Category-theoretic Reconstruction of Log Schemes from Categories of Reduced fs Log Schemes

Let $S^{\log }$ be a locally Noetherian fs log scheme and $\mathrm{⧫}\mathrm{/}{S}^{\log }$ a set of properties of fs log schemes over $S^{\log }$. In the present paper, we shall mainly be concerned with the properties “reduced”, “quasi-compact over $S^{\log }$”, “quasi-separated over $S^{\log }$”, “separated over $S^{\log }$”, and “of finite type over $S^{\log }$”. We shall write ${\mathsf{S}\mathsf{c}\mathsf{h}}_{\mathrm{⧫}\mathrm{/}{S}^{\log }}$ for the full subcategory of the category of fs log schemes over $S^{\log }$ determined by the fs log schemes over $S^{\log }$ that satisfy every property contained in $\mathrm{⧫}\mathrm{/}{S}^{\log }$. In the present paper, we discuss a purely category-theoretic reconstruction of the log scheme $S^{\log }$ from the intrinsic structure of the abstract category ${\mathsf{S}\mathsf{c}\mathsf{h}}_{\mathrm{⧫}\mathrm{/}{S}^{\log }}$.