The Stacks project

Example 14.26.7. The simplicial set $\Delta [m]$ is homotopy equivalent to $\Delta [0]$. Namely, consider the unique morphism $f : \Delta [m] \to \Delta [0]$ and the morphism $g : \Delta [0] \to \Delta [m]$ given by the inclusion of the last $0$-simplex of $\Delta [m]$. We have $f \circ g = \text{id}$. We will give a homotopy $h : \Delta [m] \times \Delta [1] \to \Delta [m]$ from $\text{id}_{\Delta [m]}$ to $g \circ f$. Namely $h$ is given by the maps

\[ \mathop{\mathrm{Mor}}\nolimits _\Delta ([n], [m]) \times \mathop{\mathrm{Mor}}\nolimits _\Delta ([n], [1]) \to \mathop{\mathrm{Mor}}\nolimits _\Delta ([n], [m]) \]

which send $(\varphi , \alpha )$ to

\[ k \mapsto \left\{ \begin{matrix} \varphi (k) & \text{if} & \alpha (k) = 0 \\ m & \text{if} & \alpha (k) = 1 \end{matrix} \right. \]

Note that this only works because we took $g$ to be the inclusion of the last $0$-simplex. If we took $g$ to be the inclusion of the first $0$-simplex we could find a homotopy from $g \circ f$ to $\text{id}_{\Delta [m]}$. This is an illustration of the asymmetry inherent in homotopies in the category of simplicial sets.

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