Kerodon

Warning 6.3.0.4. Let $\operatorname{\mathcal{C}}$ be a category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ is small, then the strict localization $W^{-1} \operatorname{\mathcal{C}}$ is also small. Beware that if $\operatorname{\mathcal{C}}$ is only assumed to be locally small (Variant 4.7.8.6), then $W^{-1} \operatorname{\mathcal{C}}$ need not be locally small. However, one can often ensure that $W^{-1} \operatorname{\mathcal{C}}$ is locally small by imposing additional assumptions on the collection of morphisms $W$.