# Joyal's CatLab Kan extensions

**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

This is work in progress!

# Contents

## Definitions

###### Definition

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

$\xymatrix{ \mathbf{A} \ar[ddr]_F \ar[rr]^{P} & & \mathbf{B} \ar[ddl]^{F'} \\ & \Rightarrow & \\ & \mathbf{M}, & }$

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

$\xymatrix{ F \ar[dr]_\beta \ar[r]^{\eta} & F' P \ar[d]^{\beta'\circ P} \\ & G P. }$

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

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

###### Definition

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

$\xymatrix{ \mathbf{A} \ar[ddr]_F \ar[rr]^{P} & & \mathbf{B} \ar[ddl]^{F'} \\ & \Leftarrow & \\ & \mathbf{M}, & }$

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

$\xymatrix{ F' P \ar[r]^{\epsilon} & F \\ \ar[u]^{\beta'\circ P} \ar[ur]_{\beta} G P. & }$

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

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

A pair $(F',\epsilon)$ is the right Kan extension of a functor $F:\mathbf{A}\to \mathbf{M}$ along a functor a functor $P:\mathbf{A}\to \mathbf{B}$ iff the pair $(F'^o,\epsilon^o)$ is the left Kan extension of the functor $F^o:\mathbf{A}^o\to \mathbf{M}^o$ along the functor $P^o:\mathbf{A}^o\to \mathbf{B}^o$.

Kan extensions can be taken in succession:

###### Proposition

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

$\xymatrix{ \mathbf{A} \ar[ddrr]_(0.6)F \ar[rr]^{P} & & \mathbf{B} \ar[dd]^(0.6){F'}^(0.4){ \quad \Rightarrow}_(0.4){\Rightarrow \quad } \ar[rr]^{Q} & & \ar[ddll]^(0.6){F''} \mathbf{C} \\ & & \quad \quad & & \\ & & \mathbf{M}. & & }$

If a natural transformation $\eta:F\to F'P$ exhibit the functor $F'$ as the left Kan extension of the functor $F$ along $P$, and a natural transformation $\eta':F'\to F''Q$ exhibit the functor $F''$ as the left Kan extension of the functor $F'$ along $Q$, then the natural transformation $(\eta'\circ P)\eta:F \to F''Q P$ exhibit the functor $F''$ as the left Kan extension of the functor $F$ along $Q P$

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