# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Definition 7.6.6.1 (Towers). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. A tower in $\operatorname{\mathcal{C}}$ is a functor $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$. We say that $\operatorname{\mathcal{C}}$ admits sequential limits if every tower in $\operatorname{\mathcal{C}}$ has a limit, and that $\operatorname{\mathcal{C}}$ admits sequential colimits if every diagram $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{C}}$ has a colimit. We say that a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves sequential limits if it preserves limits indexed by the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} )$, and that it preserves sequential colimits if it preserves colimits indexed by the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} )$.