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April 2020 Decomposition spaces and poset-stratified spaces
Shoji Yokura
Tbilisi Math. J. 13(2): 101-127 (April 2020). DOI: 10.32513/tbilisi/1593223222


In 1920s R. L. Moore introduced upper semicontinuous and lower semicontinuous decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $\mathcal D$ of a topological space $X$ such that the decomposition spaces $X/\mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/\mathcal D$ is a poset, then the decomposition map $\pi:X \to X/\mathcal D$ is a continuous map from the topological space $X$ to the poset $X/\mathcal D$ with the associated Alexandroff topology, which is nowadays called a poset-stratified space. As an application, we capture the face poset of a real hyperplane arrangement $\mathcal A$ of $\mathbb R^n$ as the associated poset of the decomposition space $\mathbb R^n/\mathcal D(\mathcal A)$ of the decomposition $\mathcal D(\mathcal A)$ determined by the arrangement $\mathcal A$. We also show that for any locally small category $\mathcal C$ the set $hom_{\mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $\mathfrak{st}^S_*: \mathcal C \to \mathcal Strat$ and a contravariant functor $\mathfrak{st}^*_T$ $\mathfrak{st}^*_T: \mathcal C \to \mathcal Strat$ from $\mathcal C$ to the category $\mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{\mathcal C}(X,Y)$.

Funding Statement

This work was supported by JSPS KAKENHI Grant Numbers 16H03936 and JP19K03468.


The author would like to thank Jim Stasheff, Dai Tamaki and Toshihiro Yamaguchi for useful comments.


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Shoji Yokura. "Decomposition spaces and poset-stratified spaces." Tbilisi Math. J. 13 (2) 101 - 127, April 2020.


Received: 8 October 2019; Accepted: 3 November 2019; Published: April 2020
First available in Project Euclid: 27 June 2020

MathSciNet: MR4117809
Digital Object Identifier: 10.32513/tbilisi/1593223222

Primary: 06A06
Secondary: 18A99, 54B15, 54B99, 55P99

Rights: Copyright © 2020 Tbilisi Centre for Mathematical Sciences


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Vol.13 • No. 2 • April 2020
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