Decomposition spaces and poset-stratified spaces

Abstract

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 $hom_{\mathcal C}(X,Y)$.