Decomposition spaces and posetstratified spaces
Shoji Yokura
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 posetstratified 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 posetstratified 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 posetstratified spaces.
We also make a remark about Yoneda's Lemmas as to posetstratified space structures of $hom_{\mathcal C}(X,Y)$.
Tbilisi Mathematical Journal, Vol. 13(2) (2020), pp. 101127
