Decomposition spaces and poset-stratified spaces

TL;DR

This paper explores the structure of decomposition spaces in topology, especially those that form poset-stratified spaces, and applies these ideas to hyperplane arrangements and category theory.

Contribution

It introduces the concept of poset-stratified spaces arising from lower semicontinuous decompositions and connects them to face posets of hyperplane arrangements and morphism sets in categories.

Findings

Decomposition spaces with poset structures can be viewed as poset-stratified spaces.

The face poset of a hyperplane arrangement is captured as a poset of a decomposition space.

Hom-sets in categories can be structured as poset-stratified spaces with associated functors.

Abstract

In 1920s R. L. Moore introduced \emph{upper semicontinuous} and \emph{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 \emph{a continuous map from the topological space $X$ to the poset $X/\mathcal D$ with the associated Alexandroff topology}, which is nowadays called \emph{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 $\frak {st}^S_*: \mathcal C \to \mathcal Strat$ and a contravariant functor $\frak {st}^*_T$ $\frak {st}^*_T: \mathcal C \to \mathcal Strat$ 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 $hom_{\mathcal C}(X,Y)$.