Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.4.6.4. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category with underlying category $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_0$, and let $f,g: X \rightarrow Y$ be a pair of morphisms of $\operatorname{\mathcal{C}}$ having the same source and target. Using Remark 1.1.6.23, we see that the following conditions are equivalent:

$(a)$

The morphisms $f$ and $g$ represent the same morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$: that is, we have $[f] = [g]$.

$(b)$

There exists a sequence of morphisms $f = f_0, f_1, f_2, \ldots , f_ n = g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ such that, for $1 \leq i \leq n$, either there exists a homotopy from $f_{i-1}$ to $f_{i}$ or a homotopy from $f_{i}$ to $f_{i-1}$ (in the sense of Definition 2.4.1.6).

If $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan, then we can replace $(b)$ by the following simpler condition:

$(c)$

There exists a homotopy from $f$ to $g$ (in the sense of Definition 2.4.1.6).

See Remark 2.4.1.9.