Kerodon

Definition 4.1.2.17. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $S$ be a collection of vertices of $\operatorname{\mathcal{C}}$. By virtue of Proposition 4.1.2.16, there exists a unique full simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ having vertex set $S$. We will refer to $\operatorname{\mathcal{C}}'$ as the full simplicial subset of $\operatorname{\mathcal{C}}$ spanned by $S$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, we will refer to $\operatorname{\mathcal{C}}'$ as the full subcategory of $\operatorname{\mathcal{C}}$ spanned by $S$.