# Kerodon

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Warning 6.3.0.10. It also is possible to contemplate a version of Definition 6.3.0.1 in the $\infty$-categorical setting. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Let us say that a functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a strict localization of $\operatorname{\mathcal{C}}$ with respect to $W$ if, for every $\infty$-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}} \} \ar [d] \\ \{ \textnormal{Functors \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}} carrying each w \in W to an isomorphism in \operatorname{\mathcal{E}}} \} }.$

However, this definition is useless. One can show that an $\infty$-category $\operatorname{\mathcal{C}}$ admits a strict localization with respect to $W$ only in the trivial case where every element of $W$ is already an isomorphism in $\operatorname{\mathcal{C}}$ (in which case we can take $F$ to be the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$). Roughly speaking, the problem is that if $w: X \rightarrow Y$ is an isomorphism in an $\infty$-category $\operatorname{\mathcal{C}}$, then the homotopy inverse isomorphism $w^{-1}: Y \rightarrow X$ is only well-defined up to homotopy (or up to a contractible space of choices), in contrast with classical category theory where the inverse isomorphism $w^{-1}$ is unique.