Difference between revisions of "Kan Extension"
m (Text replacement - "{{#ev:youtube|hP3ENPc8Jf0 |width=900 |height=500 }} " to "{{#widget:YouTube |id= hP3ENPc8Jf0 |width= 900 |height=500 }}") |
|||
| (9 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
[[wikipedia:Kan extension|Kan extension]] is a [[universal construct]] of generalized data type defined in [[Category Theory]]. | [[wikipedia:Kan extension|Kan extension]] is a [[universal construct]] of generalized data type defined in [[Category Theory]]. | ||
Kan Extension is a way to compress the idea of Lambda Calculus into a single Category Theory diagram. Using a small number of [[arrow]]s, particularly two different kinds of arrows, it is really trying to encode the universal representability of [[Lambda Calculus]]. | |||
=Proposed Application= | =Proposed Application= | ||
| Line 6: | Line 8: | ||
There are a total of 16 possible 2-input, 1-output, logic gates. They should be generalizable and represented using Kan Extension. | There are a total of 16 possible 2-input, 1-output, logic gates. They should be generalizable and represented using Kan Extension. | ||
One may want to read this paper<ref>{{:Thesis/All Concepts are Kan extensions}}</ref> by Marina on representing concepts universally. | |||
=Some useful tutorial on this subject= | =Some useful tutorial on this subject= | ||
It would be helpful to learn enough about [[Limit/Colimit]], [[Adjoint Functors]], and [[Dynamical Systems]], before studying Kan Extensions. | |||
=MathProofsable= | =MathProofsable= | ||
{{# | {{#widget:YouTube | ||
|id= hP3ENPc8Jf0 | |||
|width= 900 | |||
|height=500 | |||
}}|r4_wGxi94jg|||||}} | |||
=Richard Southwell on Kan Extensions= | =Richard Southwell on Kan Extensions= | ||
This video<ref>{{Video/}}</ref> is close to 6 hours of lengthy explanation. A large number of examples are presented in these hours. | This video<ref>{{:Video/Category Theory For Beginners: Kan Extensions}}</ref> is close to 6 hours of lengthy explanation. A large number of examples are presented in these hours. | ||
{{# | |||
This is about the time he started to formally introduce Kan Extensions. | |||
{{#widget:YouTube | |||
|id= hP3ENPc8Jf0 | |||
|width= 900 | |||
|height=500 | |||
}}|g_jEEwrpm9c|||||start=13600}} | |||
==Left Kan Extension== | |||
This is about the time he started to formally introduce Left Kan Extensions. | |||
{{#widget:YouTube | |||
|id= hP3ENPc8Jf0 | |||
|width= 900 | |||
|height=500 | |||
}}|g_jEEwrpm9c|||||start=18729}} | |||
<noinclude> | <noinclude> | ||
=References= | =References= | ||
<references/> | <references/> | ||
=Relate Pages= | |||
{{#ask: [[Category:Meta mathematics]] | |||
|format=category | |||
}} | |||
[[Category:Category Theory]] | [[Category:Category Theory]] | ||
[[Category:Universal Property]] | [[Category:Universal Property]] | ||
[[Category:Universal Construct]] | [[Category:Universal Construct]] | ||
[[Category:Kan Extension]] | [[Category:Kan Extension]] | ||
[[Category:Conceptual Spaces]] | |||
</noinclude> | </noinclude> | ||
Latest revision as of 04:39, 13 January 2024
Kan extension is a universal construct of generalized data type defined in Category Theory.
Kan Extension is a way to compress the idea of Lambda Calculus into a single Category Theory diagram. Using a small number of arrows, particularly two different kinds of arrows, it is really trying to encode the universal representability of Lambda Calculus.
Proposed Application
Conceptually, we can use Kan Extension to generalize logic gates, specifically, two inputs, one output gates.
There are a total of 16 possible 2-input, 1-output, logic gates. They should be generalizable and represented using Kan Extension.
One may want to read this paper[1] by Marina on representing concepts universally.
Some useful tutorial on this subject
It would be helpful to learn enough about Limit/Colimit, Adjoint Functors, and Dynamical Systems, before studying Kan Extensions.
MathProofsable
|r4_wGxi94jg|||||}}
Richard Southwell on Kan Extensions
This video[2] is close to 6 hours of lengthy explanation. A large number of examples are presented in these hours.
This is about the time he started to formally introduce Kan Extensions. |g_jEEwrpm9c|||||start=13600}}
Left Kan Extension
This is about the time he started to formally introduce Left Kan Extensions. |g_jEEwrpm9c|||||start=18729}}
References
- ↑ Lehner, Marina (2014). "All Concepts are Kan Extensions":Kan Extensions as the Most Universal of the Universal Constructions (PDF) (Bachelor). local page: Harvard College. Retrieved June 28, 2021.
- ↑ Southwell, Richard (Jun 28, 2021). Category Theory For Beginners: Kan Extensions. local page: Richard Southwell.