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

TL;DR

This paper develops a category-theoretic method to reconstruct a log scheme from the structure of a subcategory of fs log schemes over it, focusing on properties like reducedness and finite type.

Contribution

It introduces a purely categorical approach to reconstructing log schemes from categories of schemes satisfying specific properties, advancing the understanding of their intrinsic structures.

Findings

Reconstruction of log schemes from categorical data.

Characterization of properties like reducedness and finite type categorically.

Framework applicable to various properties of fs log schemes.

Abstract

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 $\mathsf{Sch}_{\blacklozenge/S^{\log}}$.