Simplicial set associated to a topological space

Question

Let $X$ be a topological space. We want to associate a simplicial set i.e., a functor $\Delta\rightarrow \underline{\text{Set}}$.

Here $\Delta$ is the category whose

• object set is$ \{[0],[1],[2],\cdots\}$ where $[n]=\{0,1,2\cdots,n\}$, $[0]=\{0\}$.
• by an arrow $[m]\rightarrow [n]$ we mean a map $f:\{0,1\cdots,m\}\rightarrow \{0,1,2,\cdots,n\}$ such that $f$ is non-decreasing i.e., $f(i)\leq f(j)$ if $i\leq j$.

Fix $n$.

Let $e_0=(1,0,\cdots,0)\in \mathbb{R}^{n+1},e_1=(0,1,0\cdots,0)\in \mathbb{R}^{n+1}, .... , e_n=(0,\cdots,0,1)\in \mathbb{R}^{n+1}$.

Let $\Delta_n$ be the convex combination of $\{e_0,\cdots,e_n\}\subseteq\mathbb{R}^{n+1}$. This has induced topology from $\mathbb{R}^{n+1}$.

By convention, $\Delta_0$ is just a singleton, say $\{0\}$.

For this topological space $X$, we need to define a functor $\Delta\rightarrow \underline{\text{Set}}$.

• For $[0]$, consider the set $\text{Hom}_{\text{Top}}(\Delta_0,X)$ which is collection of all continuous maps from $\{0\}$ to $X$ which is precisely the points of $X$.

• For $[1]$, consider the set $\text{Hom}_{\text{Top}}(\Delta_1,X)$ which is collection of all paths in $X$.

Similarly, for $[n]$, consider $\text{Hom}_{\text{Top}}(\Delta_n,X)$. For convenience, set $\Delta_n(X)=\text{Hom}_{\text{Top}}(\Delta_n,X)$.

Given an arrow $f:[m]\rightarrow [n]$ we have a continuous map $f_{\Delta}:\Delta_m\rightarrow \Delta_n$.

This gives maps $\Delta_n(X)\rightarrow \Delta_m(X)$

• $\theta:\Delta_n\rightarrow X$ an element of $\Delta_n(X)$ is assigned to the composition $\theta\circ f_{\Delta}:\Delta_m\rightarrow \Delta_n\rightarrow X$ an element of $\Delta_m(X)$.

This gives a contravariant functor $\Delta\rightarrow \underline{\text{Set}}$. This is the simplicial set I am associating to a topological space.

This I heard from some conversation and filled some gaps.

Question is

Did I understand till here correctly?

Answer 1

Your thoughts are correct, and it is good that you tried to fill the gaps by yourself. If you are further interested in this subject, the functor $$S : \mathbf{Top} \to \mathbf{sSet}$$ is known as the singular functor. See for example https://en.wikipedia.org/wiki/Simplicial_set, The realization functor is left adjoint to the singular functor or have a look into a good textbook.