Kerodon

Notation 10.3.1.24 (Pullback Sieves). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ be a sieve on the object $Y$. We let $f^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y} )$ denote the full subcategory of $\operatorname{\mathcal{C}}_{/X}$ spanned by those objects $e: C \rightarrow X$ for which the composition $(f \circ e): C \rightarrow Y$ belongs to $\operatorname{\mathcal{C}}^{0}_{/Y}$. By virtue of Remark 10.3.1.21, this condition is independent of the choice of composition $f \circ e$. The subcategory $f^{\ast }( \operatorname{\mathcal{C}}^{0}_{/Y} )$ is a sieve on the object $X$, which we will refer to as the pullback of $\operatorname{\mathcal{C}}^{0}_{/Y}$ along the morphism $f$.