\( \usepackage{mathrsfs} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\cHom}{\underline{Hom}} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\inj}{inj} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\coim}{coim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\eq}{eq} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\ob}{ob} \DeclareMathOperator{\coeq}{coeq} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\Ab}{\text{Ab}} \DeclareMathOperator{\Top}{\text{Top}} \DeclareMathOperator{\Rmod}{R\text{mod}} \DeclareMathOperator{\SSet}{\text{SSet}} \DeclareMathOperator{\sAb}{\text{sAb}} \DeclareMathOperator{\cA}{\mathcal{A}} \DeclareMathOperator{\cB}{\mathcal{B}} \DeclareMathOperator{\cC}{\mathcal{C}} \DeclareMathOperator{\cO}{\mathcal{O}} \DeclareMathOperator{\cM}{\mathcal{M}} \DeclareMathOperator{\sC}{\mathscr{C}} \DeclareMathOperator{\sF}{\mathscr{F}} \DeclareMathOperator{\sW}{\mathscr{W}} \DeclareMathOperator{\dd}{\partial} \DeclareMathOperator{\rd}{\mathrm{d}} % Rings \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\PP}{\mathbf{P}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \)

Section4.1Simplicial Sets

Definition4.1.1Simplex category
The simplex category\(\Delta\) has objects the sets \(\{0,\ldots,n\} = [n]\) and morphisms the non-decreasing maps between such sets. A simplicial object in a category \(\cC\) is a functor \(\Delta^{\op} \to \cC\). These objects form a category, denoted \(s(\cC)\) or just \(s\cC\), where the morphisms are natural transformations.
Example4.1.2
\(\SSet\) is the category of simplicial sets, and \(\sAb\) is the category of simplicial groups.
Example4.1.3
Given a topological space \(X\), \(\Sing X\) is the singular simplicial set. We can then form \(\ZZ\Sing X\), the free abelian group on \((\Sing X)_n\) for each \(n\).

The \(i\)th face map \(\epsilon_i\colon[n-1] \to [n]\) is the unique injection only leaving out \(i \in [n]\). The \(i\)th degeneracy map \(\eta_i\colon [n+1] \to [n]\) is the unique surjective map mapping two elements to \(i\in [n]\).

Hence for the purposes of understanding a simplicial object it is enough to understand \(A(\epsilon_i) = \dd_i\) and \(A(\eta_j) = \sigma_j\). These maps satisfy (after checking!) the relations \begin{align*} \dd_i\dd_j &= \dd_{j-1}\dd_i \text{ if } i \lt j.\\ \sigma_i\sigma_j &= \sigma_{j+1}\sigma_i \text{ if } i \leq j.\\ \dd_i\sigma_j &= \begin{cases}\sigma_{j-1}\dd_i & \text{ if } i \lt j, \\ 1 &\text{ if } i=j,j+1,\\\sigma_i\dd_{j+1} &\text{ if }i \gt j + 1.\end{cases} \end{align*}

Example4.1.5
Define the standard simplex\(\Delta[n]\) to be the image of \([n]\) under contravariant Yoneda, i.e. \(\Delta[n]_i = \Hom_{\Delta}(i,[n])\). This is universal i n the sense that \(\Delta_n = \Hom_{\SSet}(\Delta[n], A)\). We call \(A_m\) the simplices of \(\Delta\) (by Yoneda).
Example4.1.6
\(\Delta[1]_n\) is the set of maps \([n] \to [1]\), we can write these as \(0\cdots 0 1\cdots 1\) where \(0\) appears \(k\) times and \(1\) occurs \(n-k +1\) times. So \begin{align*} \Delta[1]_0 &= \{0, 1\},\\ \Delta[1]_1 &= \{0, 01, 11\},\\ \Delta[1]_2 &= \{000, 001, 011, 111\}. \end{align*} All the expressions here with repeat digits are \(\sigma_i(a)\) for some \(n\), so they all called degenerate. We only have 3 non-degenerate maps here.