This chapter is a complete reworking of previous settings, which were mostly – not exclusively – aimed at the reversible case; they have been developed in various papers, in particular [G1, G3, G4].

Starting from the basic settings of Chapter 1, we arrive in Sections 4.1 and 4.2 at the notion of a ‘symmetric dIP4-homotopical category’ (Section 4.2.6), through various steps, called dI2 and dI3-category (or dI2 and dI3-homotopical category) and their duals, dP2-category, etc. (possibly symmetric).

Special care is given to single out the results which hold in the intermediate settings, and in particular do not depend on the transposition symmetry: we have already remarked that its presence has both advantages and drawbacks (Section 1.1.5).

Some basic examples are dealt with in Sections 4.3 and 4.4; many others will follow in Chapter 5. Thus, dTop, dTop• (pointed d-spaces) and Cat are symmetric dIP4-homotopical categories. On the other hand, the category of reflexive graphs is just dIP2-homotopical (Section 4.3.3), and the category Cub of cubical sets is just dIP1-homotopical, under two isomorphic structures for left and right homotopies (Section 4.3.4). Chain complexes on an additive category form a symmetric dIP4-homotopical category, which is regular and reversible; directed chain complexes have a regular dIP4-homotopical structure, which lifts the previous one but is no longer symmetric nor reversible (Section 4.4).

In the rest of this chapter we work out the general theory of dI2, dI3 and dI4-categories. In Section 4.5 we construct the homotopy 2-category Ho2 (A) and the fundamental category functor ↑Π1: A → Cat, for a dI4-category A.