# Pair component categories for directed spaces

## TL;DR

This paper refines the theory of pair component categories for directed spaces by removing restrictions, using homology instead of homotopy, and offers an alternative to natural homology, enhancing the analysis of directed path spaces.

## Contribution

It extends previous work on directed space categories by relaxing conditions and replacing homotopy with homology, providing new tools for analyzing path space invariants.

## Key findings

- Introduces a homology-based pair component category for directed spaces.
- Provides an alternative to natural homology for computable invariants.
- Refines the concept of stable components in directed topology.

## Abstract

The notion of a homotopy flow on a directed space was introduced in \cite{Raussen:07} as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all parameter directed maps preserve the homotopy type of path spaces, such a flow (and these parameter maps) are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points and whose morphisms may be induced by these inessential d-maps. Localization with respect to subcategories with these inessential d-maps as morphisms can be combined with a path space functor into the homotopy category, the quotient pair component category has as objects pair components along which the homotopy type is invariant -- for a coherent and transparent reason. This paper follows up \cite{FGHR:04,GH:07,Raussen:07} and removes some of the restrictions for their applicability. If one uses homology equivalence instead of homotopy equivalence as the basic relation, it yields an alternative to computable versions of "natural homology" introduced in \cite{DGG:15} and elaborated in \cite{Dubut:17}. It refines, for good and for evil, the stable components introduced and investigated in \cite{Ziemianski:18}.