Kerodon

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Definition 8.5.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A retraction diagram in $\operatorname{\mathcal{C}}$ is a $2$-simplex $\sigma : \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ for which the “long” face $d_{1}(\sigma )$ is an identity morphism of $\operatorname{\mathcal{C}}$. In this case, we indicate $\sigma $ by a diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y, } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$, and we say that $\sigma $ exhibits $Y$ as a retract of $X$.