Functoriality of Kan extensions

# Functoriality of Kan extensions ## Weighted colimits $V$ a complete and cocomplete symmetric monoidal category Recall that, given $\phi:J^{op}\to V$ and $D:J\to C$, the weighted colimit is defined, if it exists, as the (unique up to iso) solution of $$C(colim_\phi D,B) \cong [C^{op},V](\phi,C(D-,B))$$ ## Left Kan extensions $K:A\to C$ and $G:A\to B$ Left Kan-extensions are defined as $$(Lan_KG)c = \int_a C(Ka,c)\bullet Ga = colim_{C(K-,c)}G$$ Spelling this out the definition of a weighted colimit, this means that $Lan_KG$ is the (up to iso) unique solution of the equation $$B((Lan_KG)c,b) \cong [A^{op},V](C(K,c),B(G,b))$$ **Remark:** The equation above is quite intuitive: Natural transformations between the cone under $c$ and the cone under $b$ are classified by $Lan_KG$. This is really the same idea as always when it comes to colimits. With $b=Lan_KG$, we obtain the counit $$C(K-,c)\to B(G,(Lan_KG)c)$$ ## Functoriality of Left Kan extensions $f:c\to c'$ induces $(Lan_KG)f : (Lan_KG)c\to (Lan_KG)c'$ defined as follows. First, we know, see [(Kelly,(1.29))](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf), that $C(Ka,-)$ is a functor. That is, there is an arrow $$C(Ka,-): C(c,c') \to [C(Ka,c),C(Ka,c')]$$ This, in turn, induces an arrow $$(Lan_KG)c \to (Lan_KG)c'$$ via (abbreviating $L=Lan_KG$) \begin{align} C(c,c') & \rightarrow [A^{op},V](C(K,c),C(K,c')) & C(K,-) \textrm{ is a functor} \\ & \to [A^{op},V](C(K,c),B(G,Lc')) & \textrm{counit} \\ & \cong B(Lc,Lc') & \textrm{Def of colimit} \end{align}  --- References: Chapter 4.1 of [Kelly](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf).