Definition 5.3.4.1. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $e$ be an edge of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$. Let us identify $e$ with a pair $(f, \overline{e} )$, where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $\overline{e}$ is an edge of the simplicial set $\mathscr {F}(C)$. We will say that the edge $e = (f, \overline{e} )$ is horizontal if $\overline{e}$ is a degenerate edge of $\mathscr {F}(C)$.
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