# Realization of simplicial sets

Consider a simplicial set $X$, i.e. a contravariant functor from $\mathbf{\Delta} \to \mathbf{Set}$ where $\mathbf{\Delta}$ has as objects $[n]:=\{0, \cdots, n \}$ for all $n \in \mathbb{N}$ and as morphisms order-preserving set maps. Denote the simplicial set represented by $[n]$ as $\Delta[n]$.

There is a functor $F:\mathbf{\Delta} \to \mathbf{Top}$ which sends $[n] \mapsto \Delta^n$ where $\Delta^n$ is the standard $n$-simplex. We would like to extend $F$ to a functor $||$ on simplicial sets, where extend means that $||$ agrees with $F$ on the image of the Yoneda embedding $Y$. There are two ways to do this, and I would like to prove that they are the same.

1: By general nonsense, there is a natural bijection (given by restriction and Yoneda) between

\begin{align} \{ Functors \;\mathbf{C} \to \mathbf{D} \} \leftrightarrow \{\text{Left adjoint functors } \hat{\mathbf{C}} \to \mathbf{D} \} \end{align}

where $\hat{C}$ is $\operatorname{Fun}(\mathbf{C},\mathbf{Set})$, the category of presheaves on $C$. To be more precise given a simplicial set $X$, write it as a colimit of representable functors

\begin{align} X \cong \varprojlim_{\substack{\Delta[n] \to X \\ \text{in} \; \mathbf{\Delta} \; \downarrow X}} \Delta[n] \end{align}

and define

\begin{align} |X|\cong \varprojlim_{\substack{\Delta[n] \to X \\ \text{in} \; \mathbf{\Delta} \; \downarrow X}} F(\Delta[n]) \end{align}

2: Given a simplicial set $X$, define

\begin{align} |X|'&=\left(\coprod X_n \times \Delta^n \right)/ \sim \end{align} where $X_n:=X[n]$ is given the discrete topology and where $(\sim)$ is an equivalence relation given by $(x,p) \sim (y,q)$ if either \begin{align} d_i(x)&=y \qquad \text{and } \qquad d^i(q)=p; \; \text{or} \\ s_j(x)&=y \qquad \text{and } \qquad s^j(q)=p. \end{align} where $d^i, s^i$ are the face and degeneracy maps induced by $F$ and $d_i, s_i$ are the face and degeneracy maps induced by $X$.

I can prove that $|\Delta[n]|'\cong \Delta^n$ but the rest of the proof eludes me. Is it enough to prove that $||'$ preserves colimits?

• It might be easier to directly prove that a left Kan extension (your first definition) can be computed by a coend (your second definition when you take into account that $\Delta$ is generated by cofaces and codegeneracies). Commented Sep 1, 2015 at 15:08
• Two years after but still worth mentioning since someone else might pass by to get a look since it's a good question. You have a typo where you denote accidentally $\varprojlim$ instead of $\varinjlim$.
– user321268
Commented Jun 26, 2017 at 22:09

Yes, it is enough to prove ||' preserves colimits, as then both || and ||' would extend the same functor $[n]\to \Delta^n$ to $Fun(\Delta^{op} ,Top)$ and any two such functors must be naturally isomorphic. You can show that ||' preserves colimits by showing that ||' is left adjoint to the singular functor $S$.