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I'm starting to learn a bit about simplicial sets and related concepts. I have seen a few definitions of the $n$-skeleton of a simplicial set, and I'm trying to relate them, of course they're equivalent but it's not direct as to why, intuitively I sort of see why they should be, but that's about it.

First definition / properties (from Goerss & Jardine p.8) : For a simplicial set $X$, "define $sk_nX$ to be the subcomplex of $X$ which is generated by the simplices of $X$ of degree $\leq n$." They also give the following pushout diagram, where $NX_n$ is the set of non degenerate simplices of $X_n$.

$$\begin{array}{ccc} \coprod_{x \in NX_n} \partial \Delta^n & \longrightarrow & sk_{n-1}(X) \\ \downarrow{} & & \downarrow{} \\ \coprod_{x \in NX_n}{\Delta}^n & \longrightarrow & sk_n(X), \end{array} $$

I've also seen this diagram being given as a definition, i.e. take $sk_0X = X_0$ and defining $sk_n$ inductively by the pushout diagram above.

Second definition (taken from the nLab, or wikipedia) : Let $\Delta_n$ be the full subcategory of $\Delta$ consisting of the objects $[0], [1],\ldots, [n]$, we have the inclusion $\Delta_{\leq n}\hookrightarrow \Delta$ which induces a truncation functor $$ tr_{\leq n} : sSet:=[\Delta^{op},Set]\to [\Delta_{\leq n}^{op},Set]:=sSet_{\leq n}$$ This functor admits a left adjoint by Kan extension, hence we get a functor : $$sk_n : sSet_{\leq n}\to sSet $$ called the n-skeleton. Finally we denote $\mathbf{sk}_n:=sk_n\circ tr_{\leq n}:sSet\to sSet$.

I'm guessing that $sk_nX$ from the first definition is $\mathbf{sk}_nX$ from the second, considering $X$ is a simplicial set and the source of $sk_n$ isn't $sSet$ but that of $\mathbf{sk}_n$ is.

I'm sort of used to the first definition, using some intuition from CW complexes, but it's not clear how to see that the two definitions say the same thing (this is partly due to "generated" doing some heavy lifting in the first definition, I'm not sure what is meant here).

Taking the adjunction from the second definition we get an isomorphism $$Hom_{sSet}(\mathbf{sk}_nX,Y):=Hom_{sSet}(sk_n\circ tr_{\leq n}X,Y)\simeq Hom_{sSet_{\leq n}}(tr_{\leq n}X,tr_{\leq n}Y). $$ I suppose this does indeed say that $\mathbf{sk}_nX$ is solely determined by the simplices of degree $\leq n$, but given that I don't know what Goerss and Jardine really mean with "generated by the simplices of X of degree $\leq n$" I'm not sure if more can be said.

Furthermore, taking either definition, I simply cannot really see why we have the push out diagram above. It certainly doesn't help that I don't quite understand what the maps $$ \coprod_{x \in NX_n} \partial \Delta^n \to sk_{n-1}X, \quad \coprod_{x \in NX_n} \partial \Delta^n\to \coprod_{x \in NX_n} \Delta^n$$ are supposed to be. I know this diagram can be proven at higher level of generality as done in Goerss & Jardine p.355 but I sure hope there's a simpler way to see this, because at that point in the book they've developed a bit more machinery to deal with that.

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$\newcommand{\sk}{\operatorname{Sk}}$I found that the Kerodon was a useful companion text when I was struggling through the first chapter of Goerss-Jardine. Some warning: Lurie defines certain key concepts differently, e.g. the notion of weak equivalence, and it takes some abstract nonsense and further knowledge to justify the equality of these definitions. But for some experience with the "combinatorics" and handling of things like skeleta, it was useful.

Given a simplicial set $X$, we can explicitly define its $n$-skeleton $\sk_nX$ as the simplicial set with $m$-simplices: $$\sk_nX_m:=\{\sigma\in X_m:\exists k\le n,\,f\in\Delta(m,k),\,\tau\in X_k,\,\sigma=X_f(\tau)\}$$That is, $\sk_nX_m\subseteq X_m$ consists of all the $m$-simplices which can be reached from dimension $n$ or lower. $\sk_nX_m=X_m$ if $0\le m\le n$. $\sk_nX$ inherits the action on arrows from $X$ and subset inclusions make $\sk_nX$ a subobject of $X$.

Then you can have lots of "fun" trying to verify the adjunctions stated in nLab and verifying the pushout square, alleged by Lurie, G/J, nLab etc. is actually a pushout. I found this last part especially tedious, but I did prove it in my notes (every text online treated it as a trivial/uninteresting detail and skipped over it, save for the Kerodon whose proof was so spectacularly terse I still have no idea what it meant). I asked and self-answered several questions to that effect in November last year.

As for your specific question about what the arrows in this pushout diagram even mean, I answered this a while ago. Take $X=\emptyset$ in that post to get your case as a special case.

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"Generated by" means what it usually does: there is a unique smallest sub-simplical set containing all the simplices of dimension less than $n$. Explicitly, its $m$-simplices for $m>n$ are those that can be written as a degenerate image $s_{i_1} \circ s_{i_2} \circ \ldots \circ s_{i_{m - n}} (x) $ of an $n$-simplex $x$, with the simplicial identities showing this is indeed a sub-object.

Recall that, by the Yoneda Lemma, $Δ^{n}$ can be viewed as the free simplicial set on an $n$-simplex, so maps from $\amalg_{x\in NX_{n}}Δ^{n}$ are tantamount to a choice of $n$-simplices for each element of $NX_{n}$. There is an obvious such map $\amalg_{x\in NX_{n}}Δ^{n} \rightarrow X$ and the restriction to $\amalg_{x\in NX_{n}}∂Δ^{n}$ lands in $sk_{n - 1}X$, giving the horizontal map. The vertical one is simply the coproduct of the inclusions. With this in mind, the pushout diagram says the following: from $sk_{n - 1}X$, freely add one $n$-simplex for each $n$-simplex of $X$ that has not been added yet (is non-degenerate) and mod out by identifying the $n-1$ skeleta of these as they are attached in $X$. The fact that no more relations are needed follows from the ability to write degenerate simplicies uniquely as the degenerate image of a non-degenerate simplex (see Hatcher Proposition A.19).

The Kan extension definition is the usual adjoint you want for an object generated by a set. It is clear that there is at most one extension of a map of truncated simplicial sets and one can be defined well by the unique writing property mentioned above.

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