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Definition Let $\operatorname{\mathcal{C}}$ be a simplicial set. We say that a simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is full if it satisfies the following condition:

  • Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be a simplex of $\operatorname{\mathcal{C}}$ having the property that, for each integer $0 \leq i \leq n$, the vertex $\sigma (i) \in \operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}'$. Then $\sigma $ belongs to $\operatorname{\mathcal{C}}'$.

If this condition is satisfied, then the inclusion map $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration. In particular, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\operatorname{\mathcal{C}}'$ is a subcategory of $\operatorname{\mathcal{C}}$; in this case, we will say that $\operatorname{\mathcal{C}}'$ is a full subcategory of $\operatorname{\mathcal{C}}$.