Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.

An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors.

In Categories for the Working Mathematician Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that

The notion of Kan extensions subsumes all the other fundamental concepts of category theory.

Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on constrained optimization.

Definition

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A Kan extension proceeds from the data of three categories

 

and two functors

 ,

and comes in two varieties: the "left" Kan extension and the "right" Kan extension of   along  .

Abstractly, the functor   gives a pullback map  . When they exist, the left and right adjoints to   applied to   gives the left and right kan extensions. Spelling the definition of adjoints out, we get the following definitions;

The right Kan extension amounts to finding the dashed arrow and the natural transformation   in the following diagram:

Formally, the right Kan extension of   along   consists of a functor   and a natural transformation   that is couniversal with respect to the specification, in the sense that for any functor   and natural transformation  , a unique natural transformation   is defined and fits into a commutative diagram:

where   is the natural transformation with   for any object   of  

The functor R is often written  .

As with the other universal constructs in category theory, the "left" version of the Kan extension is dual to the "right" one and is obtained by replacing all categories by their opposites.

The effect of this on the description above is merely to reverse the direction of the natural transformations.

(Recall that a natural transformation   between the functors   consists of having an arrow   for every object   of  , satisfying a "naturality" property. When we pass to the opposite categories, the source and target of   are swapped, causing   to act in the opposite direction).

This gives rise to the alternate description: the left Kan extension of   along   consists of a functor   and a natural transformation   that are universal with respect to this specification, in the sense that for any other functor   and natural transformation  , a unique natural transformation   exists and fits into a commutative diagram:

where   is the natural transformation with   for any object   of  .

The functor L is often written  .

The use of the word "the" (as in "the left Kan extension") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique up to unique isomorphism. In this case, that means that (for left Kan extensions) if   are two left Kan extensions of   along  , and   are the corresponding transformations, then there exists a unique isomorphism of functors   such that the second diagram above commutes. Likewise for right Kan extensions.

Properties

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Kan extensions as (co)limits

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Suppose   and   are two functors. If A is small and C is cocomplete, then there exists a left Kan extension   of   along  , defined at each object b of B by

 

where the colimit is taken over the comma category  , where   is the constant functor. Dually, if A is small and C is complete, then right Kan extensions along   exist, and can be computed as the limit

 

over the comma category  .

Kan extensions as (co)ends

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Suppose   and   are two functors such that for all objects a and a of A and all objects b of B, the copowers   exist in C. Then the functor X has a left Kan extension   along F, which is such that, for every object b of B,

 

when the above coend exists for every object b of B.

Dually, right Kan extensions can be computed by the end formula

 

Limits as Kan extensions

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The limit of a functor   can be expressed as a Kan extension by

 

where   is the unique functor from   to   (the category with one object and one arrow, a terminal object in  ). The colimit of   can be expressed similarly by

 

Adjoints as Kan extensions

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A functor   possesses a left adjoint if and only if the right Kan extension of   along   exists and is preserved by  . In this case, a left adjoint is given by   and this Kan extension is even preserved by any functor   whatsoever, i.e. is an absolute Kan extension.

Dually, a right adjoint exists if and only if the left Kan extension of the identity along   exists and is preserved by  .

Applications

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The codensity monad of a functor   is a right Kan extension of G along itself.

References

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  • Cartan, Henri; Eilenberg, Samuel (1956). Homological algebra. Princeton Mathematical Series. Vol. 19. Princeton, New Jersey: Princeton University Press. Zbl 0075.24305.
  • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.
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