##  [Kan Extension](/kan-extension) 

  ##  [Kan Extension](https://mathlogic.quantumdictionary.io/kan-extension-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A universal construction that produces the best approximation of a functor F along another functor K, yielding a left Kan extension Lan_K F (colimit-based) or a right Kan extension Ran_K F (limit-based) characterized by universal natural transformations.

 

 

 

 

 





 

 



 ##  [Kan Extension](https://puremath.quantumdictionary.io/kan-extension-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A universal construction that extends a functor along another functor in the best possible way, producing left or right Kan extensions characterized by universal mapping properties that generalize limits, colimits and adjoints.