Kerodon

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Definition 5.1.6.1. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^{F} \ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}, & } \]

where $U$ and $V$ are inner fibrations. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We say that $G$ is a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$ if the following conditions are satisfied:

  • The composition $U \circ G$ is equal to $V$: that is, the diagram

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [ll]_{G} \ar [dl]^{V} \\ & \operatorname{\mathcal{E}}& } \]

    is commutative.

  • The composite morphisms $G \circ F $and $F \circ G$ are isomorphic to $\operatorname{id}_{\operatorname{\mathcal{C}}}$ and $\operatorname{id}_{\operatorname{\mathcal{D}}}$ as objects of the $\infty $-categories $\operatorname{Fun}_{ / \operatorname{\mathcal{E}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}_{/\operatorname{\mathcal{E}}}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, respectively.

We say that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ if there exists a morphism of simplicial sets $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which is a homotopy inverse of $F$ relative to $\operatorname{\mathcal{E}}$. We say that inner fibrations $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ are equivalent if there exists a morphism of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ (so that, in particular, we have $U = V \circ F$).