I'm trying to understand the construction of the geometric realization of a simplicial set. The goal is to solve the following problem:
For each simplicial set $S_\bullet$, find a space $|S_\bullet|$, called a geometric realization of $S_\bullet$, and a natural isomorphism $\hom(S_\bullet, \mathrm{Sing_\bullet}(-))\cong \hom(|S_\bullet|, -)$.
Here, $\mathrm{Sing}_\bullet$ is the singular simplicial set functor.
I have several questions:
- The solution can be summarized as follows: firstly, show that whenever a class of simplicial sets has a geometric realization, then each simplicial set that can be built as a colimit of these simplicial sets has a geometric realization too. Secondly, the $\Delta^n$s have a geometric realization by the Yoneda lemma. Thirdly, by the co-Yoneda lemma every simplicial set is the colimit of $\Delta^n$s. I have a hard time making the first step precise. The obvious way of writing it down is as follows:
Let $F\colon J\to \mathbf{sSet}$ and suppose each $F(j)$ has a geometric realization $|F(j)|$. Then $\mathrm{colim}_j |F(j)|$ is a geometric realization of $\mathrm{colim}_j F(j)$.
But what is $\mathrm{colim}_j |F(j)|$? It should be the colimit of some functor with domain $J$. But what functor? I'm tempted to say $|-|\circ F\colon J\to \mathbf{Top}$ -- but we are only in the process of constructing $|-|$!
- Suppose the above problem is solved, i.e., we have constructed $|S_\bullet|$ for each simplicial set $S_\bullet$. And in each case we have constructed a natural bijection $$\alpha_{S_\bullet}\colon \hom(S_\bullet, \mathrm{Sing_\bullet}(-))\cong \hom(|S_\bullet|, -).$$ How can we now built a left adjoint of $\mathrm{Sing}_\bullet$? (We already did it on objects and now we do it on morphisms.)
I heard that there is always a unique way to define the morphisms of the left adjoint. But I'm confused, for the following reason: let $S_\bullet \to T_\bullet$ be a map of simplicial set. We want to construct a morphism $|S_\bullet|\to |T_\bullet|$. By Yoneda, it suffices to construct a mapping $$\hom(|T_\bullet|, -)\to \hom(|S_\bullet|, -).$$ Here is how we do it: for each $X$, given $|T_\bullet|\to X$, use $\alpha_{T_\bullet}^{-1}$ to obtain a morphism $T_\bullet\to \mathrm{Sing}_\bullet(X)$. Now precompose with $S_\bullet\to T_\bullet$ to get a morphism $S_\bullet \to \mathrm{Sing}_\bullet(X)$. Finally, apply $\alpha_{S_\bullet}$ to get a morphism $|S_\bullet|\to X$.
But is this way of defining the functor $|-|$ on morphisms really unique? After all, it might depend on $\alpha_{S_\bullet}$!
- Is there a more elegant way to describe $|-|$ on morphisms? This reminds me of a quote I read in Riehl's A Leisurely Introduction to Simplicial Sets:
Uniqueness of the universal property will imply that $L$ is functorial, as is always the case when one uses a colimit construction to define a functor.
I have no idea what this means, but I'm very curious. Can you formulate the general principle behind that quote precisely?