The Stacks project

Lemma 14.32.2. Let $f^0, f^1 : V \to U$ be maps of simplicial sets. Let $n \geq 0$ be an integer. Assume

  1. The maps $f^ j_ i : V_ i \to U_ i$, $j = 0, 1$ are equal for $i < n$.

  2. The canonical morphism $U \to \text{cosk}_ n \text{sk}_ n U$ is an isomorphism.

  3. The canonical morphism $V \to \text{cosk}_ n \text{sk}_ n V$ is an isomorphism.

Then $f^0$ is homotopic to $f^1$.

First proof. Let $W$ be the $n$-truncated simplicial set with $W_ i = U_ i$ for $i < n$ and $W_ n = U_ n / \sim $ where $\sim $ is the equivalence relation generated by $f^0(y) \sim f^1(y)$ for $y \in V_ n$. This makes sense as the morphisms $U(\varphi ) : U_ n \to U_ i$ corresponding to $\varphi : [i] \to [n]$ for $i < n$ factor through the quotient map $U_ n \to W_ n$ because $f^0$ and $f^1$ are morphisms of simplicial sets and equal in degrees $< n$. Next, we upgrade $W$ to a simplicial set by taking $\text{cosk}_ n W$. By Lemma 14.32.1 the morphism $g : U \to W$ is a trivial Kan fibration. Observe that $g \circ f^0 = g \circ f^1$ by construction and denote this morphism $f : V \to W$. Consider the diagram

\[ \xymatrix{ \partial \Delta [1] \times V \ar[rr]_{f^0, f^1} \ar[d] & & U \ar[d] \\ \Delta [1] \times V \ar[rr]^ f \ar@{-->}[rru] & & W } \]

By Lemma 14.30.2 the dotted arrow exists and the proof is done. $\square$

Second proof. We have to construct a morphism of simplicial sets $h : V \times \Delta [1] \to U$ which recovers $f^ i$ on composing with $e_ i$. The case $n = 0$ was dealt with above the lemma. Thus we may assume that $n \geq 1$. The map $\Delta [1] \to \text{cosk}_1 \text{sk}_1 \Delta [1]$ is an isomorphism, see Lemma 14.19.15. Thus we see that $\Delta [1] \to \text{cosk}_ n \text{sk}_ n \Delta [1]$ is an isomorphism as $n \geq 1$, see Lemma 14.19.10. And hence $V \times \Delta [1] \to \text{cosk}_ n \text{sk}_ n (V \times \Delta [1])$ is an isomorphism too, see Lemma 14.19.12. In other words, in order to construct the homotopy it suffices to construct a suitable morphism of $n$-truncated simplicial sets $h : \text{sk}_ n V \times \text{sk}_ n \Delta [1] \to \text{sk}_ n U$.

For $k = 0, \ldots , n - 1$ we define $h_ k$ by the formula $h_ k(v, \alpha ) = f^0(v) = f^1(v)$. The map $h_ n : V_ n \times \mathop{\mathrm{Mor}}\nolimits _{\Delta }([k], [1]) \to U_ n$ is defined as follows. Pick $v \in V_ n$ and $\alpha : [n] \to [1]$:

  1. If $\mathop{\mathrm{Im}}(\alpha ) = \{ 0\} $, then we set $h_ n(v, \alpha ) = f^0(v)$.

  2. If $\mathop{\mathrm{Im}}(\alpha ) = \{ 0, 1\} $, then we set $h_ n(v, \alpha ) = f^0(v)$.

  3. If $\mathop{\mathrm{Im}}(\alpha ) = \{ 1\} $, then we set $h_ n(v, \alpha ) = f^1(v)$.

Let $\varphi : [k] \to [l]$ be a morphism of $\Delta _{\leq n}$. We will show that the diagram

\[ \xymatrix{ V_{l} \times \mathop{\mathrm{Mor}}\nolimits ([l], [1]) \ar[r] \ar[d] & U_{l} \ar[d] \\ V_{k} \times \mathop{\mathrm{Mor}}\nolimits ([k], [1]) \ar[r] & U_{k} } \]

commutes. Pick $v \in V_{l}$ and $\alpha : [l] \to [1]$. The commutativity means that

\[ h_ k(V(\varphi )(v), \alpha \circ \varphi ) = U(\varphi )(h_ l(v, \alpha )). \]

In almost every case this holds because $h_ k(V(\varphi )(v), \alpha \circ \varphi ) = f^0(V(\varphi )(v))$ and $U(\varphi )(h_ l(v, \alpha )) = U(\varphi )(f^0(v))$, combined with the fact that $f^0$ is a morphism of simplicial sets. The only cases where this does not hold is when either (A) $\mathop{\mathrm{Im}}(\alpha ) = \{ 1\} $ and $l = n$ or (B) $\mathop{\mathrm{Im}}(\alpha \circ \varphi ) = \{ 1\} $ and $k = n$. Observe moreover that necessarily $f^0(v) = f^1(v)$ for any degenerate $n$-simplex of $V$. Thus we can narrow the cases above down even further to the cases (A) $\mathop{\mathrm{Im}}(\alpha ) = \{ 1\} $, $l = n$ and $v$ nondegenerate, and (B) $\mathop{\mathrm{Im}}(\alpha \circ \varphi ) = \{ 1\} $, $k = n$ and $V(\varphi )(v)$ nondegenerate.

In case (A), we see that also $\mathop{\mathrm{Im}}(\alpha \circ \varphi ) = \{ 1\} $. Hence we see that not only $h_ l(v, \alpha ) = f^1(v)$ but also $h_ k(V(\varphi )(v), \alpha \circ \varphi ) = f^1(V(\varphi )(v))$. Thus we see that the relation holds because $f^1$ is a morphism of simplicial sets.

In case (B) we conclude that $l = k = n$ and $\varphi $ is bijective, since otherwise $V(\varphi )(v)$ is degenerate. Thus $\varphi = \text{id}_{[n]}$, which is a trivial case. $\square$

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