# Kerodon

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Definition 7.4.3.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and let ${\bf 1}$ denote the cone point of the simplicial set $\operatorname{\mathcal{C}}^{\triangleright } \simeq \operatorname{\mathcal{C}}\star \{ {\bf 1} \}$. Suppose that we are given a cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$, and set

$\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}} \quad \quad \overline{\operatorname{\mathcal{E}}}_{{\bf 1}} = \{ {\bf 1} \} \times _{\operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}.$

We will say that a morphism $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ is a covariant refraction diagram if there exists a morphism of simplicial sets $H: \Delta ^1 \times \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}$ satisfying the following conditions:

• The restriction $H|_{ \{ 0\} \times \operatorname{\mathcal{E}}}$ is the identity morphism from $\operatorname{\mathcal{E}}$ to itself.

• The restriction $H|_{ \{ 1\} \times \operatorname{\mathcal{E}}}$ is equal to $\mathrm{Rf}$.

• For every vertex $X \in \operatorname{\mathcal{E}}$, the restriction $H|_{ \Delta ^1 \times \{ X\} }$ is a $\overline{U}$-cocartesian edge of $\overline{\operatorname{\mathcal{E}}}$.