Kerodon

Warning 10.3.1.27. The terminology of Definition 10.3.1.26 has the potential to create confusion. Since the forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ creates colimits (Proposition 7.1.4.20), a sieve $\operatorname{\mathcal{C}}^{0}_{/X} \subseteq \operatorname{\mathcal{C}}_{/X}$ which is dense in the sense of Definition 10.3.1.26 is also dense when regarded as a full subcategory of $\operatorname{\mathcal{C}}_{/X}$ (in the sense of Definition 8.4.1.5). Beware that the converse is false in general (Example 10.3.1.28). However, it is true if $\operatorname{\mathcal{C}}$ admits finite products (Proposition 10.3.1.29).