Ok so a simplicial set is a functor $\Delta^{op}\to\mathsf{Set}$ where $\Delta$ is the category whose objects are finite sets $[n]=\{1,\ldots,n\}$ and whose morphisms are orderer preserving maps
In $\Delta$ we have $n-1$ injective coface maps $d^i\colon[n-1]\to[n]$ with $d^i(k)=k$ for $k<i$ and $d^i(k)=k+1$ for $k\geq i$
And we have $n+1$ surjective codegeneracy maps $s^i\colon [n+1]\to[n]$ with $s^i(k)=k$ for $k\leq i$ and $s^i(k)=k-1$ for $k>i$
And their compositions satisfy a bunch of identities
$\Delta^n$, the $n$-simplex is the embedding of $[n]$ in $\mathrm{Func}(\Delta^{op},\mathsf{Set})$ given by Yoneda
So $\Delta^n=\mathrm{Hom}_{\Delta}(-,[n])$
And I know how the nerve of a category is defined
That summarizes my knowledge of simplicial sets quite accurately
@Alessandro Nice, yeah, the functorial definition is the cleanest but obfuscates what the geometry is. I'd define it using the face and degeneracy maps as follows
Take $X : \Delta^{op} \to Set$ a functor, and let $X_n = X([n])$. I'll call $\{X_n\}$ the collection of $n$-simplices of the simplicial set, to be thought as the $n$-dimensional faces of a simplicial complex
The coface and codegeneracy maps give rise to face and degeneracy maps $d_n : X_n \to X_{n-1}$ and $s_n : X_n \to X_{n+1}$.
If I think of $\Delta_n\subseteq\Bbb R^{n+1}$ as $(x_0,\ldots,x_n)$ such that $\sum x_i=1$ then $d^i:\Delta_n\to\Delta_{n+1}$ has as image $(x_0,\ldots,x_{n+1})$ such that $\sum x_j=1$ and $x_i=0$ I think
Yikes, I wouldn't use that convex combination definition. Just label the vertices so that you can talk about the faces as being "opposite" to the i-th vertex
Then the faces have a natural numbering as well
Namely, the $i$-th face of $[0, 1, \cdots, n]$ is $[0, 1, \cdots, \hat{i}, \cdots, n]$.
The face of $\Delta^n$ which sits opposite to the vertex $[i]$
@Alessandro So the "collapse map" $\Delta^2 \to \Delta^1$ which sends $0$ to $0$, $1$ to $1$ and $2$ to $1$ is an order-non-decreasing map. This is sort of what $s_1 : X_2 \to X_1$ should do.
This is the crucial difference between a simplicial complex and a simplicial set
Namely, $X_n$ not only consists of $n$-simplexes, but all degenerate $n$-simplices as well, and which $(n-1)$-simplex they degenerate to is kept track by $s_n : X_n \to X_{n-1}$.
Degenerate $n$-simplices are to be envisioned as $n$-simplices but hiding inside an $(n-1)$-simplex. They do not appear when you take geometric realization. It loses information.
Take any topological space $Z$. Define $\text{Sing}_\bullet$ to be the simplicial set such that $\text{Sing}_n$ is the set of all continuous maps $\Delta^n \to Z$.
The face map $\text{Sing}_n \to \text{Sing}_{n-1}$ comes from composing with the inclusion of the $i$-th face $\Delta^{n-1}_i \hookrightarrow \Delta^n$
The degeneracy map $\text{Sing}_n \to \text{Sing}_{n+1}$ comes from composing with the $i$-th degeneracy $\Delta_i^{n+1} \twoheadrightarrow \Delta^n$
So this is the singular simplicial set obtained from $Z$. Kinda like the singular chain complex, but that doesn't carry information about the degeneracies, again, so there's some loss of information in thinking like that
Namely, consider two paths $\gamma_1, \gamma_2$ in $Z$. These can be thought as elements of $\text{Sing}_1$.
$\gamma_1 * \gamma_2$ is also an element of $\text{Sing}_1$
There's a degenerate element of $\text{Sing}_2$ which you can think of as a triangle whose three edges maps to $\gamma_1, \gamma_2$ and $\gamma_1 * \gamma_2$
And the whole triangle gets collapsed to something 1-dimensional
Oh ok try $\gamma_2$ to be a constant path at the endpoint of $\gamma_1$. Then there's a degenerate element in $\text{Sing}_2$ you can think of as being represented by the triple $(\gamma_1, \text{const}, \gamma_1 * \text{const})$
It's like $[0, 1, 2] \to [0, 1, 1]$
My previous analogy was bad because you should preserve vertices while collapsing.
This actually represents the homotopy between $\gamma_1$ and $\gamma_1 * \text{const}$.
@Alessandro So the "collapse map" $\Delta^2 \to \Delta^1$ which sends $0$ to $0$, $1$ to $1$ and $2$ to $1$ is an order-non-decreasing map. This is sort of what $s_1 : X_2 \to X_1$ should do.
@Alessandro So, to be precise, if $\gamma$ is a path in $Z$ between $x$ and $y$, $c$ is the constant path at $y$, then the homotopy $\gamma * c \sim \gamma$ is represented by a degenerate 2-simplex in $\text{Sing}_2$.
This is alright? It's quite important to understand how the homotopical business seeps in
This is fine, but it feels weird, like some homotopies between elements of $\mathrm{Sing}_1$ can be represented by elements of $\mathrm{Sing}_2$, but not all of them
That's because you're doing "simplicial" homtopies, and these need not compose to maps from a unified simplex.
Say $\gamma_1 \sim \gamma_2$ be two paths in $Z$
You can represent $\gamma_1 \sim \gamma_1 * c$ by an element of $\text{Sing}_2$. In fact, a degenerate element. You can also represent $\gamma_1 * c \sim \gamma_2$ by an element of $\text{Sing}_2$.
So you can always break a homotopy so that they are maps from 2-simplices in this fashion
This is why degenerate simplices are so important for the formalism
This should become clear when we discuss geometric realization.
That's basically the morally correct answer and how one might do this in general
But there is an extension answer here
The nontrivial central extension $0 \to \Bbb Z \to \Bbb Z \to \Bbb Z/2 \to 0$ reduces to the nontrivial central extension $0 \to \Bbb Z/2 \to \Bbb Z/4 \to \Bbb Z/2 \to 0$
Therefore the classifying class of the extension in $H^2(\Bbb Z/2;\Bbb Z)$ reduces to the classifying class of the second extension in $H^2(\Bbb Z/2; \Bbb Z/2)$
It actually suffices to compute $H^0(\Bbb Z/2; \Bbb Z)$ and $H^1(\Bbb Z/2; \Bbb Z)$ to compute $H^*(\Bbb Z/2; \Bbb Z)$ because of the Gysin sequence. You recover the full ring structure as well I think.
Yeah the major thing is that if $A$ is abelian the "action" $G \to \text{Aut}(A)$, which is only defined as a map to $\text{Out}(A)$, becomes an honest action
So $A$ is a $G$-module and you can do the schtick with $H^2$
I wonder what all this means in terms of obstruction theory of $BA \to BE \to BG$. I vaguely understand existence of section being determined by $H^1$ and $H^2$ as the fact that we only need a section at the level of the presentation complex or whatever it's called, the 2-skeleton of $BG$.
