I recently encountered the definitions of functors between Simplicial set and topological space. But I dont understand these definitons . For functor: Sset-> Top, we first have this def : The geometric realization |X| of a simplicial set X is the topological space constructed in the following way. Give X the discrete topology and let $\hat{X} = \cup_{n\geq0}X_n\times \bigtriangleup^n$ ,where $\bigtriangleup^n$ is the standard topological n-simplex.
Now define an equivalence relation by: $(d_ix_n, u_{n-1}) ~ (x_n, \partial_iu_n-1)$ for $x_n\in X_n$ and $u_{n-1}\in\bigtriangleup^{n-1}$.
$(s_ix_n, u_{n+1}) ~ (xn, {\sigma}{i}u{n+1})$ for
$x_n\in X_n$ and $u_{n-1}\in \bigtriangleup^n-1$. Now |x|= $\hat{X}$ /~.
Why |x| is a topological space ? And how to construct the functor: Sset-> Top from this def?
For the functor in reverse direction.The singular complex functor is the functor Sing : Top -> sSet that sends a topological space A to the graded set: Sing(A) ={f:$\bigtriangleup^n$->A|f continuous }
I am wondering if there is something wrong with it as this defition of Sing(A) seems to send graded set to A (not A to graded set , which is what we want ...)
Could anyone help me and give a detailed explanation of these two functors ? Thanks in advance
