Warning 8.4.1.10. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category. Consider the following conditions on a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense, in the sense of Definition 8.4.1.5.
- $(2)$
Every object $X \in \operatorname{\mathcal{D}}$ can be realized as the colimit of a diagram taking values in the full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated by $\operatorname{\mathcal{C}}$ under colimits. That is, if $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is a full subcategory which contains $\operatorname{\mathcal{C}}$ and is closed under the formation of colimits in $\operatorname{\mathcal{D}}$, then $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$.
It follows immediately from the definitions that $(1) \Rightarrow (2) \Rightarrow (3)$. Beware that neither of this implications is reversible. See Exercises 8.4.1.11 and 8.4.1.12.