The homotopical structures we are studying are ‘categorically algebraic’, in the sense that they are based on endofunctors and ‘operations’ on them (natural transformations between their powers), much in the same way as in the theory of monads.

This is why such a structure can generally be lifted from a ground category A to a categorical construction on the latter, yielding a second category E equipped with a forgetful functor U : E → A, or with a family of functors Ui : E → A.

We treat thus: categories of diagrams and sheaves (Section 5.1), slice categories (Section 5.2), categories of algebras for a monad (Sections 5.3 and 5.4) and categories of differential graded algebras (Sections 5.5–5.7).

Applying – for instance – these results to dTop, the symmetric dIP4-homotopical category of d-spaces, we obtain that any category of diagrams dTopS and any slice category dTop\A or dTop/B is a symmetric dIP4-homotopical category (Sections 5.1.5 and 5.2.6). Furthermore, any category of sheaves Shv(S, dTop), over any site S, is a symmetric dP4-homotopical category (Section 5.1.5).

The same is also true of any category of algebras dTopT, for every monad T on dTop which is made consistent with the path functor, in a natural sense. This yields the homotopical structure of d-topological semigroups, or d-topological groups, or d-spaces equipped with an action of a fixed d-topological group, etc. (Section 5.4).

Similarly, the homotopy structure of Cat can be lifted to the category of strict monoidal categories (Section 5.4.6).

In classical algebraic topology, homotopy equivalence between ‘spaces’ gives rise to a plain equivalence of their fundamental groupoids; therefore, the categorical skeleton provides a minimal model of the latter.

But the study of homotopy invariance in directed algebraic topology is far richer and more complex. Our directed structures have a fundamental category ↑Π1 (X), and this must be studied up to appropriate notions of directed homotopy equivalence of categories, which are more general than categorical equivalence.

We shall use two (dual) directed notions, which take care, respectively, of variation ‘in the future’ or ‘from the past’: a future equivalence in Cat is a future homotopy equivalence (Section 1.3.1) satisfying two conditions of coherence; it can also be seen as a symmetric version of an adjunction, with two units. Its dual, a past equivalence, has two counits. Then we study how to combine these two notions, so as to take into account both kinds of invariance. Minimal models of a category, up to these equivalences, are then introduced to better understand the ‘shape’ and properties of the category we are analysing, as well as of the process it represents.

Within category theory, the study of future (and past) equivalences is a sort of ‘variation on adjunctions’: they compose as the latter (Section 3.3.3) and, moreover, two categories are future homotopy equivalent if and only if they can be embedded as full reflective subcategories of a common one (Theorem 3.3.5); therefore, a property is invariant for future equivalences if and only if it is preserved by full reflective embeddings and by their reflectors.

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.

Homology theories of directed ‘spaces’ will take values in directed algebraic structures. We will use preordered abelian groups, letting the preorder express most (not all) of the information codified in the original distinguished directions. One could also use abelian monoids, following a procedure developed by A. Patchkoria [P1, P2] for the homology semimodule of a ‘chain complex of semimodules’, but this would give here less information (see Section 2.1.4).

Directed homology of cubical sets, the main subject of this chapter, has interesting features, also related to non-commutative geometry.

Indeed, it may happen that the quotient S/˜ of a topological space has a trivial topology, while the corresponding quotient of its singular cubical set □S keeps relevant topological information, identified by its homology and agreeing with the interpretation of such ‘quotients’ in noncommutative geometry. This relationship, briefly explored here, should be further clarified.

Let us start from the classical results on the homology of an orbit space S/G, for a group G acting properly on a space S; these results can be extended to free actions if we replace S with its singular cubical set and take the quotient cubical set (□S)/G (Corollary 2.4.4 and Theorem 2.4.5). Thus, for the group Gν = Z + ϑZ (ϑ irrational), the orbit space R/Gϑ has a trivial topology (the indiscrete one), but can be replaced with a non-trivial cubical set, X = (□R)/Gϑ, whose homology is the same as the homology of the group Gϑ ≅ Z2, and coincides with the homology of the torus T2 (see (2.76)).

As we have seen, directed algebraic topology studies ‘directed spaces’ with ‘directed algebraic structures’ produced by homotopy or homology functors: on the one hand the fundamental category (and possibly its higher dimensional versions), and on the other preordered homology groups. Its general aim is modelling non-reversible phenomena.

We now sketch an enrichment of this subject: we replace the truthvalued approach of directed algebraic topology (where a path is licit or not) with a measure of costs, taking values in the interval [0,∞] of extended (weakly) positive real numbers. The general aim is, now, measuring the cost of (possibly non-reversible) phenomena.

Weighted algebraic topology will study ‘weighted spaces’, like (generalised) metric spaces, with ‘weighted’ algebraic structures, like the fundamental weighted (or normed) category, defined here, and the weighted homology groups, developed in [G10] for weighted (or normed) cubical sets.

Lawvere's generalised metric spaces, endowed with a possibly nonsymmetric distance taking values in [0,∞] (already considered in Section 1.9.6), are a basic setting where weighted algebraic topology can be developed (see Section 6.1). This approach is based on the standard generalised metric interval δI, with distance δ(x, y) = y–x, if x ≤ y, and δ(x, y) = ∞ otherwise; the resulting cylinder functor I(X) = X ⊗δI has the l1-type metric (Section 6.2). We define the fundamental weighted category wΠ1 (X) of a generalised metric spaces, and begin its study (Sections 6.3 and 6.4).

We work with elementary and extended homotopies, which are given by 1-Lipschitz maps (i.e. weak contractions) and Lipschitz maps, respectively (see Section 6.2).

In this book, category theory is used extensively, if at an elementary level. The notions of category, functor and natural transformation are used throughout, together with standard tools like limits, colimits and adjoint functors.

The brief review of this appendix is also meant to fix the notation used here. Proofs can be found in the texts mentioned in Section A1.1, except for some non-standard points at the end of this chapter.

Basic notions

Smallness

Something must be said about set-theoretical aspects, to make precise the meaning of the category of ‘all' sets’, or ‘all’ topological spaces, and so on.

We work within the NBG (von Neumann–Bernays–Göodel) theory, where there are sets and classes, and the class of all sets or all spaces makes sense. In a category A the objects form a class ObA and the morphisms form a class MorA; but, for every pair X, Y of objects, we assume that the morphisms X → Y, also called maps or arrows, form a setA(X, Y). The category is said to be small if the class ObA is a set, and large otherwise.

Aims and applications

Directed algebraic topology is a recent subject which arose in the 1990s, on the one hand in abstract settings for homotopy theory, like [G1], and, on the other hand, in investigations in the theory of concurrent processes, like [FGR1, FGR2]. Its general aim should be stated as ‘modelling non-reversible phenomena’. The subject has a deep relationship with category theory.

The domain of directed algebraic topology should be distinguished from the domain of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. While the classical domain of topology and algebraic topology is a reversible world, where a path in a space can always be travelled backwards, the study of non-reversible phenomena requires broader worlds, where a directed space can have non-reversible paths.

The homotopical tools of directed algebraic topology, corresponding in the classical case to ordinary homotopies, the fundamental group and fundamental n-groupoids, should be similarly ‘non-reversible’: directed homotopies, the fundamental monoid and fundamental n-categories. Similarly, its homological theories will take values in ‘directed’ algebraic structures, like preordered abelian groups or abelian monoids. Homotopy constructions like mapping cone, cone and suspension, occur here in a directed version; this gives rise to new ‘shapes’, like (lower and upper) directed cones and directed spheres, whose elegance is strengthened by the fact that such constructions are determined by universal properties.

Applications will deal with domains where privileged directions appear, such as concurrent processes, rewrite systems, traffic networks, space-time models, biological systems, etc. At the time of writing, the most developed ones are concerned with concurrency: see [FGR1, FGR2, FRGH, Ga1, GG, GH, Go, Ra1, Ra2].

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.