Difference between revisions of "Kan Extensions"
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The notion of Kan extensions subsumes all the other fundamental concepts of category theory. | The notion of Kan extensions subsumes all the other fundamental concepts of category theory. | ||
=Kan Extensions are partial colimits= | =Kan Extensions are partial colimits= | ||
[[Paolo Perrone]] has a few talks on explaining [[Kan Extensions]] as partial [[colimit]]s. | [[Paolo Perrone]] has a few talks on explaining [[Kan Extensions]] as partial [[colimit]]s<ref>{{:Video/Kan extensions are partial colimits, Paolo Perrone, 11/02/2021</ref><ref>{{:Video/Perrone - Kan extensions are partial colimits}}</ref><ref>{{:Video/Paolo Perrone: Kan extensions are partial colimits}}</ref>. | ||
{{#ask: [[Presented by::Paolo Perrone]] | {{#ask: [[Presented by::Paolo Perrone]] | ||
|format=table | |format=table | ||
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}} | }} | ||
A Kan extension is a mathematical object<ref>{{:Thesis/All Concepts are Kan extensions}}</ref> that can be used to represent [[Concept|concepts]] or [[Idea|ideas]]. | |||
=Left and Right Kan Extensions= | |||
There are two kinds of Kan Extensions, left and right. They may be compared to the notion of horizontal and vertical composition in [[Category Theory]]. | |||
=Left and Right Adjoint= | |||
Left and Right Kan Extensions can be thought of as Left and Right [[Adjoint Functors]] when the target is mapped by an [[Identity Functor]]. | |||
==List of Videos== | |||
* [[Video/Kan Lifts and Kan Extensions, part 1|Kan Lifts and Kan Extensions, Part 1]] | |||
* [[Video/Kan Extensions and Kan Lifts, Part 2|Kan Extensions and Kan Lifts, Part 2]] | |||
* [[Video/Kan Extensions|Kan Extensions by MathProofsable]] | |||
* [[Video/Category Theory For Beginners: Kan Extensions|Category Theory For Beginners: Kan Extensions]] | |||
<noinclude> | |||
=References= | =References= | ||
<references/> | <references/> | ||
=Related Pages= | |||
[[Category:Kan extension]] | |||
[[Category:Adjoint Functors]] | |||
</noinclude> | |||
Latest revision as of 13:11, 24 March 2022
On page 248 of Categories for the Working Mathematician[1], Saunders Mac Lane stated:
The notion of Kan extensions subsumes all the other fundamental concepts of category theory.
Kan Extensions are partial colimits
Paolo Perrone has a few talks on explaining Kan Extensions as partial colimits[2][3][4].
| Content Link |
|---|
| Video/Kan extensions are partial colimits, Paolo Perrone, 11/02/2021 |
| Video/Paolo Perrone: Kan extensions are partial colimits |
| Video/Perrone - Kan extensions are partial colimits |
A Kan extension is a mathematical object[5] that can be used to represent concepts or ideas.
Left and Right Kan Extensions
There are two kinds of Kan Extensions, left and right. They may be compared to the notion of horizontal and vertical composition in Category Theory.
Left and Right Adjoint
Left and Right Kan Extensions can be thought of as Left and Right Adjoint Functors when the target is mapped by an Identity Functor.
List of Videos
- Kan Lifts and Kan Extensions, Part 1
- Kan Extensions and Kan Lifts, Part 2
- Kan Extensions by MathProofsable
- Category Theory For Beginners: Kan Extensions
References
- ↑ Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (2nd ed.). local page: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001. , 248
- ↑ {{:Video/Kan extensions are partial colimits, Paolo Perrone, 11/02/2021
- ↑ Perrone, Paolo (Feb 28, 2022). Perrone - Kan extensions are partial colimits. local page: Category Theory CT20->21.
- ↑ Perrone, Paolo (Jun 12, 2020). Paolo Perrone: Kan extensions are partial colimits. local page: Topos Institute.
- ↑ 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.