Let Δ be the category of finite ordinal numbers with order-preserving maps, i.e., Δ consists of objects strings
A morphism f:n→[m] is an order-preserving function (a functor) and we can think of the morphism like diagrams where arrows don't cross.
Along all the functors in Δ there are two special classes: Coface and codegeneracy map which satisfy the cosimplicial identities and are a set of relations and generators for Δ in the sense that every arrow f:n→[m] can be uniquely written. And with this information we get that a simplicial set is the same thing as a graded set Yn, with the face and degeneracy map that satisfy the simplicial identities. This gives us the classical way to write a simplicial set. Here is the list of the identities in the pictures:
I need help undestanding what these maps do, the coface, codegeneracy, face and degeneracy maps. And why they need to satisfy the identities. In general ¿what information does it give us? Why it gives us Hom_Δ(_,n Thank you.
Let's give some geometric intuition - I'll assume you're familiar with, and have an intuitive grasp of, the geometric notions of simplicial complexes (or, slightly more generally, delta-complexes, which already are somewhat combinatorial). Simplicial sets are just a generalization of this notion which combinatorializes it so we don't talk about maps of actual simplices to actual spaces, which it happens have a very nice categorical description as a contravariant functor $X:\Delta^{\text{op}}\to \textbf{Set}$. The category of these (nothing but the category of presheaves on $\Delta$) is then the category of simplicial sets.
So working backwards, what nice maps might we want to see on the category of simplicial sets? Well, on simplicial complexes, we have the familiar boundary map $\partial$, but we want to be able to access each face of the boundary of a complex/set directly - hence the face maps $\partial_1,\partial_2,\ldots,\partial_n$ which map a simplex to the face which lacks the $i$th vertex on a simplex. We have $n$ face maps on a $n$-simplex, but the face maps all map to a $n-1$-simplex, suggesting a simplicial set $X$ needs to be "graded" somehow - this is the reason we think of simplex category of ordinals: $[n]$ will correspond to a $n$-simplex, and the face map will send it to something corresponding to $[n-1]$.
But $\Delta$ itself only has one copy of each ordinal; we need to associate a set of simplices to each ordinal: suddenly, the idea appears natural to take a functor $X$ into $\textbf{Set}$, so that $X[n]$ would be a set, whose elements could then correspond to the $n$-simplices. Furthermore, let's look at the initial object $\emptyset$. This maps into the empty simplicial set. If we take compositions of face maps as the only natural type of "dimension-lowering" map, then we find that there should "effectively" only be one map into $\emptyset$, since any combination of face maps on a complex will delete the vertices one-by-one in some order, so the initial object becomes terminal. Indeed, in general, inclusion of smaller ordinals in larger ones seems to fit most naturally with "inclusions" of sets of bigger simplices into the larger sets of their lower-dimensional faces, which gives the intuition behind a simplicial set being given by a contravariant functor (a presheaf).
Now, remarkably, it all begins to work out: all the $n$ inclusions $[n-1]\to [n]$ have a very nice correspondence with intuitive geometric idea of taking faces of the simplexes in the set $X[n]$ to map into a subset of $X[n-1]$: "skipping over" the $i$th element in $[n]$ corresponds to removing the $i$th vertex. We hence call the former morphisms of $\Delta$ the coface morphisms (because of contravariance), and denote them by $\delta^i$, in the tradition of duality notation. Then the classic face map identity $\delta_i\delta_j=\delta_{j-1}\delta_i$ ("removing the $j$th and then $i$th vertex is the same as removing the $i$th and then the $j-1$st when $j$ comes after $i$ in the ordering of the simplex," since the numbering of $j$ gets moved one back when $i$ is removed) becomes dualized when pulled back through the contravariant functor to $\delta^j\delta^i=\delta^i\delta^j$, which we call the coface identity. We can inspect to see this is obviously true in $\Delta$ by just working out the maps on an ordinal.
Finally, from a combinatorial perspective, the category $\Delta$ is clearly generated by the coface morphisms above, and morphisms of the form $s_i:[n]\to[n-1]$ which "double" up at element $i$. This fills the lack of any dimension-raising maps in our simplicial set category: the images $Xg:G[n-1]\to G[n]$ of these maps then correspond to creating a "degenerate" $n$ simplex by "doubling up" at vertex $i$ via creating a "loop" there. (Note that these maps are what distinguish simplicial sets from delta sets, in which degenerate simplices and maps do not exist.) Hence the $i$th and $i+1$st faces of the $i$th degeneracy of a simplex $S$ are $S$ itself. We call the morphisms in $\Delta$ codegeneracy maps $s^i$, and their image morphisms under the functor degeneracy maps $s_i$, in analogy with the face/coface nomenclature.
For degeneracy maps, then, we see that $s_is_j=s_{j+1}s_i$ is combinatorially obvious in much the same way that the face map identity was, so dualizing, we get $s^js^i=s^is^{j+1}$ for $i\le j$, the coface identity - again it works out correctly in $\Delta$. The "geometric" meanings of the face/degeneracy identities in terms of removing/adding vertices/edges can be similarly worked out, and dualized to give the coface/codegeneracy identities, which you can check are true in $\Delta$. It'd be instructive to do this yourself. Hence we have constructed the category $\textbf{sSet}$ of simplicial sets.
The Yoneda embedding is just a functor from a category into its category of presheaves with some nice properties, so it can be written $h:\Delta\to \textbf{sSet}$. The image $h[n]$ is then the representable presheaf $\text{hom}_{\Delta}(-,[n])$, which we call the standard $n$-simplex. The Yoneda lemma then tells us that the $n$-simplices of a simplicial set $X$ are in bijection with $\text{hom}(h[n],X)=\text{hom}(\text{hom}_{\Delta}(-,[n]),X)$, and that this bijection is natural in both $X$ and $[n]$. In particular, the bijection identifies the morphism of $h[n]$ (which consist of a single non-degenerate $n$-simplex) into $X$ taking its simplex to simplex $S\in X[n]$, with exactly that simplex $S[n]$.
I admit I don't quite understand the degeneracy map stuff on a level deeper than "this makes the categorical formulation possible and/or much nicer," though having a nice categorical formulation is, I think the point of all of this - it's the motivation for basically all the applications of simplicial sets, in very abstract settings like classifying spaces for categories and higher homotopy theory.