Joyal's CatLab Kan extensions

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Category theory

Category theory

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Main definitions

Definitions

Definition

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

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

We shall say that the natural transformation η:F→F′P\eta:F\to F'P exibit F′F' 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′,U∘η)(U F',U\circ \eta) is the left Kan extension of the functor UFU F for any functor U:M→NU:\mathbf{M}\to \mathbf{N}.

Definition

Dually, the right Kan extension of a functor F:A→MF:\mathbf{A}\to \mathbf{M} along a functor P:A→BP:\mathbf{A}\to \mathbf{B} is defined to be a functor F′:B→MF':\mathbf{B}\to \mathbf{M} equipped with a natural transformation ϵ:F′∘P→F\epsilon:F'\circ P \to F

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

We shall say that the natural transformation ϵ:F′∘P→F\epsilon:F'\circ P \to F exibit F′F' 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′,U∘ϵ)(U F',U\circ \epsilon) is the right Kan extension of the functor UFU F for any functor U:M→NU:\mathbf{M}\to \mathbf{N}.

A pair (F′,ϵ)(F',\epsilon) is the right Kan extension of a functor F:A→MF:\mathbf{A}\to \mathbf{M} along a functor a functor P:A→BP:\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 o→M oF^o:\mathbf{A}^o\to \mathbf{M}^o along the functor P o:A o→B 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:F→F′P\eta:F\to F'P exhibit the functor F′F' as the left Kan extension of the functor FF along PP, and a natural transformation η′:F′→F″Q\eta':F'\to F''Q exhibit the functor F″F'' as the left Kan extension of the functor F′F' along QQ, then the natural transformation (η′∘P)η:F→F″QP(\eta'\circ P)\eta:F \to F''Q P exhibit the functor F″F'' as the left Kan extension of the functor FF along QPQ P

Revised on November 20, 2020 at 21:56:01 by Dmitri Pavlov