arXiv subjects

Kento Ogawa

Publications and source records attributed to Kento Ogawa.

4 recordsLinked to original sources

arXiv2026

Some categorical remarks on coarse subspaces of coarse spaces

In this paper, we provide a categorical framework for understanding coarse subspaces of coarse spaces. First, we introduce the notion of a controlled total relation between coarse spaces and show that the category whose morphisms are closeness classes of controlled total relations is isomorphic to the conventional category of coarse spaces defined using closeness classes of controlled maps. Next, we show that the assignment associating to each coarse space the finite-join partially ordered set of its coarse subspaces is functorial, and prove that this partially ordered set is naturally isomorphic to the poset of subobjects in the category of coarse spaces. Furthermore, we formulate asymptotic disjointness between coarse subspaces and show that mono-morphisms preserve this relation. These results provide a categorical interpretation of the framework of coarse subspaces introduced by Leitner--Vigolo [Lecture Notes in Math.~(2023)] and characterize coarse subspaces as objects intrinsic to the category of coarse spaces. They also provide a foundation for a coarse-geometric interpretation of the properness criterion established by Kobayashi [Math.~Ann.~(1989); J.~Lie Theory (1996)] and Benoist [Ann.~of Math.~(1996)] (cf.~Nagaya--Ogawa--Okuda [Proc.~Japan Acad.~Ser.~A (2025)]).

math.CT↗

arXiv2023

A proof of Kobayashi's properness criterion from a viewpoint of metric geometry

Let $G$ be a locally-compact group and $(H,L)$ a pair of closed subgroups of $G$. For the cases where $G$ is a real linear reductive Lie group, T. Kobayashi [Math. Ann. '89, J. Lie Theory '96] established a criterion for properness of the $L$-action on the homogeneous space $G/H$ in terms of Cartan's KAK-decomposition of $G$. In this paper, we show that a similar theorem also holds if $G$ is a locally-compact group admitting a suitable isometric action on a metric space, and give a proof of Kobayashi's criterion in terms of CAT(0) metric geometry on non-compact Riemannian symmetric spaces.

math.DG↗

arXiv2022

Wreath products and projective system of non Schurian association schemes

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, with an explicit description of primitive idempotents.

math.CO↗

arXiv2022

Functoriality of Bose-Mesner algebras and profinite association schemes

We show that taking the set of primitive idempotents of commutative association schemes is a functor from the category of commutative association schemes with surjective morphisms to the category of finite sets with surjective partial functions. We then consider projective systems of commutative association schemes consisting of surjections (which we call profinite association schemes), for which Bose-Mesner algebra is defined, and describe a Delsarte theory on such schemes. This is another method for generalizing association schemes to those on infinite sets, related with the approach by Barg and Skriganov. Relation with $(t,m,s)$-nets and $(t,s)$-sequences is studied. We reprove some of the results of Martin-Stinson from this viewpoint.

math.CO↗