Simplicial sets

On the set of vertices of an $n$-simplex we have an ordering coming from the order on the set $[n]=\{0, \hdots, n\}$.

./Chapter1/1simplex_ordered.eps

./Chapter1/2simplex_ordered.eps

./Chapter1/3simplex_ordered.eps

Degeneracy map which does not preserve the ordering on vertices is not allowed. For example if $n=2$ we have two allowed degeneracies $s_0$, $s_1$

./Chapter1/degeneracy1.eps

./Chapter1/degeneracy2.eps

./Chapter1/degeneracy3.eps

The face and degeneracy maps satisfy the following identities
$$\begin{align*} \dl_j\dl_i &= \dl_i\dl_{j-1},\quad i<j \\ \sg_j\sg_i &= \sg_i\sg... ...i<j \\ \id & i=j,\;i=j+1 \\ \dl_{i-1}\sg_j & i>j+1\end{cases}} \\ \end{align*}$$

Now suppose we have a simplicial set $K_{\bullet}$. For all $x\in K_n$ we take a simplex $\Dl^n$ and we will build a topological space out of these data.

The geometric realization of a simplicial set is the following topological space

$$\big\vert X_{\bullet}\big\vert:=\coprod_{n\geq 0}X_n\times \Dl^n/ \sim,$$

where the equivalence relation $\sim$ is defined as follows. We identify $(x, \dl_i t)\in X_n\times\Dl^n$ with $(d_i x, t)\in X_{n-1}\times \Dl^{n-1}$ for any $x\in X_n$, $t\in \Dl^{n-1}$ and $(x, \sg_j t)\in X_n\times \Dl^n$ with $(s_jx, t)\in X_{n+1}\times \Dl^{n+1}$ for any $x\in X_{n-1}$ and $t\in \Dl^{n+1}$. The topology on $\vert X_{\bullet}\vert$ is the quotient topology.

There exists a simplicial category $\Dl$, whose objects are finite ordered sets $[n]=\{0, \hdots, n\}$, and morphism $\Mor([n], [m])$ are nondecreasing set maps.

The category $\Dl$ can be described by generators and relations. As generators we take face and degeneracy maps
$$\begin{align*} \dl_i\:[n-1]&\to [n] \\ \sg_j\:[n+1]&\to [n] \end{align*}$$
and relations are as before
$$\begin{align*} \dl_j\dl_i &= \dl_i\dl_{j-1},\quad i<j \\ \sg_j\sg_i &= \sg_i\sg... ...i<j \\ \id & i=j,\;i=j+1 \\ \dl_{i-1}\sg_j & i>j+1\end{cases}} \\ \end{align*}$$






If all $X_n$ are topological spaces, and the face and degeneracy maps are continuous, then we call $X_{\bullet}$ a simplicial space. Then the geometric realization is defined as before, but we keep track of the topology of $X_n$ in the construction.

$$\big\vert X_{\bullet}\big\vert:=\coprod_{n\geq 0}X_n\times \Dl^n/ \sim,$$

$$\begin{align*} (x, \dl_i t)&\sim (d_i x, t) \\ (x, \sg_j t)&\sim (s_jx, t) \end{align*}$$