Kan Extensions
Table of Contents
The general idea of Kan extensions is to universally extend a functor \( F \colon \mathcal C \rightarrow \mathcal D \) along a functor \( i \colon \mathcal C \rightarrow \mathcal D \):
\begin{equation*} \xymatrix{ \mathcal C \ar[r]^F \ar[d]_i & \mathcal D \\ \mathcal C ' \ar@{-->}[ur]} \end{equation*}Since there are to ways to ask for a universal extension, we get the notion of a left Kan extension and of a right Kan extension. This extension problem (or maybe we abstractly define an extension problem as such a lifting problem) is a special lifting problem:
\begin{equation*} \xymatrix{ \mathcal C \ar[r]^F \ar[d]_i & \mathcal D \ar[d] \\ \mathcal C ' \ar@{-->}[ur] \ar[r] & 1} \end{equation*}Hence we can also ask for the dual notion and arrive a the notion of a Kan lift:
\begin{equation*} \xymatrix{ \emptyset \ar[r] \ar[d] & \mathcal D ' \ar[d] \\ \mathcal C \ar@{-->}[ur] \ar[r] & \mathcal D} \end{equation*}Of course we again get left Kan lifts and right Kan lifts. In full generality we might ask for an extension of \( F \colon \mathcal C \rightarrow \mathcal D \) along \( i \colon \mathcal C \rightarrow \mathcal C ' \) that also lifts \( F' \colon \mathcal C ' \rightarrow \mathcal D ' \) along \( p \colon \mathcal D \rightarrow \mathcal D ' \):
\begin{equation*} \xymatrix{ \mathcal C \ar[r]^F \ar[d]_i & \mathcal D \ar[d]^p \\ \mathcal C ' \ar@{-->}[ur] \ar[r] _{F'} & \mathcal D} \end{equation*}But in the following we only discuss the extension problems.
1. Definition of Kan Extensions
We give the definition using universal properties (we think we are biased towards limits and colimits and in general universal properties).
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( i \colon \mathcal C \rightarrow \mathcal C ' \) be functors. A left Kan extension of \( F \) along \( i \) consists of
- a functor \( \Lan_i F \colon \mathcal C ' \rightarrow \mathcal D \),
- a natural transformation \( \eta \colon F \Rightarrow \Lan_i F \circ i \),
such that
for every functor \( G \colon \mathcal C ' \rightarrow \mathcal D \) and natural transformation \( \alpha \colon F \Rightarrow G \circ i \) there is a unique natural transformation \( \beta \colon \Lan_i F \Rightarrow G \) such that \( \alpha = (\beta \triangleright i) \bullet \eta \) or in diagrams:
\begin{equation*} \xymatrix{ \mathcal C \ar[r]^F="F" \ar[d]_i & \mathcal D \\ \mathcal C ' \ar[ur]_G \ar@{=>}"F";"2,1"} = \xymatrix{ \mathcal C \ar[r]^F="F" \ar[d]_i & \mathcal D \\ \mathcal C ' \ar[ur]^{}="Lan" \ar@/_1.5pc/[ur] _G="G" \ar@{=>}"F";"2,1"^{\eta } \ar@{=>}"Lan";"G"^{\exists!}} \end{equation*}
The following lemma is just a reformulation of the universal property in terms of representable functors.
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( i \colon \mathcal C \rightarrow \mathcal C ' \) be functors. Then \( \Lan_i F \) is a left Kan extension of \( F \) along \( i \) if and only if \( \Hom(F,i^*(-)) \) is (co)represented by \( \Lan_i F \), ie
\begin{equation*} \Hom _{\Fun(\mathcal C ',\mathcal D)}(\Lan_i F,-) \cong \Hom _{\Fun(\mathcal C,\mathcal D)}(F,i^*(-)) \end{equation*}The natural transformation \( \eta \colon F \Rightarrow \Lan_i F \circ i \) induces the bijection by \( 1 _{\Lan_i F} \mapsto \eta \). Given the bijection \( \eta \) is recovered as the image of \( 1 _{\Lan_i F}\).
Analogously we define right Kan extensions.
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( i \colon \mathcal C \rightarrow \mathcal C ' \) be functors. A right Kan extension of \( F \) along \( i \) consists of
- a functor \( \Ran_i F \colon \mathcal C ' \rightarrow \mathcal D \),
- a natural transformation \( \eta \colon \Ran_i F \circ i \Rightarrow F \),
such that
for every functor \( G \colon \mathcal C ' \rightarrow \mathcal D \) and natural transformation \( \alpha \colon G \circ i \Rightarrow F \) there is a unique natural transformation \( \beta \colon G \Rightarrow \Ran_i F \) such that \( \alpha = \eta \bullet (\beta \triangleright i) \) or in diagrams:
\begin{equation*} \xymatrix{ \mathcal C \ar[r]^F="F" \ar[d]_i & \mathcal D \\ \mathcal C ' \ar[ur]_G \ar@{=>}"2,1";"F"} = \xymatrix{ \mathcal C \ar[r]^F="F" \ar[d]_i & \mathcal D \\ \mathcal C ' \ar[ur]^{}="Ran" \ar@/_1.5pc/[ur] _G="G" \ar@{=>}"2,1";"F"^{\eta } \ar@{=>}"G";"Ran"^{\exists!}} \end{equation*}
Again we can reformulate this using representable functors.
Let \( F \colon \mathcal C \rightarrow \mathcal D \) and \( i \colon \mathcal C \rightarrow \mathcal C ' \) be functors. Then \( \Ran_i F \) is a right Kan extension of \( F \) along \( i \) if and only if \( \Hom(i^*(-),F) \) is (co)represented by \( \Ran_i F \), ie
\begin{equation*} \Hom _{\Fun(\mathcal C ',\mathcal D)}(-,\Ran_i F) \cong \Hom _{\Fun(\mathcal C,\mathcal D)}(i^*(-),F) \end{equation*}The natural transformation \( \eta \colon \Ran_i F \circ i \Rightarrow F \) induces the bijection by \( 1 _{\Ran_i F} \mapsto \eta \). Given the bijection \( \eta \) is recovered as the image of \( 1 _{\Ran_i F}\).
Above we defined the left/right Kan extension along \( i \colon \mathcal C \rightarrow \mathcal C ' \) for one functor \( F \colon \mathcal C \rightarrow \mathcal D \). We can also ask for left/right Kan extensions along \( i \) for all functors from \( \mathcal C \) to \( \mathcal D \) simultaneously. Hence we arrive a the notion of a (global) left/right Kan extension along \( i \).
Let \( i \colon \mathcal C \rightarrow \mathcal C ' \) be a functor. Let \( \mathcal D \) be a category. A left Kan extension along \( i \) consists of
- a functor \( \Lan_i \colon \Fun(\mathcal C,\mathcal D) \rightarrow \Fun(\mathcal C ', \mathcal D) \),
such that
- \( \Lan_i \) is a left adjoint of \( i^* \colon \Fun(\mathcal C ',\mathcal D) \rightarrow \Fun(\mathcal C,\mathcal D) \), ie