Kerodon

Construction 4.6.5.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ denote the fiber product $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} $, and we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ denote the fiber product $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$. We will be primarily interested in these constructions in the situation where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, we refer to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ as the left-pinched space of morphisms from $X$ to $Y$ and to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ as the right-pinched space of morphisms from $X$ to $Y$.