Kan extensions are universal constructs in category theory, a branch of mathematics.
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 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.
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
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
that is couniversal with respect to the specification, in the sense that for any functor
is defined and fits into a commutative diagram: where
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.
This gives rise to the alternate description: the left Kan extension of
that are universal with respect to this specification, in the sense that for any other functor
exists and fits into a commutative diagram: where
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
are the corresponding transformations, then there exists a unique isomorphism of functors
Likewise for right Kan extensions.
If A is small and C is cocomplete, then there exists a left Kan extension
, defined at each object b of B by where the colimit is taken over the comma category
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
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 The limit of a functor
possesses a left adjoint if and only if the right Kan extension of
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