Kerodon

Example 9.2.3.8. Let $R$ be an associative ring, let $\operatorname{\mathcal{C}}$ be (the nerve of) the category of left $R$-modules, and let $\operatorname{\mathcal{C}}_{\mathrm{fin}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the finitely presented left $R$-modules. Then the inclusion functor $\operatorname{\mathcal{C}}_{\mathrm{fin}} \hookrightarrow \operatorname{\mathcal{C}}$ exhibit $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}_{\mathrm{fin}}$. This follows from Corollary 9.2.3.6, since every finitely presented $R$-module $M$ is a compact object of $\operatorname{\mathcal{C}}$ (Proposition 9.2.0.1), and every $R$-module can be realized as a filtered colimit of finitely presented $R$-modules. This assertion has counterparts for many other mathematical structures studied in algebra; see Remark 9.2.0.3.