Kerodon

Warning 9.2.8.10. The converse of Proposition 9.2.8.9 is false in general. For example, let $K = \Delta ^1 / \operatorname{\partial \Delta }^1$ be the simplicial circle (Example 1.5.7.11) and let $\operatorname{\mathcal{C}}$ be (the nerve of) the category $\operatorname{ Ab }$ of abelian groups. Then $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ can be identified with (the nerve of) the category of pairs $(M,u)$, where $M$ is an abelian group and $u: M \rightarrow M$ is an endomorphism of $M$. In this case, $(M,u)$ is a compact object of $\operatorname{\mathcal{C}}$ if and only if $M$ is finitely generated as a module over the polynomial ring $\operatorname{\mathbf{Z}}[u]$. (see Proposition 9.2.0.1). This condition does not guarantee that $M$ is finitely generated as an abelian group (for example, the polynomial ring $\operatorname{\mathbf{Z}}[u]$ itself is not finitely generated as an abelian group).