The characterization for the sobriety of $L$-convex spaces

TL;DR

This paper characterizes the conditions under which stratified $L$-convex spaces are sober, linking sobriety to sobrification and injectivity within the category of stratified $L$-convex spaces, using commutative integral quantales.

Contribution

It provides a new characterization of sobriety in stratified $L$-convex spaces through sobrification and categorical injectivity.

Findings

A stratified sober $L$-convex space is a sobrification of another if a quasihomeomorphism exists.

Sober stratified $L$-convex spaces are exactly the strictly injective objects in their category.

The study extends the understanding of sobriety in the context of $L$-convex spaces with a commutative integral quantale.

Abstract

With a commutative integral quantale $L$ as the truth value table, this study focuses on the characterizations of the sobriety of stratified $L$-convex spaces, as introduced by Liu and Yue in 2024. It is shown that a stratified sober $L$-convex space $Y$ is a sobrification of a stratified $L$-convex space $X$ if and only if there exists a quasihomeomorphism from $X$ to $Y$; a stratified $L$-convex space is sober if and only if it is a strictly injective object in the category of stratified $S_0$ $L$-convex spaces.