Let $S^{ \log}$ be a locally Noetherian fs log scheme and $\blacklozenge/S^{ \log}$ 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 $\mathsf{ Sch}_{ \blacklozenge/S^{ \log}}$ 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 $\blacklozenge/S^{ \log}$. 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 $\mathsf{ Sch}_{ \blacklozenge/S^{ \log}}$.
Categories: math.AG math.CT