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Example 7.6.5.7. Fix a prime number $p$. For every integer $n \geq 0$, let $p^{n} \operatorname{\mathbf{Z}}$ denote the cyclic subgroup of $\operatorname{\mathbf{Z}}$ generated by $p^{n}$, so that we have a tower of classifying simplicial sets

7.74
\begin{equation} \begin{gathered}\label{equation:bad-inverse-limit} \xymatrix@C =50pt@R=50pt{ \cdots \ar [r] & B_{\bullet }( p^{3} \operatorname{\mathbf{Z}}) \ar [r] & B_{\bullet }( p^2 \operatorname{\mathbf{Z}}) \ar [r] & B_{\bullet }( p \operatorname{\mathbf{Z}}) \ar [r] & B_{\bullet }(\operatorname{\mathbf{Z}}). } \end{gathered} \end{equation}

Then:

  • The tower (7.74) has a limit in the ordinary category of simplicial sets, given by the simplicial set $\Delta ^0$ (which we can identify with the classifying simplicial set for the trivial group $(0) = \bigcap _{n \geq 0} p^{n} \operatorname{\mathbf{Z}}$).

  • The simplicial set $\Delta ^0$ is also a limit of the tower (7.74) in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

  • In the $\infty $-category $\operatorname{\mathcal{S}}$, the tower (7.74) has a different limit, which has uncountably many connected components (see Remark ).