arXiv subjects
Publications and source records attributed to Tomoki Yuji.
In this paper, we introduce a Grothendieck topology on the category of totally bounded metric spaces and develop a theory of stacks with respect to this topology. We further define the fine moduli stack of compact metric spaces and prove that its coarse moduli space is isometric to the Gromov--Hausdorff space.
In the present paper, we prove that a topological space admits a functorial Lindel\"ofification if and only if its realcompactification is Lindel\"of. To investigate the functorial Lindel\"ofifiability of a topological space, for each topological property $\mathsf{P}$, we introduce the notion of "functorial $\mathsf{P}$-ification" and give an explicit construction of the functorial $\mathsf{P}$-ification. Moreover, for a discrete space $X$, we discuss the functorial $|X|$-Lindel\"ofifiability of $X$ and study relationships with properties of the cardinal $|X|$. Finally, we apply our results concerning functorial $\kappa$-Lindel\"ofifiability (for some cardinal $\kappa$) to the space of ordinals and construct several functorial $\kappa$-Lindel\"ofifiable spaces.
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 $\mathsf{Sch}_{\blacklozenge/S^{\log}}$.
Let $S$ be a locally Noetherian normal scheme and $\blacklozenge/S$ a set of properties of $S$-schemes. Then we shall write Sch$_{\blacklozenge/S}$ for the full subcategory of the category of $S$-schemes Sch$_{/S}$ determined by the objects $X\in {\rm Sch}_{\blacklozenge/S}$ that satisfy every property of $\blacklozenge/S$. In the present paper, we shall mainly be concerned with the properties "reduced", "quasi-compact over $S$", "quasi-separated over $S$", and "separated over $S$". We give a functorial category-theoretic algorithm for reconstructing $S$ from the intrinsic structure of the abstract category Sch$_{\blacklozenge/S}$. This result is analogous to a result of Mochizuki \cite{Mzk04} and may be regarded as a partial generalization of a result of de Bruyn \cite{deBr19} in the case where $S$ is a locally Noetherian normal scheme.