# Pushout diagram of the $n$-skeleton of a simplicial set

In Goerss, Jardine book "Simplicial Homotopy Theory" at the very beginning of the book, where they prove that given a simplicial set $X$ the geometric realization is a CW-complex, there is a pushout diagram constructed $$\begin{array} A \coprod_{x \in NX_n} \partial \Delta^n & \stackrel{}{\longrightarrow} & sk_{n-1}(X) \\ \downarrow{} & & \downarrow{} \\ \coprod_{x \in NX_n}{\Delta}^n & \stackrel{}{\longrightarrow} & sk_n(X), \end{array}$$ where $NX_n$ is the set of non-degenerate simplices of $X_n$. In my head intuitively I want to believe that, $sk_n(X)$ is kind of forgetful functor that prunes $X$ from $n+1$ and upwards. For those who prefer a more rigorous definition, we can think the functor $\mathsf{sk_n}:\mathsf{Set}^{\mathsf{\Delta^{op}}} \rightarrow\mathsf{Set}^{\mathsf{\Delta^{op}}},$ as the composition of the functors induced by the inclusion $\mathsf{\Delta}_{n \geq 0} \hookrightarrow \mathsf{\Delta}$ on the functor categories after applying $[ - , \mathsf{Set}]$, along with its induced Left Kan Extension, which turns out to be left adjoint. So this compotion gives an elegenant way to forget everything over $n$.

So, my question has to do with why/how we come up with this diagram and what's the idea behind that? Why the pushout of this diagram is the $n$-skeleta of $X$, and if this is somehow related with the functor $\mathsf{sk_{n}}$?

• This is covered later on in the book (pg 355, just after proposition 1.4). It's part of the more general theory of Reedy categories. You might like to check out the section in Riehl's book "Categorical Homotopy Theory", or her paper with Verity "The Theory and practice of Reedy Categories". – Tyrone Jun 27 '17 at 7:24