TITLE: Right Kan extension of $\mathcal{F} : \mathsf{\Delta} \rightarrow \mathsf{Top}$.
QUESTION [9 upvotes]: My question arose while studying something about Kan Extensions.
We know that we have the following diagram 
$$ 
\begin{array}{ccc} &&\mathsf{\Delta} & \xrightarrow{\mathcal{F}} & \mathsf{Top}\\ &&\mathcal{y} \searrow& 
& \nearrow Lan_{\mathcal{y}\mathcal{F}}\\
  &&& \mathsf{sSet}, &&&& 
\end{array} 
$$
where $Lan_{\mathcal{y}\mathcal{F}}$, stands for the left Kan Extension of $\mathcal{F}$ along the Yoneda Embedding $\mathcal{y}$, of the simplex category $\mathsf{\Delta}$ into the category of its presheaves. Now, because the category of topological spaces is co-complete and the simplex category is small we know the left Kan Extesnsion exists. However for $\mathcal{F}$ being the functor $$[n] \mapsto |\Delta^n|,$$
we get a rather well-known functor denoted by $|-|$, called geometric realization, which is left adjoint of the singularization functor,
$$ S : \mathsf{Top} \rightarrow \mathsf{sSet} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Y \mapsto S_Y = \mathsf{hom}_\mathsf{Top}( \mathcal{F}[-],Y).$$
Now, because of the particular structure these categories possess, means that the right Kan Extension of $\mathcal{F}$ along the Yoneda Embedding $\mathcal{y}$ does exist as well, and moreover we have a certain adjunction of the form $Lan_{\mathcal{y}} \dashv - \circ \mathcal{y} \dashv Ran_{\mathcal{y}}.$ With all the above it is clear what the left Kan extension is, however I'm not sure about the right one. Can you explain what is this functor? (If I haven't done something wrong so far :))
Also for the first adjunction I mentioned, $|-| \dashv S$, is there any categorical theoretic insight (an argument arising from Kan Extension theory for instance) that imposes these functors being adjoints?

REPLY [3 votes]: I'll try to enlight a little the obscure point. (Co)Ends are very powerful because it allows to make quick computation but it sometimes hides the meat.
The first thing to recall is that every presheaf $P$ on the category $\mathsf C$ can be written as the colimit of the Yoneda embedding over the category of its elements (it is a corollary of Yoneda's lemma):
$ \DeclareMathOperator*{\colim}{colim} $
$$ P \simeq \colim_{x:a \to P} y(a) $$
So if $Q$ is a presheaf on $\mathsf D$ that commutes with limits, we have the following:
$$\begin{split} \hat{\mathsf C}(P,Q\mathcal F) \simeq \lim_{x:a \to P} \hat{\mathsf C}(y(a),Q\mathcal F) \simeq \lim_{x:a \to P} Q(\mathcal F(a)) \\\simeq Q(\colim_{x:a \to P}\mathcal F(a)) \simeq Q(\operatorname{Lan}_y\mathcal F(P)) \end{split}$$
where the last isomorphism is from the usual formula of left Kan extension.
Now apply that with $Q$ being the functor $\mathsf D(-,d)$ (which commutes with limits), and you end up with: 
$$ \hat{\mathsf C}(P,N_\mathcal F(d)) \simeq \mathsf D(\operatorname{Lan}_y\mathcal F(P),d)$$

This is more or less exactly the same as the (co)end chain, but I hope it brings some light to people not at ease with (co)end yoga.
Fun facts. Left Kan extensions really are everywhere in this framework:

The nerver $N_{\mathcal F}$ is the left kan extension of $y$ along $\mathcal F$.
The fact that every presheaf is a canonical colimit can be express by the following: the identity functor on $\hat{\mathsf C}$ is the left Kan extension of $y$ along $y$.

This can be a clue of why the right Kan extension of $\mathcal F$ does not have good property in general compared to the left one.