Inessential directed maps and directed homotopy equivalences

A directed space is a topological space X together with a subspace P⃗ (X)⊂XI 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 P⃗ (X)+− of directed paths between a source − and a target + - up to homotopy. Such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of such 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 the pair component categories of directed spaces introduced in \cite{Raussen:18} are invariant under directed homotopy equivalences, as well.