Pair component categories for directed spaces

Abstract

The notion of a homotopy flow on a directed space was introduced in [21] as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all directed maps along such a 1-parameter deformation preserve the homotopy types of path spaces, such a flow and the parameter maps are called inessential.For a directed space, one may consider various categories whose objects are pairs of reachable points to which a functor associates the space of directed paths between them. The monoid of all inessential maps acts on such a category by endofunctors leaving the associated path spaces invariant up to homotopy. We construct a pair component category as quotient category: it has as objects pair components along which the homotopy type is invariant -for a coherent and transparent reason.This paper follows up [8,16,21] and removes some of the restrictions for their applicability. At least in several examples, it gives reasonable results for spaces with non-trivial directed loops. 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 [5] and elaborated in [3]. It refines, for good and for evil, the stable components introduced and investigated in [25].1991 Mathematics Subject Classification. 18B35, 55P60, 55U40, 68Q85.

Inessential directed maps and directed homotopy equivalences

“…In § 4, we show that directed topological complexity of d-spaces [9] is invariant under a sharp version of directed homotopy equivalence. In the final § 5, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of [15]. Simple non-trivial examples illustrate the concepts.…”

“…x1 for every pair of points (x 1 , x 2 ) ∈ X 2 ; cf definition 1.2(1). For reasonable d-spaces, like the -spaces from definition 1.4, these path spaces depend only mildly on the pair of points; they are stable within so-called components [15,17]. If we wish that a d-map not only relates the topology of its domain and target, but also the topology of all assembled path spaces, we need to add the following requirement:…”

“…commutes up to homotopy, cf [13,15]. The map L f • t has a fibre homotopy inverse by mapping, for x y, a path σ ∈ P (∂ n )…”

“…It was the main aim of [15] (consult that paper for motivations and details) to organize essential information about the relations between spaces of directed paths (with varying end points) within a given d-space in appropriate categories; to investigate, and if possible, to simplify and/or to compress these categories. Point of departure is the extension category E π 1 (X) of the fundamental category [4, 10] π 1 (X) (objects: points in X; morphisms: homotopy classes of d-paths between source and target).…”

“…All endod-maps give rise to the morphisms of a category d(X) (under composition) on the object set X 2 . The 'mixed' category dE π 1 (X), cf [15], has again X 2 as set of objects; its morphisms are freely generated by those of E π 1 (X) and of d(X) modulo the relations generated by the following two commutative diagrams (fx, fy)…”