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 A Kan extension proceeds from the data of three categories
A , B , C {\displaystyle \mathbf {A} ,\mathbf {B} ,\mathbf {C} }
and two functors
X : A → C , F : A → B {\displaystyle X:\mathbf {A} \to \mathbf {C} ,F:\mathbf {A} \to \mathbf {B} } , and comes in two varieties: the "left" Kan extension and the "right" Kan extension of X {\displaystyle X} along F {\displaystyle F} . Abstractly, the functor F {\displaystyle F} gives a pullback map F ∗ : [ B , C ] → [ A , C ] {\displaystyle F^{*}:[\mathbf {B} ,\mathbf {C} ]\to [\mathbf {A} ,\mathbf {C} ]} . When they exist, the left and right adjoints to F ∗ {\displaystyle F^{*}} applied to X {\displaystyle X} 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 ϵ {\displaystyle \epsilon } in the following diagram:
Formally, the right Kan extension of X {\displaystyle X} along F {\displaystyle F} consists of a functor R : B → C {\displaystyle R:\mathbf {B} \to \mathbf {C} } and a natural transformation ϵ : R F → X {\displaystyle \epsilon :RF\to X} that is terminal with respect to this specification, in the sense that for any functor M : B → C {\displaystyle M:\mathbf {B} \to \mathbf {C} } and natural transformation μ : M F → X {\displaystyle \mu :MF\to X} , a unique natural transformation δ : M → R {\displaystyle \delta :M\to R} is defined and fits into a commutative diagram:
where δ F {\displaystyle \delta _{F}} is the natural transformation with δ F ( a ) = δ ( F a ) : M F ( a ) → R F ( a ) {\displaystyle \delta _{F}(a)=\delta (Fa):MF(a)\to RF(a)} for any object a {\displaystyle a} of A . {\displaystyle \mathbf {A} .}
The functor R is often written Ran F X {\displaystyle \operatorname {Ran} _{F}X} . 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.
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