Kerodon

Warning 9.2.9.3. The converse of Corollary 9.2.9.2 is false in general: that is, the $\infty $-category $\operatorname{\mathcal{D}}$ might have compact objects which do not have compact image in each $\mathscr {F}(C)$. For example, let $\operatorname{\mathcal{C}}= \Delta ^1 / \operatorname{\partial \Delta }^1$ be the simplicial circle and let $\mathscr {F}$ be the constant functor whose value is (the nerve of) the category of abelian groups. In this case, we can identify $\operatorname{\mathcal{D}}= \varprojlim (\mathscr {F} )$ with (the nerve of) the category of pairs $(M, u)$, where $M$ is an abelian group and $u: M \xrightarrow {\sim } M$ is an automorphism of $M$. In this case, $(M,u)$ is a compact object of $\operatorname{\mathcal{D}}$ if and only if $M$ is finitely generated as a module over the Laurent polynomial ring $\operatorname{\mathbf{Z}}[u^{\pm 1} ]$. This condition does not guarantee that $M$ is finitely generated as an abelian group (compare with Warning 9.2.8.10).