Joyal's CatLab
Kan extensions

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Contents

Main definitions

Definitions

Definition

Recall that the left Kan extension of a functor F:AMF:\mathbf{A}\to \mathbf{M} along a functor P:ABP:\mathbf{A}\to \mathbf{B} is defined to be a functor F:BMF':\mathbf{B}\to \mathbf{M} equipped with a natural transformation η:FFP\eta:F\to F'P,

which is universal in the following sense: for every functor G:BMG:\mathbf{B}\to \mathbf{M} and every natural transformation β:FGP\beta: F\to G P, there exists a unique natural transformation β:FG\beta':F'\to G such that (βP)η=β(\beta'\circ P)\eta=\beta,

We shall say that the natural transformation η:FFP\eta:F\to F'P exibit FF' as the left Kan extension of FF along PP.

Recall that a left Kan extension (F,η)(F',\eta) is said to be absolute if it stays a left Kan extension after postcomposing it with any functor, that is, if the pair (UF,Uη)(U F',U\circ \eta) is the left Kan extension of the functor UFU F for any functor U:MNU:\mathbf{M}\to \mathbf{N}.

Definition

Dually, the right Kan extension of a functor F:AMF:\mathbf{A}\to \mathbf{M} along a functor P:ABP:\mathbf{A}\to \mathbf{B} is defined to be a functor F:BMF':\mathbf{B}\to \mathbf{M} equipped with a natural transformation ϵ:FPF\epsilon:F'\circ P \to F

which is universal in the following sense: for every functor G:BMG:\mathbf{B}\to \mathbf{M} and every natural transformation β:GPF\beta: G\circ P \to F, there exists a unique natural transformation β:GF\beta': G\to F' such that ϵ(βP)=β\epsilon (\beta'\circ P)=\beta,

We shall say that the natural transformation ϵ:FPF\epsilon:F'\circ P \to F exibit FF' as the right Kan extension of FF along PP.

Recall that a right Kan extension (F,ϵ)(F',\epsilon) is said to be absolute if it stays a right Kan extension after postcomposing it with any functor, that is if the pair (UF,Uϵ)(U F',U\circ \epsilon) is the right Kan extension of the functor UFU F for any functor U:MNU:\mathbf{M}\to \mathbf{N}.

A pair (F,ϵ)(F',\epsilon) is the right Kan extension of a functor F:AMF:\mathbf{A}\to \mathbf{M} along a functor a functor P:ABP:\mathbf{A}\to \mathbf{B} iff the pair (F o,ϵ o)(F'^o,\epsilon^o) is the left Kan extension of the functor F o:A oM oF^o:\mathbf{A}^o\to \mathbf{M}^o along the functor P o:A oB oP^o:\mathbf{A}^o\to \mathbf{B}^o.

Kan extensions can be taken in succession:

Proposition

Suppose that we have a diagram of categories, functors and natural transformations,

If a natural transformation η:FFP\eta:F\to F'P exhibit the functor FF' as the left Kan extension of the functor FF along PP, and a natural transformation η:FFQ\eta':F'\to F''Q exhibit the functor FF'' as the left Kan extension of the functor FF' along QQ, then the natural transformation (ηP)η:FFQP(\eta'\circ P)\eta:F \to F''Q P exhibit the functor FF'' as the left Kan extension of the functor FF along QPQ P

Revised on April 3, 2013 at 02:09:13 by Zhen Lin