The Stacks project

Lemma 14.28.3. Let $\mathcal{C}$ be a category. Suppose that $U$ and $V$ are two cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of cosimplicial objects. Recall that $U$, $V$ correspond to simplicial objects $U'$, $V'$ of $\mathcal{C}^{opp}$. Moreover $a, b$ correspond to morphisms $a', b' : V' \to U'$. The following are equivalent

  1. There exists a homotopy $h = \{ h_{n, \alpha }\} $ from $a$ to $b$ as in Remark 14.28.2.

  2. There exists a homotopy $h = \{ h_{n, i}\} $ from $a'$ to $b'$ as in Remark 14.26.4.

Thus $a$ is homotopic to $b$ as in Remark 14.28.2 if and only if $a'$ is homotopic to $b'$ as in Remark 14.26.4.

Proof. In case $\mathcal{C}$ has finite products, then $\mathcal{C}^{opp}$ has finite coproducts and we may use Definitions 14.28.1 and 14.26.1 instead of Remarks 14.28.2 and 14.26.4. In this case $h : U \to \mathop{\mathrm{Hom}}\nolimits (\Delta [1], V)$ is the same as a morphism $h' : \mathop{\mathrm{Hom}}\nolimits (\Delta [1], V)' \to U'$. Since products and coproducts get switched too, it is immediate that $(\mathop{\mathrm{Hom}}\nolimits (\Delta [1], V))' = V' \times \Delta [1]$. Moreover, the “primed” version of the morphisms $e_0, e_1 : \mathop{\mathrm{Hom}}\nolimits (\Delta [1], V) \to V$ are the morphisms $e_0, e_1 : V' \to \Delta [1] \times V$. Thus $e_0 \circ h = a$ translates into $h' \circ e_0 = a'$ and similarly $e_1 \circ h = b$ translates into $h' \circ e_1 = b'$. This proves the lemma in this case.

In the general case, one needs to translate the relations given by ( into the relations given in Lemma 14.26.2. We omit the details.

The final assertion is formal from the equivalence of (1) and (2). $\square$

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