I know that it is a trivial question. And it can be found in any text on simplicial sets. (So, sorry about that). But I'm studying the Jardine's Book on Simplicial Homotopy Theory (Jardine and Goerss). And I couldn't understand the construction that they made there.
How can we describe the simplicial n-sphere $ \partial \Delta ^n $ as a coequalizer?
Thank you in advance!
I suppose you are referring to the remark before Lemma 3.1 in Chapter I. The point is that $\partial \Delta^n$ is obtained by gluing $n + 1$ copies of $\Delta^{n-1}$ along $\frac{1}{2} (n + 1) n$ copies of $\Delta^{n-2}$. Draw some pictures for $n = 2$ and $n = 3$ if that doesn't seem obvious to you. The coequaliser is just a formal way of expressing this geometric construction.
Incidentally, $\partial \Delta^n$ can also be described as the largest simplicial subset of $\Delta^n$ that does not contain the element $\mathrm{id} : [n] \to [n]$.