# Inessential directed maps and directed homotopy equivalences

## TL;DR

This paper investigates inessential symmetries of directed spaces, explores their algebraic and topological properties, and establishes invariance of directed topological complexity and component categories under a new notion of directed homotopy equivalence.

## Contribution

It introduces a new concept of directed homotopy equivalence that differs from previous definitions and proves its invariance properties for directed topological complexity and component categories.

## Key findings

- Directed homotopy equivalences satisfy a 2-out-of-3 property.
- Directed topological complexity is invariant under the new equivalence.
- Directed homotopy equivalences induce isomorphisms on component categories.

## Abstract

A directed space is a topological space $X$ together with a subspace $\vec{P}(X)\subset X^I$ of \emph{directed} paths on $X$. A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec{P}(X)_-^+$ of directed paths between a source ($-$) and a target ($+$) - up to homotopy. If it is, moreover, homotopic to the identity map -- in a directed sense -- such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of inessential d-maps. Comparing two d-spaces $X$ and $Y$ "up to symmetry" yields the notion of a directed homotopy equivalence between them. Under appropriate conditions, all directed homotopy equivalences are shown to satisfy a 2-out-of-3 property. Our notion of directed homotopy equivalence does not agree completely with the one defined in \cite{Goubault:17} and \cite{GFS:18}; the deviation is motivated by examples. Nevertheless, directed topological complexity, introduced in \cite{GFS:18} is shown to be invariant under our notion of directed homotopy equivalence. Finally, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of directed spaces introduced in \cite{Raussen:18}.