. It can be the intellectual foundation to conduct [[universal data abstraction]] in the developmental efforts of [[PKC]] and its applications. That means it has direct application to the compilation and interpretation of complex information systems, that covers almost any engineered system of practical interesting. To learn the formal definition of [[Monoidal Category]], [[Richard Borcherds]] has a video on [[Monoidal Category]]<ref>{{:Video/Categories 6 Monoidal categories}}</ref>. | |||
=Monoidal Category as a Two Dimensional Algebra?= | =Monoidal Category as a Two Dimensional Algebra?= | ||
{{:Monoidal Category as a Two Dimensional Algebra}} | |||
}} | |||
=Monoidal Categories in Visual Representations= | |||
[[Peter Selinger]] has a paper called: [[Paper/A survey of graphical languages for monoidal categories|A survey of graphical languages for monoidal categories]]<ref>{{:Paper/A survey of graphical languages for monoidal categories}}</ref>. | |||
There are a few variations of monoidal categories: | There are a few variations of monoidal categories: | ||
=Symmetrical Monoidal Category= | =Symmetrical Monoidal Category= | ||
{{:Symmetrical Monoidal Category}} | |||
=Braided Monoidal Category= | =Braided Monoidal Category= | ||
{{:Braided Monoidal Category}} | |||
<noinclude> | |||
=References= | =References= | ||
<references/> | <references/> | ||
=Related Pages= | =Related Pages= | ||
[[Category:Category Theory]] | [[Category:Category Theory]] | ||
[[Category:Universal Component]] | |||
[[Category:Monoidal Category]] | |||
</noinclude> | </noinclude> | ||
Latest revision as of 04:42, 23 March 2022
Monoidal category(Q1945014) (單項範疇/么半範疇/么乘範疇/張量範疇) is a category that admits tensor products, in other words, it is the formal mathematical data structure to represent hyperlinked entities. Bob Coecke claims that Monoidal Category is the universal construct that can be used to construct everything, physical or informational[1]. It can be used as the building block for all languages, including natural languages, see Quantum Natural Language Processing[2]. Bob Coecke's argument about Monoidal Category is closely related to the concept of monad as illustrated in Leibniz's Monadology. It can be the intellectual foundation to conduct universal data abstraction in the developmental efforts of PKC and its applications. That means it has direct application to the compilation and interpretation of complex information systems, that covers almost any engineered system of practical interesting. To learn the formal definition of Monoidal Category, Richard Borcherds has a video on Monoidal Category[3].
Monoidal Category as a Two Dimensional Algebra?
Daniel Tubbenhauer's VisualMath also has a video on What are…monoidal categories?[4]. At the end of the video, he stated that Monoidal Category can be used as a way to model a Two-Dimensional Algebra.
Content related to Monoidal Category:
Monoidal Categories in Visual Representations
Peter Selinger has a paper called: A survey of graphical languages for monoidal categories[5]. There are a few variations of monoidal categories:
Symmetrical Monoidal Category
Braided Monoidal Category
References
- ↑ Coecke, Bob; Kissinger, Aleks (2017). Picturing Quantum Processes. local page: Cambridge University Press. ISBN 978-1316219317.
- ↑ Coecke, Bob (Dec 6, 2021). Bob Coecke, From Quantum Linguistics to Spacetime Linguistics, and Cognition. local page: The Quantum Information Structure of Spacetime.
- ↑ Borcherds, Richard (Oct 10, 2021). Categories 6 Monoidal categories. local page: Richard E. BORCHERDS.
- ↑ Tubbenhauer, Daniel (Feb 27, 2022). What are…monoidal categories?. local page: VisualMath.
- ↑ Selinger, Peter (Aug 23, 2009). A survey of graphical languages for monoidal categories (PDF). local page: arXiv.