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

Section4.3Topological Spaces and More Examples

Example4.3.1
Let \(\Delta^n\) be the geometric \(n\) simplex \[\Delta^n = \{(t_0,\ldots,t_n) \in \RR_{\ge 0}^{n+1} : \sum t_i = 1\}.\] Any \([m]\xrightarrow{\alpha} [n] \in \Delta\) induces a set map on the vertices which extends linearly to \(\Delta^m \xrightarrow{\alpha_*} \Delta^n\). This makes \(\Delta^{\bullet}\) into a cosimplicial topological space. Then \(\Hom_{\Top}(\Delta^{\bullet}, X)\) is naturally a simplicial set. It is \(\Sing X\). So we have a functor \(\Sing \colon \Top \to \SSet\) and we want an adjoint.
Definition4.3.2Geometric realisation
There is a functor \(|\cdot |\colon \SSet \to \Top\) defined by \[ |A_n| = \coprod_n A_n \times \Delta^n / \sim. \] Where for \(\alpha \colon [m] \to [n]\) we identify \(A_m \times \Delta^n \ni (\alpha^* x, y)\) with \((x, \alpha_* y) \in A_n \times \Delta^n\).
Example4.3.3
\[|\Delta[n]| = \Delta^n.\]

We are going to restrict these functors.

Example4.3.4
Let \(G\) be a group, now let \(BG_n = G^{\times n}\) and \[\dd_i(g_1,\ldots,g_n) = \begin{cases}(g_2,\ldots,g_n) & i = 0, \\(g_1\ldots,g_ig_{i+1},\ldots,g_n) & i = 1,\ldots,n-1,\\(g_1,\ldots,g_{n-1}) & i = n.\end{cases}\]\[\sigma_i(g_1,\ldots,g_n) = (g_1,\ldots,1,\ldots,g_n)\] where the \(1\) goes in the \(i\)th place. \(|BG|\) is called the classifying space of \(G\) and it is a \(K(G,1)\).
Example4.3.5
A group is just a category with only one object and where all arrows are isomorphisms. So for a small category \(\cC\) we let \(B\cC_0\) be \(\ob \cC\) and \(B\cC_{n\ge 1}\) be all compatible \(n\)-tuples of morphisms. We define the face and degeneracy maps by composition and identity as above.