1 - Directed structures and first-order homotopy properties

Summary

We begin by studying basic homotopy properties, which will be sufficient to introduce directed homology in the next chapter.

Section 1.1 explores the transition from classical to directed homotopy, comparing topological spaces with the simplest topological structure where privileged directions appear: the category pTop of preordered topological spaces.

In Sections 1.2 and 1.3 we begin a formal study of directed homotopy in a dI1-category, i.e. a category equipped with an abstract cylinder endofunctor I endowed with a basic structure. Dually, we have a dP1-category, with a cocylinder (or path) functor P, while a dIP1-category has both endofunctors, under an adjunction I ⊣ P. The higher order structure, developed in Chapter 4, will make substantial use of the ‘second-order’ functors, I² or P².

Sections 1.4 and 1.5 introduce our main directed world, the category dTop of spaces with distinguished paths, or d-spaces, which – with respect to preordered spaces – also contains objects with non-reversible loops, like the directed circle ↑S¹. Then, in Section 1.6, we explore the category Cub of cubical sets and their left or right directed homotopy structures.

Coming back to the general theory, in Section 1.7, we deal with dI1-homotopical categories, i.e. dI1-categories which have a terminal object and all homotopy pushouts, and therefore also mapping cones and suspensions. This leads to the (lower or upper) cofibre sequence of a map, whose classical counterpart for topological spaces is the well-known Puppe sequence [Pu].

These results are dualised in Section 1.8, which is concerned with dP1-homotopical categories, homotopy pullbacks and the fibre sequence of a map. Pointed dIP1-homotopical categories combine both aspects and cover pointed preordered spaces and pointed d-spaces.