Theory and Applications of Categories, Vol. 10, No. 14, 2002, pp. 369–391.

DIRECTED HOMOTOPY THEORY, II. HOMOTOPY CONSTRUCTS

MARCO GRANDIS

ABSTRACT.

Directed Algebraic Topology studies phenomena where privileged directions appear, derived from the analysis of concurrency, traffic networks, space-time models, etc. This is the sequel of a paper, ‘Directed homotopy theory, I. The fundamental category’, where we introduced directed spaces, their non reversible homotopies and their fundamental category. Here we study some basic constructs of homotopy, like homotopy pushouts and pullbacks, mapping cones and homotopy fibres, suspensions and loops, cofibre and fibre sequences.

Introduction

Directed Algebraic Topology is a recent subject, arising from domains where privileged directions appear, like concurrent processes, traffic networks, space-time models, etc. (cf. [2, 3, 4, 5, 9, 10]). Its domain should be distinguished from classical Algebraic Topology by the principle that directed spaces have privileged directions, and directed paths therein need not be reversible. Its homotopical tools will also be ‘non-reversible’: directed homotopies and fundamental category instead of ordinary homotopies and fundamental groupoid.

This is a sequel of a paper which will be cited as Part I [11]; I.1 (resp. I.1.2, or I.1.2.3) will refer to Section 1 of Part I (resp. its Subsection 1.2, or item (3) in the latter).

In Part I, we introduced directed spaces, their directed homotopies and their fundamental category, including a ‘Seifert-van Kampen’ type theorem, to compute it. The notion of ‘directed space’ which we are using is a topological space X equipped with a family dX of ‘directed paths’ [0, 1] → X, containing all constant paths and closed under increasing reparametrisation and concatenation. Such objects, called directed spaces or d-spaces, with the obvious d-maps - preserving the assigned paths - form a category dTop which has general properties similar to Top. (The prefixes d, ↑ are used to distinguish a directed notion from the corresponding ‘reversible’ one.) Relations of d-spaces with preordered spaces, locally preordered spaces, bitopological spaces and generalised metric spaces with ‘asymmetric’ distance have been discussed in I.1 and I.4.

Work supported by MIUR Research Projects. The author gratefully acknowledges accurate comments and suggestions from the referee.

Received by the editors 2001 December 5 and, in revised form, 2002 August 8.

Transmitted by Ronald Brown. Published on 2002 August 27.

2000 Mathematics Subject Classification: 55P99, 18G55.

Key words and phrases: homotopy theory, homotopical algebra, directed homotopy, homotopy pushouts, homotopy pullbacks, mapping cones, homotopy fibres.

\begin{matrix}c \\ ◯ \end{matrix} Marco Grandis, 2002. Permission to copy for private use granted.

Directed homotopies are based on the standard directed interval ↑I, the directed cylinder functor, ↑I(X) = X×↑I, and its right adjoint, the directed path functor, ↑P (X) = X^{↑I}. Such functors, with a structure consisting of faces, degeneracy, connections and interchange, satisfy the axioms of an IP-homotopical category, as studied in [7] for a different case of directed homotopy, cochain algebras; moreover, for d-spaces, paths and homotopies can be concatenated.

We develop here a study of homotopy pushouts and pullbacks (Section 1); the main results deal with their 2-dimensional universal property and its consequences, and are similar to certain general results of [G1] (for categories equipped with ‘formal homotopies’), yet more delicate because here we have to take care of the direction of homotopies. In Sections 2-3, mapping cones and suspensions are dealt with, as well as homotopy fibres and loop-objects (for pointed d-spaces, of course); higher homotopy monoids ↑π_{n}(X, x) are introduced in 3.4. Combining the present results with the general theory developed in [7], we obtain the cofibre sequence of a map and the fibre sequence of a pointed map, including their ‘exactness property’ (Theorem 2.5), by comparison with sequences of iterated mapping (co)cones, alternatively lower or upper. Note that, even if paths in a d-space X cannot be reversed, generally, they can nevertheless be reflected in the opposite object RX = X^{op}; thus, lower and upper cones determine each other (2.1). The fibre sequence of a pointed map (3.3) produces a sequence of higher homotopy monoids which is not exact, generally (3.5); but it has already been observed in Part I that the homotopy monoids ↑π_{1}(X, x) contain only a fragment of the fundamental category ↑Π_{1}(X): higher dimensional properties should probably be studied by higher fundamental categories (as introduced in [9] for simplicial sets). Finally, in Section 4, we shall see how, on ‘comma categories’ dTop\A of d-spaces under a discrete object, the d-homotopy invariance of the fundamental functor ↑Π_{1} is strict ‘on the base points’.

As in Part I, category theory intervenes at an elementary level. Some basic facts are of frequent use: all (categorical) limits (generalising cartesian products and projective limits) can be constructed from products and equalisers; dually, all colimits (generalising sums and injective limits) can be constructed from sums and coequalisers. Left adjoint functors preserve all the existing colimits, while right adjoints preserve limits; F ⊣ G means that F is left adjoint to G. Comma categories are only used in the last section (see [12, 1]). A map between topological spaces is a continuous mapping. A homotopy φ between maps f, g: X → Y can be written as φ: f → g: X → Y, or φ: f → g, or φ: X ⇒ Y. An order relation is reflexive, transitive and anti-symmetric; a mapping which preserves such relations is said to be increasing (always used in the weak sense). The index α takes values 0, 1, written −, + in superscripts.

1. Directed homotopy pushouts and pullbacks

Homotopy pushouts of d-spaces can be constructed in the usual way, from the directed cylinder. We shall always work with the standard ones, determined up to isomorphism by strict universal properties. The main results, here, concern their 2-dimensional properties.

1.1. A review of directed spaces. A directed topological space X = (X, dX), or dspace (I.1.1), is a topological space equipped with a set dX of (continuous) maps a: I → X, defined on the standard interval I = [0, 1]; these maps, called directed paths or d-paths, must contain all constant paths and be closed under (weakly) increasing reparametrisation and concatenation. The d-space X is thus equipped with a path-preorder x ⪯ x′, defined by the existence of a directed path from x to x′.

A directed map f: X → Y (or d-map, or map of d-spaces) is a continuous mapping between d-spaces which preserves the directed paths: if a ∈ dX, then fa ∈ dY.

The category of d-spaces is written as dTop. It has all limits and colimits, constructed as in Top and equipped with the initial or final d-structure for the structural maps; for instance a path I → ΠX_{i} is directed if and only if all its components I → X_{i} are so. The forgetful functor U: dTop → Top preserves thus all limits and colimits; a topological space is generally viewed as a d-space by its natural structure, where all paths are directed (via the right adjoint to U, I.1.1).

Reversing d-paths, by the involution r(t) = 1 − t, yields the reflected, or opposite, d-space RX = X^{op}, where a ∈ d(X^{op}) if and only if a^{op} = ar ∈ dX. A d-space is symmetric if it is invariant under reflection. More generally, it is reflexive, or self-dual, if it is isomorphic to its reflection.

The directed real line, or d-line ↑R, is the Euclidean line with directed paths given by the (weakly) increasing maps I → R. Its cartesian power in dTop, the n-dimensional real d-space ↑R^{n} is similarly described (with respect to the product order, x ≤ y if x_{i} ≤ y_{i} for all i). The standard d-interval ↑I = ↑[0, 1] has the subspace structure of the d-line; the standard d-cube ↑I^{n} is its n-th power, and a subspace of ↑R^{n}. These d-spaces are not symmetric (for n > 0), yet reflexive. The standard directed circle ↑S^{1} = ↑I/∂I has the obvious ‘counter-clockwise’ d-structure; but we also consider the natural circle S^{1} and the ordered circle ↑O^{1} ⊂ R×↑R (I.1.2); for higher spheres, see 2.3, 3.2.

The directed interval ↑I = ↑[0, 1] is a lattice in dTop; its structure (I.2.1) consists of two faces (∂^{−}, ∂^{+}), a degeneracy (e), two connections or main operations (g^{−}, g^{+}) and an interchange (s)

\xymatrix{\{∗\} \ar[r]^{∂^{α}} \ar[r] & ↑I \ar[l]^{e} & ↑I^{2} \ar[l]_{g^{α}} \ar[l] & ↑I^{2} \ar[r]^{s} & ↑I^{2}}  (1)

\begin{matrix}∂^{−}(∗) = 0, & ∂^{+}(∗) = 1, \\ g^{−}(t, t′) = max(t, t′), & g^{+}(t, t′) = min(t, t′), & s(t, t′) = (t′, t). \end{matrix}

As a consequence, the (directed) cylinder endofunctor of d-spaces, ↑I(X) = X ×↑I, has natural transformations, which are denoted by the same symbols and names

\xymatrix{1 \ar[r]^{∂^{α}} \ar[r] & ↑I \ar[l]^{e} & ↑I^{2} \ar[l]_{g^{α}} \ar[l] & ↑I^{2} \ar[r]^{s} & ↑I^{2}}  (2)

and satisfy the axioms of a cubical monad with interchange ([8], Section 2).

The directed interval ↑I is exponentiable (Theorem I.1.7): this means that the cylinder functor ↑I = −×↑I has a right adjoint, the (directed) path functor, or cocylinder ↑P

↑P: dTop → dTop,  ↑P (Y) = Y^{↑I},  (3)

where the d-space Y^{↑I} is the set of d-paths dTop(↑I, Y) with the usual compact-open topology and the d-structure where a map c: ↑I → dTop(↑I, Y) is directed if and only if, for all increasing maps h, k: ↑I → ↑I, the derived path t ↦ c(h(t))(k(t)) is in dY (I.2.2).

The lattice structure of ↑I in dTop produces - contravariantly - a dual structure on ↑P (a cubical comonad with interchange [8]); the derived natural transformations (faces, etc.) will be named and written as above, but proceed in the opposite direction and satisfy dual axioms (note that ↑P^{2}(Y) = (Y^{↑I})^{↑I} = Y^{↑I^{2}}, by composing adjunctions)

\xymatrix{1 \ar[r]_{e} & ↑P \ar[l] \ar[l]_{∂^{α}} \ar[r]^{g^{α}} \ar[r] & ↑P^{2} & ↑P^{2} \ar[r]^{s} & ↑P^{2}}  (4)

\begin{matrix}∂^{−}(a) = a(0), ∂^{+}(a) = a(1), & e(x)(t) = x, \\ g^{−}(a)(t, t′) = a(max(t, t′)), … \end{matrix}

Now, a (directed) homotopy φ: f → g: X → Y is defined as a d-map φ: ↑IX = X×↑I → Y whose two faces ∂^{±}(φ) = φ.∂^{±}: X → Y are f and g, respectively. Equivalently, it is a d-map X → ↑P (Y) = Y^{↑I}, with faces as above. A (directed) path a: ↑I → X is the same as a d-path a ∈ dX, and amounts to a homotopy between two points, a: x → x′: {∗} → X. The structure of d-homotopies (I.2.3) essentially consists of the following operations (for u: X′ → X, v: Y → Y′, ψ: g → h)

(a) whisker composition of maps and homotopies:

v^{∘}φ^{∘}u: vfu → vgu (v^{∘}φ^{∘}u = v.φ.↑Iu: ↑IX′ → Y′),

(b) trivial homotopies:

0_{f}: f → f (0_{f} = fe: ↑IX → Y),

(c) concatenation of homotopies:

φ + ψ: f → h (defined via the concatenation of d-paths).

(The whisker composition will also be written by juxtaposition, when this is not ambiguous.) The category of d-spaces is an IP-homotopical category ([7], 2.7); loosely speaking, it has:

- adjoint endofunctors ↑I ⊣ ↑P, with the required structure (faces, etc., satisfying the axioms);

- all pushouts (preserved by the cylinder) and all pullbacks (preserved by the cocylinder);

- initial and terminal object.

Therefore, all results of [7] for such a structure apply (as for cochain algebras). Moreover, here we can concatenate paths and homotopies.

Let us also briefly recall that ‘d-homotopy relations’ require some care (cf. I.2.4, I.2.7). First, we have a d-homotopy preorder f ⪯ g, defined by the existence of a homotopy f → g (and extending the path-preorder of points); it is consistent with composition but nonsymmetric (f ⪯ g being equivalent to Rg ⪯ Rf). Second, we write f ≃ g the equivalence relation generated by ⪯: there is a finite sequence f ⪯ f_{1} ⪰ f_{2} ⪯ f_{3} … g (of d-maps between the same objects); it is a congruence of categories on dTop.

To conclude this review of Part I, a d-homotopy equivalence is a d-map f: X → Y having a d-homotopy inverse g: Y → X, in the sense that gf ≃ idX, fg ≃ idY; then we say that X and Y have the same d-homotopy type, or are d-homotopy equivalent (in n steps if n instances of the homotopy preorder ⪯ are sufficient for each of the previous ≃-relations, cf. I.2.7). In particular, if idX ⪯ gf and idY ⪯ fg, we say that X and Y are immediately d-homotopy equivalent, in the future; if, further, idX = gf, then f embeds X as a future deformation retract of Y.

1.2. Homotopy pushouts. Let f: X → Y and g: X → Z be two morphisms of directed spaces, with the same domain. The standard (directed) homotopy pushout, or h-pushout, from f to g is a four-tuple (A; u, v; λ), as in the left diagram, where λ: uf → vg: X → A is a homotopy satisfying the following universal property (of cocomma squares)

[figure: X; g; f; Z; v; X; id; id; X; ∂^{+}; Y; u; λ; A; X; ∂^{−}; _{ev_{X}}; ↑IX]  (5)

- for every λ′: u′f → v′g: X → A′, there is precisely one map h: A → A′ such that u′ = hu, v′ = hv, λ′ = h^{∘}λ.

The existence of the solution is proved below; its uniqueness up to isomorphism is obvious. The object A, a ‘double mapping cylinder’, will be denoted as ↑I(f, g). Note that ↑I(g, f) = R(↑I(Rf, Rg)), where RX = X^{op} is the reflected d-space (1.1).

As shown in the right diagram, the cylinder itself ↑I(X) = X×↑I, equipped with the obvious homotopy (cylinder evaluation, represented by the identity of the cylinder)

ev^{X}: ∂^{−} → ∂^{+}: X → ↑IX,  ev^{X}(x, t) = (x, t),  (6)

is the h-pushout of the pair (idX, idX): by the very definition of homotopies, it establishes a bijection between maps h: ↑IX → W and homotopies h^{∘}ev^{X}: h∂^{−} → h∂^{+}: X → W. On the other hand, every homotopy pushout in dTop can be constructed from the cylinder and ordinary pushouts, by the colimit of the following diagram (which amounts to two ordinary pushouts)

\xymatrix{Y & X \ar[l]_{f} \ar[r]^{∂^{−}} & ↑IX & X \ar[l]_{∂^{+}} \ar[r]^{g} & Z}  (7)

i.e. as the quotient of the sum (↑IX) + Y + Z, under the equivalence relation identifying (x, 0) with f(x) and (x, 1) with g(x). The forgetful functor U: dTop → Top preserves cylinders and pushouts, hence h-pushouts as well.

1.3. Homotopy pullbacks. Dualising 1.2, the h-pullback from f: Y → X to g: Z → X is a four-tuple (A; u, v; λ), as in the left diagram, where λ: fu → gv: A → X satisfies the following universal property (of comma squares)

[figure: Y; f; X; X; id; X; λ; _{ev_{X}}; A; v; u; Z; g; ↑P X; ∂^{+}; ∂^{−}; X; id]  (8)

- for every λ′: fu′ → gv′: A′ → X, there is exactly one map h: A′ → A such that u′ = uh, v′ = vh, λ′ = λ^{∘}h.

The object A will be denoted as ↑P (f, g). Again, ↑P (g, f) = R(↑P (Rf, Rg)). As shown in the right diagram above, the path-object ↑P X is the h-pullback of the pair (idX, idX), via the obvious homotopy ev_{X}: ∂^{−} → ∂^{+}: ↑P X → X (path evaluation, represented by id(↑P X)). All homotopy pullbacks in dTop can be constructed from paths and ordinary pullbacks, by the following limit (which amounts to two ordinary pullbacks)

\xymatrix{Y \ar[r]^{f} & X & ↑P X \ar[l]_{∂^{−}} \ar[r]^{∂^{+}} & X & Z \ar[l]_{g}}  (9)

i.e. as the following d-subspace of the product (↑P X)×Y ×Z

↑P (f, g) = {(a, y, z) ∈ (↑P X)×Y ×Z | a(0) = f(y), a(1) = g(z)}.  (10)

Note that the forgetful functor U: dTop → Top does not preserve path-objects (nor h-pullbacks): U(↑P X) is a subspace of P (UX), and a proper one unless ↑P X is just a ‘space’.

In the rest of this section we shall study h-pushouts, in a rather formal way (which could be easily extended to an abstract IP-homotopical category with concatenation and ‘accelerations’, cf. I.2.6.3)); also the dual properties, for h-pullbacks, hold.

1.4. Theorem. [The higher property] The h-pushout A = ↑I(f, g) satisfies also a 2dimensional universal property. Precisely, given two maps a, b, two homotopies σ, τ and a double homotopy Φ (I.2.5) with the following boundaries

[figure: Y; u; au; bu; ↓σ; W; λ; auf; aλ; σf; avg; τg; X; f; g; A; a; b; W; Φ; buf; bλ; bvg; Z; v; av; bv; ↓τ; W]  (11)

a, b: A → W,  σ: au → bu,  τ: av → bv,  Φ: X ×↑I^{2} → W,
∂_{1}^{−}(Φ) = Φ.(X ×∂^{−}×↑I) = σ^{∘}f, ∂_{2}^{−}(Φ) = Φ.(X ×↑I×∂^{−}) = a^{∘}λ, …

there is some homotopy φ: a → b such that φ^{∘}u = σ, φ^{∘}v = τ (and precisely one which also satisfies φ.(λ×↑I) = Φ)).

Proof. By the adjunction ↑I ⊣ ↑P, we can view the data as d-maps with values in ↑P W, namely

\begin{matrix}σ′: Y → ↑P W, & τ′: Z → ↑P W, \\ Φ′ : X ×↑I → ↑P W, & ∂^{−}Φ′ = σ′f, ∂^{+}Φ′ = τ′g. \end{matrix}  (12)

There is thus one map φ′: A → ↑P W such that φ′u = σ′, φ′v = τ′, φ′^{∘}λ = Φ′. This is the same as a homotopy φ: ↑IA → W satisfying our conditions. Moreover, its lower face is a (and the upper one is b) because

\begin{matrix}(∂^{−}φ′).u = ∂^{−}σ′ = au, & (∂^{−}φ′).v = ∂^{−}τ′ = av, \\ (∂^{−}φ′)^{∘}λ = Φ.∂^{−}(X ×↑I) = a^{∘}λ. \end{matrix}  (13)

1.5. Theorem. [The h-pushout functor] The double mapping cylinder ↑I(f, g) acts functorially on the variables f, g (precisely, it is a functor dTop^{v} → dTop, where v is the category formed by two diverging arrows: • ← • → •) and turns coherent d-homotopies into d-homotopies, as specified below.

(a) Given a morphism (x, y, z): (f, g) → (f′, g′), consisting of two commutative squares in dTop

\xymatrix{Y \ar[d]_{y} & X \ar[l]_{f} \ar[r]^{g} \ar[d]^{x} & Z \ar[d]^{z} \\ Y′ & X′ \ar[l]^{f′} \ar[r]_{g′} & Z′}  (14)

there is precisely one map a = ↑I(x, y, z): ↑I(f, g) → ↑I(f′, g′) coherent with the hpushouts, i.e. such that (as in the left cube below)

au = u′y,  av = v′z,  a^{∘}λ = λ′^{∘}x;  (15)

[figure: Y; u; y; Y; u; η; λ; λ; X; f; g; x; A; a; X; f; g; ξ; A; φ; Z; v; z; Z; v; ζ; Y′; u′; Y′; u′; λ′; λ′; X′; f′; g′; A′; X′; f′; g′; A′; Z′; v′; Z′; v′]  (16)

(b) Given a coherent system of homotopies, from the pair (f, g) to the pair (f′, g′), as in the right cube above (where the double arrow at ξ stands for ξ: x → x′: X → X′; etc).

\begin{matrix}(ξ, η, ζ): (x, y, z) → (x′, y′, z′): (f, g) → (f′, g′), \\ ξ: x → x′,  η: y → y′,  ζ: z → z′, \\ f′^{∘}ξ = η^{∘}f,  g′^{∘}ξ = ζ^{∘}g, \end{matrix}  (17)

there is some homotopy φ: a → a′: ↑I(f, g) → ↑I(f′, g′) which completes coherently the cube

φ: a → a′, φ^{∘}u = u′^{∘}η, φ^{∘}v = v′^{∘}ζ,  (18)

and precisely one such φ if we also ask that φ.(λ×↑I) = λ′.(ξ×↑I).s.

Proof. (a) Immediate, from the first universal property of the h-pushout λ of (f, g).

(b) Follows from the 2-dimensional property of the same h-pushout, with respect to the double homotopy Φ = λ′.(ξ×↑I).s: X ×↑I^{2} → X′×↑I → A′

[figure: Y; u; au; a′u; ↓u′η; A′; λ; auf; aλ; u′ηf; avg; v′ζg; X; f; g; A; a; a′; A′; Φ; a′uf; a′λ; a′vg; Z; v; av; a′v; ↓u′ζ; A′]  (19)

∂_{1}^{−}(Φ) = u′f′^{∘}ξ = u′^{∘}η^{∘}f, ∂_{2}^{−}(Φ) = λ′^{∘}x = a^{∘}λ, …

There is thus some homotopy φ: a → a′ such that φ^{∘}u = u′^{∘}η, φ^{∘}v = v′^{∘}ζ; and precisely one which also satisfies φ.(λ×↑I) = Φ.

1.6. Theorem. [Pasting of h-pushouts] Let ξ, η, ζ be standard homotopy pushouts

[figure: X; f; x; Y; g; y; Z; z; Z; b; T; u; ξ; A; v; w; η; B; T; a; ζ; C]  (20)

Then B and C are immediately d-homotopy equivalent, in the future (1.1), by canonical maps and homotopies.

Precisely, define w: A → C, h: B → C and k: C → B through the universal property of ξ, η, ζ, respectively (and the concatenation of homotopies, for k).

\begin{matrix}w.u = a: T → C, & w.y = bg: Y → C, & w^{∘}ξ = ζ: ax → bgf, \\ h.v = w: A → C, & h.z = b: Z → C, & h^{∘}η = 0: wy → bg, \\ k.a = vu: T → B, & k.b = z: Z → B, & k^{∘}ζ = (v^{∘}ξ + η^{∘}f): vux → zgf. \end{matrix}  (21)

Then h, k form an immediate homotopy equivalence, with idC ⪯ hk and idB ⪯ kh; the following higher coherence relations can also be obtained (note we are not saying that ψ^{∘}v = 0).

\begin{matrix}φ: idC → hk, & φ^{∘}a = 0_{a}, & φ^{∘}b = 0_{b}, \\ ψ: idB → kh, & ψ^{∘}z = 0_{z}, & ψ^{∘}vu = 0_{vu}, & ψ^{∘}vy = 0 + η. \end{matrix}  (22)

By reflection, if we paste d-homotopy pushouts (placed as above) with the opposite direction of homotopies, we get an immediate d-homotopy equivalence in the past.

Proof. (This result will be essential for the sequel, e.g. to prove the homotopical exactness of the cofibre sequence, in Theorem 2.5. It is a refinement of a similar result in [6], 3.4.)

First, the higher universal property of ζ (1.4) yields a homotopy φ: idC → hk (with φ^{∘}a = 0_{a}, φ^{∘}b = 0_{b}), provided by the acceleration double homotopy ζ → ζ + 0 (I.2.6.3), denoted by #

\xymatrix{ax \ar[rr]^{ζ} \ar[dd]_{0_{a}x} &  & bgf \ar[dd]^{0_{b}gf} & hka = hvu = wu = a, \\  & # &  & hkb = hz = b, \\ hkax \ar[rr]_{hkζ} &  & hkbgf &  & hk^{∘}ζ = hv^{∘}ξ + h^{∘}η^{∘}f = w^{∘}ξ + 0 = ζ + 0.}  (23)

We exploit now the higher universal property of ξ to link the maps v, kw: A → B; consider the following three double homotopies (acceleration, degeneracy and upper connection)

\xymatrix{vux \ar[rrrr]^{vξ} \ar[dd]_{0x} &  &  &  & vyf \ar[dd]_{0f} \ar[dddd]^{(0+η)f} \\  &  & # &  &  & kwu = ha = vu, \\ vux \ar[rr]^{vξ} \ar[dd]_{0x} &  & vyf \ar[rr]^{0} \ar[dd]^{0} &  & vyf \ar[dd]_{ηf} \\  & # &  & # &  &  & kw^{∘}ξ = k^{∘}ζ = v^{∘}ξ + η^{∘}f. \\ kwux \ar[rr]_{vξ} &  & vyf \ar[rr]_{ηf} &  & kwyf}  (24)

Their pasting yields a homotopy ρ: v → kw such that ρ^{∘}u = 0, ρ^{∘}y = 0 + η. Finally, the higher property of η produces a homotopy ψ: idB → kh (such that ψ^{∘}v = ρ, ψ^{∘}z = 0), through a double homotopy deriving from degeneracy and lower connection

[figure: vy; η; 0; ρy; zg; 0g; #; ρ^{∘}y = 0 + η,; vy; η; η; zg; 0g; #; kh^{∘}η = 0: kwy → kbg.; zg; khη; zg]  (25)

1.7. Cofibrations. Say that a d-map u: X → A is a lower d-cofibration if (see the left diagram below), for every d-space W and every d-map h: A → W, every d-homotopy ψ: h′ = hu → k′ can be ‘extended’ to a d-homotopy φ: h → k on A (so that φ^{∘}u = ψ, whence ku = k′)

[figure: X; u; h′; k′; ↓ψ; W; W; h; k; ↓φ; X; p; A; h; k; ↓φ; W; W; h′; k′; ↓ψ; B]  (26)

The opposite notion (for every k and ψ: h′ → k′ = ku there is some φ such that φ^{∘}u = ψ) is called upper d-cofibration; a bilateral d-cofibration has to satisfy both conditions.

The right diagram above shows the definition of a lower d-fibration p: X → B: for every h: W → X and ψ: h′ = ph → k′ there is some φ: h → k which lifts y (p^{∘}φ = ψ).

1.8. Theorem. (a) In every h-pushout A = ↑I(f, g) (as in (5)), the first ‘injection’ u: Y → A is an upper d-cofibration, while the second v: Z → A is a lower one. (a*) In every h-pullback, the first ‘projection’ is an upper d-fibration, while the second is a lower one.

Proof. It is sufficient to verify the second statement of (a). Take a d-map h: A → W and a d-homotopy ψ: h′ = hv → k′. Then, there is one map k: A → W such that

ku = hu, kv = k′, k^{∘}λ = h^{∘}λ + ψ^{∘}g: huf → k′g: X → W.  (27)

Moreover, by the last relation, we can construct a double homotopy as in the right diagram below (as in (24), by acceleration, degeneracy and upper connection)

[figure: Y; u; hu; ku; ↓0; W; λ; huf; hλ; 0; hvg; ψg; X; f; g; A; h; k; W; #; kuf; kλ; kvg; Z; v; hv; kv; ↓ψ; W]  (28)

and this produces a homotopy φ: h → k such that φ^{∘}v = ψ.

1.9. Theorem. [h-pushouts of cofibrations] If f: X → Y is a lower d-cofibration and g: X → Z any map, the obvious map h: A → V from the h-pushout A = ↑I(f, g) to the ordinary pushout V is an immediate d-homotopy equivalence, in the past

[figure: Y; u; u′; λ; hu = u′, hv = v′,; X; f; g; A; h; V; h^{∘}λ = 0: u′f → v′g.; Z; v; v′]  (29)

Proof. The extension property of f ensures that the homotopy λ: uf → vg can be extended to some homotopy φ: u → w: Y → A; thus φ^{∘}f = λ and wf = vg. There is then one map k: V → A such that ku′ = w, kv′ = v.

First, 1_{A} ⪯ kh, by the 2-dimensional property of λ, using a lower connection (since φ^{∘}f = λ and kh^{∘}λ = 0)

[figure: Y; u; u; w; ↓φ; A; λ; uf; λ; φf; vg; 0; X; f; g; A; 1; kh; A; #; khuf; khλ; khvg; Z; v; v; v; ↓0; A]  (30)

Second, 1_{V} ⪯ hk, by the 2-dimensional property of the ordinary pullback V (which trivially holds since homotopies are represented by a cocylinder). Since the pair of homotopies h^{∘}φ: u′ → hw, 0: v′ → v′ is coherent with f, g (h^{∘}φ^{∘}f = h^{∘}λ = 0 = 0^{∘}g), there is one homotopy ψ such that ψ^{∘}u′ = h^{∘}φ and ψ^{∘}v′ = 0_{v′}; finally, ψ: 1_{V} → hk, since

∂^{−}(h^{∘}φ) = hu = u′, ∂^{+}(h^{∘}φ) = hw = hk.u′, ∂^{±}(0_{v′}) = v′.

2. Mapping cones and the cofibre sequence

Mapping cones (i.e., homotopy cokernels) and suspensions are particular instances of homotopy pushouts. The cofibre sequence of a map has strong properties of ‘homotopical exactness’: it is homotopy equivalent to a sequence of iterated mapping cones.

2.1. Mapping cones. In contrast with ordinary homotopy, the lack of a reversion for directed homotopies produces two mapping cones, generally non isomorphic, yet linked by reflection.

Every d-map f: X → Y has an upper h-cokernel, or upper mapping cone ↑C^{+}f = ↑I(f, t_{X}), the h-pushout from f to the terminal map t_{X}: X → {∗}, as in the left diagram below; it can be obtained as the quotient of the sum ↑IX +Y +{∗} under the equivalence relation identifying (x, 0) with f(x) and (x, 1) with {∗}, for all x ∈ X (cf. (7)); its vertex v^{+} ‘is in the future’: it can be reached from every point (it is a maximum in the path preorder, and actually the only one)

[figure: X; t_{X}; f; {∗}; v^{+}; X; f; t_{X}; Y; c^{+}; Y; c^{−}; γ; ↑C^{+}f; {∗}; v^{−}; γ; ↑C^{−}f]  (31)

(This is called the ‘lower’ mapping cone in [7]; the present choice of terms, based on the vertex rather than the basis, derives from the analysis of contractility, at the end of 2.2).

Symmetrically, one obtains the lower mapping cone ↑C^{−}f = ↑I(t_{X}, f) = R↑C^{+}(Rf)), as in the right diagram; the equivalence relation identifies now (x, 0) with ∗ and (x, 1) with f(x); the vertex v^{−} ‘is in the past’: it can reach every point (is a minimum in the path preorder).

As an easy consequence of the homotopy invariance of the h-pushout functor (Theorem 1.5), the mapping cone functor is d-homotopy invariant as well, in the same sense

↑C^{α}: dTop^{2} → dTop;  (32)

(2 is ‘the one-arrow category’ • → •; an object of dTop^{2} is a d-map, while a morphism is a commutative square of d-maps, as in (14) with Z = {∗}).

Note that h-cokernels are based on the terminal object {∗}. Working dually with the initial object ∅ would give trivial results: all h-kernels would be empty; as in the ordinary case, one has to move to the pointed case to get h-kernels of interest; this will be considered in the next section.

2.2. Cones. Applying these constructs to an identity, we have the upper d-cone ↑C^{+}X = ↑C^{+}(idX) of an object, from the basis (in the past) to the vertex (in the future); and the lower one, ↑C^{−}X. The functors ↑C^{α}: dTop → dTop are d-homotopy invariant.

In Top, the cone of the circle is the compact disc. In dTop, we get six different dspaces by letting ↑C^{α} act on S^{1}, ↑S^{1} and ↑O^{1}. Thus, the natural circle S^{1} has an upper d-cone ↑C^{+}S^{1}, where a path has to move - anyhow - towards the centre (the vertex), at least in the weak sense, and a lower d-cone ↑C^{−}S^{1} = R(↑C^{+}S^{1}), where paths proceed the other way and the centre is the only point which can reach all the others. In the d-spaces ↑C^{α}(↑S^{1}), a ‘pointlike vortex’ appears at the centre (showing that such d-spaces cannot be defined by a local preorder, cf. I.1.6);

[figure: ↑C^{+}S^{1}; ↑C^{−}S^{1}; ↑C^{+}↑S^{1}; ↑C^{−}↑S^{1}; .; ↓; ↑; →; ←; .; ↑; ↓; ←; →; .; ↓; ↑; →; ←; .; ↑; ↓; ←; →] (33)

the fundamental category of these d-spaces has been computed in I.3.5, I.3.7.

In Top, again, the projective plane P^{2} is the ordinary mapping cone of the endomap f: S^{1} → S^{1} of degree 2. In dTop, the same mapping, viewed as f: S^{1} → S^{1} and g: ↑S^{1} → ↑S^{1}, yields four directed versions of the projective plane: the quotients of the previous disks under the usual equivalence relation, namely ↑C^{+}f, ↑C^{−}f ≅ R(↑C^{+}f), ↑C^{+}g, ↑C^{−}g ≅ R(↑C^{+}g). The last isomorphism follows from ↑S^{1} being reflexive, i.e. isomorphic to R(↑S^{1}).

A d-space X is said to be future contractible if it has a future deformation retract at some point (1.1; I.2.7). This happens if and only if the basis c^{−}: X → ↑C^{+}X has a retraction h: ↑C^{+}X → X (apply the universal property of ↑C^{+}X). The cone ↑C^{+}X itself is future contractible (to its vertex v^{+}, and only to this point) by means of the homotopy induced on ↑C^{+}X ×↑I by the lower connection (constant at t = 1 and at t′ = 1)

g^{−}: ↑C^{+}X ×↑I → ↑C^{+}X,  g^{−}[x, t, t′] = [x, max(t, t′)].  (34)

2.3. Suspension. The (directed, non-pointed) suspension ↑ΣX is a lower and upper d-cone, at the same time

[figure: X; t_{X}; t_{X}; {∗}; v^{+}; ↑ΣX = ↑I(t_{X}, t_{X}) = ↑C^{−}(t_{X}) = ↑C^{+}(t_{X}),; R.↑Σ = ↑Σ.R.; {∗}; v^{−}; γ; ↑ΣX]  (35)

Concretely, it is the quotient of the sum ↑IX + {v^{−}} + {v^{+}} which identifies the lower basis with a lower vertex v^{−}, and the upper basis with an upper vertex v^{+}. It is equipped with a homotopy (suspension evaluation)

ev^{X}: v^{−}t_{X} → v^{+}t_{X}: X → ↑ΣX,  (x, t) ↦ [x, t],  (36)

which is universal for homotopies between constant maps. In particular, ↑Σ({∗}) = ↑I and ↑Σ(∅) = S^{0} (note that the latter is not a quotient of ↑I(∅) = ∅ !).

The suspension ↑Σ is an endofunctor of dTop (by 1.5): given f: X → Y, the suspended map ↑Σf: ↑ΣX → ↑ΣY is the unique morphism which satisfies the conditions

↑Σf.v^{−} = v^{−}, ↑Σf.v^{+} = v^{+}, (↑Σf)^{∘}ev^{X} = ev^{Y} ^{∘}f;  (37)

moreover, by 1.5, this functor is homotopy invariant: given a homotopy φ: f → g, there is some homotopy ψ: ↑Σf → ↑Σg (in fact, there is precisely one such that ψ.(ev^{X} ×↑I) = ev^{Y} .(φ×↑I).s). Therefore, ↑Σ preserves immediate homotopy equivalences, and n-step homotopy equivalences as well.

The (unpointed!) suspension of S^{0} is the quotient of ↑I + ↑I which identifies lower and upper endpoints, separately. This coincides with the d-structure induced by R×↑R on the standard circle (or any circle), called the ordered circle (in I.1.2.5), because it is of (partial) order type

↑Σ(S^{0}) = ↑O^{1} ⊂ R×↑R.  (38)

More generally, one can define the ordered n-sphere ↑O^{n} = ↑Σ^{n}(S^{0}). It is isomorphic to the structure induced on the standard n-sphere by R×↑R^{n}, as well as to the pasting of two ordered discs ↑B^{n} ⊂ ↑R^{n} along their boundary; the latter description shows that ↑O^{n} is indeed of order type (while R×↑R^{n} is just of preorder type, since the natural R has the chaotic preorder, cf. 1.1).

2.4. The cofibre sequence. Every d-map f: X → Y has a lower cofibre sequence, produced by lower h-cokernels (as well as an upper one)

\xymatrix{X \ar[r]^{f} & Y \ar[r]^{x} & ↑C^{−}f \ar[r]^{d} & ↑ΣX \ar[r]^{↑Σf} & ↑ΣY \ar[r]^{↑Σx} & ↑Σ(↑C^{−}f) \ar[r]^{↑Σd} & ↑Σ^{2}X …}  (39)

x = c^{+}: Y → ↑C^{−}f;
d.v^{−} = v^{−}: {∗} → ↑ΣX,  d.x = v^{+}.t_{Y}: Y → ↑ΣX,
d^{∘}γ = ev^{X}: v^{−}.t_{X} → v^{+}.t_{X}: X → ↑ΣX.

(More generally, this works for any category with cylinder functor, terminal object and pushouts; cf. [7], Section 1.) Moreover, as sketched below, in 2.5 (and proved - more generally - in [7], 1.10, 3.4), this sequence can be linked, via a homotopically commutative diagram, to a sequence of iterated h-cokernels of f, where each map is, alternatively, the lower or upper h-cokernel of the preceding one

[figure: X; f; Y; x; ↑C^{−}f; d; ↑ΣX; ↑Σf; ↑ΣY; ↑Σx; ↑Σ(↑C^{−}f); ↑Σd; ↑Σ^{2}X …; ≃; ≃; ≃; X; f; Y; x; ↑C^{−}f; x_{2}; ↑C^{+}x; x_{3}; h_{1}; ↑C^{−}x_{2}; x_{4}; h_{2}; ↑C^{+}x_{3}; x_{5}; h_{3}; ↑C^{−}x_{4} …; h_{4}]  (40)

If f is an upper d-cofibration (1.7), we can replace ↑C^{−}f with the ordinary pushout of f along X → {∗}, i.e. the quotient Y/f(X) (Theorem 1.9), which is immediately d-homotopy equivalent to the former (in the future).

2.5. Theorem. [Homotopical exactness of the cofibre sequence] Every comparison h_{n} in the diagram (40) is a d-homotopy equivalence in m steps (possibly less), where m is the integral part of (n + 2)/3.

Proof. (Note that this need not be true in the setting of [7], where the concatenation of homotopies is missing; and actually, it is not true for cochain algebras, where the comparisons are just homotopy equivalences of cochain complexes.)

In fact, the map h_{1}: ↑C^{+}x → ↑ΣX derives from pasting the h-pushouts ↑C^{−}f, ↑C^{+}x and comparing them to the suspension (it is constructed as h in (21)); Theorem 1.6 ensures that it is an immediate d-homotopy equivalence, in the future

[figure: X; f; t_{X}; Y; t_{Y}; x; {∗}; v^{+}; {∗}; v^{+}; {∗}; v^{−}; γ; ↑C^{−}f; x_{2}; d; δ; ↑C^{+}x; {∗}; v^{−}; ev; ↑ΣX]  (41)

\begin{matrix}h_{1}.x_{2} = d: ↑C^{−}f → ↑ΣX, & h_{1}.v^{+} = v^{+}: \{∗\} → ↑ΣX, \\ h_{1}.δ = 0: dx → v^{+}.t_{Y}: Y → ↑ΣX. \end{matrix}

The maps h_{2} and h_{3} are similarly defined, and are immediate d-homotopy equivalences as well, in the past and the future, respectively. Then h_{4} = ↑Σh_{1}.k_{4} is a 2-step d-homotopy equivalence, obtained by composing ↑Σh_{1} (an equivalence in the future, by the homotopy invariance of suspension, in 2.3) with a map k_{4}: ↑C^{−}x_{4} → ↑Σ(↑C^{+}x) obtained as the previous three comparisons (an equivalence in the past). One proceeds this way, adding one step every third index.

Finally, by the homotopy invariance of the mapping cone functors (32), the cofibre sequence acts functorially on the variable f (in dTop^{2}), and this functor preserves dhomotopies.

2.6. An example. Let X = c_{0}I be the Euclidean interval with the d-discrete structure (where d-paths are constant) and Y = ↑I; let f: X → Y be the d-map provided by idI. We want to compute the comparison h_{1}: ↑C^{+}x → ↑ΣX.

First, ↑ΣX is the quotient of c_{0}I×↑I which identifies the lower (resp. upper) basis with a point v^{−} (resp. v^{+}); its d-paths a: v^{−} → v^{+} move up the fibres (more precisely, they are all maps t ↦ [t_{0}, a(t)] where a: ↑I → ↑I is a d-path from 0 to 1), and are homotopic (with fixed endpoints) if and only if they pertain to the same fibre (same t_{0} ∈ c_{0}I); thus, the set of homotopy classes ↑Π_{1}(↑ΣX)(v^{−}, v^{+}) can be identified with the interval [0, 1]

[figure: ↑ΣX; ↑C^{−}f; ↑C^{+}x.; 0; 1; v^{−}; v^{+}; •; t_{0}; 0; 1; v^{−}; ∂^{+}(↑I); 0; 1; v^{−}; v^{+}] (42)

On the other hand, ↑C^{+}x = ↑C^{+}(∂^{+}: Y → ↑C^{−}f) has a coarser structure on its ‘upper half’, deriving from the cone on Y = ↑I. The comparison h_{1}: ↑C^{+}x → ↑ΣX is now the obvious map which collapses this upper half.

3. Pointed d-spaces and the fibre sequence

Pointed d-spaces have a homotopy structure similar to the unpointed case. Their homotopy fibres (i.e., homotopy kernels) and loop-objects form a fibre sequence with exactness properties dual to the ones of the cofibre sequence. But the derived sequence of higher homotopy monoids is not exact.

3.1. Pointed d-spaces. The category dTop_{∗} of pointed d-spaces behaves (with respect to dTop) much as its ordinary counterpart, the category Top_{∗} of pointed spaces (with respect to Top).

Limits and coequalisers in dTop_{∗} are computed as in dTop (and conveniently pointed); the sum is a quotient of the unpointed sum, modulo the equivalence relation which identifies all base points; but it will be preferably realised in the usual form of a join, as a pointed subspace of the corresponding product. Thus, in the binary case

(X, x_{0}) ∨ (Y, y_{0}) = ((X ×{y_{0}}) ∪ ({x_{0}}×Y), (x_{0}, y_{0})) ⊂ (X ×Y, (x_{0}, y_{0})).  (43)

Also here, the smash product, formed by collapsing the join in the product

(X, x_{0}) ∧ (Y, y_{0}) = ((X ×Y)/((X ×{y_{0}}) ∪ ({x_{0}}×Y)), [(x_{0}, y_{0})]),  (44)

yields a symmetric monoidal structure, whose identity is (S^{0}, 1).

A (pointed directed) homotopy of pointed d-maps, φ: f → g: (X, x_{0}) → (Y, y_{0}) is defined as a dTop-homotopy such that φ(x_{0}, t) = y_{0} (for all t). To represent such homotopies, the (pointed directed) cylinder of (X, x_{0}) is a quotient of the cylinder of X, where the fibre at the base point, {x_{0}}×↑I, is collapsed to a base point

↑I(X, x_{0}) = (X ×↑I)/({x_{0}}×↑I) = (X, x_{0}) ∧ ↑I_{∗},  (45)

and can be obtained as the smash product with the pointed d-interval ↑I_{∗}, where an isolated base-point has been added (applying the left adjoint (−)_{∗}: dTop → dTop_{∗} of the forgetful functor).

More simply, the pointed path functor is the same as the unpointed one, pointed at the constant path at the base point

↑P (X, x) = (↑P (X), 0_{x}),  ↑I ⊣ ↑P (on dTop_{∗}).  (46)

3.2. Pointed h-pushouts. All pointed homotopy pushouts can be constructed from the pointed cylinder described above, via the usual colimit (7) calculated in dTop_{∗}. Thus, the pointed d-suspension can be obtained as a quotient of the unpointed cylinder

↑Σ(X, x_{0}) = (X ×↑I)/(({x_{0}}×↑I) ∪ (X ×S^{0})) = (X, x_{0}) ∧ (↑S^{1}, [0]),  (47)

and also as the smash product with the directed circle ↑S^{1} = ↑I/∂I pointed at [0] = [1].

As a relevant difference from the ordinary case, the pointed d-suspension of a d-sphere is different from the unpointed one (cf. (38)): here we obtain ↑Σ(S^{0}, 1) = (↑S^{1}, [0]). Actually, it is easy to deduce from the definition of higher directed spheres, ↑S^{n} = (↑I^{n})/(∂I^{n}) (I.1.2.4), that

↑Σ^{n}(S^{0}, 1) = (↑S^{n}, [0]).  (48)

The cofibre sequence of a pointed d-map f: X → Y behaves as in the unpointed case (Section 2).

3.3. Pointed h-pullbacks. When a pointed object is denoted by a single letter, say X, its base point will be written as ∗_{X} or ∗.

All pointed h-pullbacks can be constructed as pointed limits (cf. (9)), from the pointed cocylinder (46). Dualising 2.1, the upper homotopy kernel of a pointed d-map f: X → Y, or upper homotopy fibre ↑K^{+}f = ↑P (f, i_{Y}), is the h-pullback from f to the initial map i_{Y}: {∗} → Y, as in the left diagram below; similarly one has the upper cocone of an object, ↑K^{+}X, as in the right diagram

[figure: X; f; Y; X; id; X; κ; κ; ↑K^{+}f; {∗}; ↑K^{+}X; {∗}]  (49)

\begin{matrix}↑K^{+}f = ↑P (f, i_{Y}) = \{(x, b) ∈ X ×↑P Y | b(0) = f(x), b(1) = ∗_{Y} \}, \\ ↑K^{+}X = ↑K^{+}(idX) = \{a ∈ ↑P X | a(1) = ∗_{X}\}, \end{matrix}

pointed at (∗_{X}, 0_{∗_{Y}}) and 0_{∗_{X}}, respectively. The lower analogues are written ↑K^{−}f and ↑K^{−}X.

Dualising 2.3, the (directed) loop-object ↑ΩX is a lower and upper h-kernel, at the same time

[figure: {∗}; f; X; _{ev_{X}}; ↑ΩX; {∗}]  (50)

↑ΩX = ↑P (i_{X}, i_{X}) = {c ∈ ↑P X | c(0) = ∗_{X} = c(1)},  ∗ = 0_{∗_{X}}.

It comes equipped with a d-homotopy ev_{X}: 0 → 0: ↑ΩX → X (loop evaluation), universal for d-endohomotopies of null maps in X: for every φ: 0 → 0: A → X there is one d-map f: A → ↑ΩX such that φ = ev_{X}^{∘}f. The functors produced by h-kernel, cocone and loop-objects (on dTop_{∗}^{2} and dTop_{∗}) are homotopy invariant, as proved previously for the dual cases.

A pointed d-map f: X → Y has a lower fibre sequence, produced by lower h-kernels

\xymatrix{… ↑Ω^{2}Y \ar[r]^{↑Ωd} & ↑Ω(↑K^{−}f) \ar[r]^{↑Ωp} & ↑ΩX \ar[r]^{↑Ωf} & ↑ΩY \ar[r]^{d} & ↑K^{−}f \ar[r]^{p} & X \ar[r]^{f} & Y}  (51)

and satisfying dual properties with respect to the cofibre sequence (Section 2). In particular, it is d-homotopy equivalent to the sequence of iterated lower and upper h-kernels; and acts functorially on the variable f, preserving (pointed) d-homotopies of d-maps. If f is a (pointed) upper d-fibration (cf. 1.7), we can replace ↑K^{−}f with the ordinary kernel of f, the fibre F = f^{−1}{∗_{Y} }, which is d-homotopy equivalent.

3.4. Higher homotopy monoids. A pointed d-space A produces a sequence of dhomotopy invariant functors (n ≥ 0)

↑π_{n}^{A}: dTop_{∗} → Set_{∗},  ↑π_{n}^{A}(X) = [↑Σ^{n}A, X]_{∗} = [A, ↑Ω^{n}X]_{∗},  (52)

where [A, X]_{∗} is the pointed set of classes of maps A → X in dTop_{∗}, up to pointed d-homotopy. Plainly, the concatenation of loops (↑ΩX) × (↑ΩX) → ↑ΩX produces a structure of monoid on all such sets, for n ≥ 1, which is commutative for n ≥ 2 (by the middle-four interchange of pasting, as usual; cf. I.2.5.3).

The superscript A will be omitted when A = S^{0} = {−1, 1}, pointed at 1. Then, ↑π_{0}(X) is the set of d-path components ↑Π_{0}(X) = |X|/ ≃ (I.1.5.2), pointed at the component of the base point, while the remaining functors ↑π_{n} yield the homotopy monoids of a pointed d-space

\begin{matrix}↑π_{0}(X) = [↑S^{0}, X]_{∗} = (|X|/ ≃, [∗_{X}]), \\ ↑π_{n}(X) = [↑S^{n}, X]_{∗} = [S^{0}, ↑Ω^{n}X]_{∗} = ↑π_{0}(↑Ω^{n}X) & (n ≥ 1). \end{matrix}  (53)

In particular, ↑π_{1}(X) is (as previously defined, in I.3.3) the monoid of endoarrows [c]: ∗_{X} → ∗_{X} in the fundamental category ↑Π_{1}(X)

↑π_{1}(X) = [↑S^{1}, X]_{∗} = ↑π_{0}(↑ΩX) = ↑Π_{1}(X)(∗, ∗);  (54)

it is (strictly) invariant up to pointed d-homotopy and can be computed via ↑Π_{1}(X) (by the methods developed in I.3, especially the van Kampen-type pasting theorem).

Replacing ↑S^{1} with the reversible circle I˜ /∂I (I.1.3c), we get the group of invertible loops Inv(↑π_{1}(X)). The ordered circle ↑O^{1} (cf. (38)), pointed at its minimum or maximum, yields two other d-homotopy invariants of interest, [(↑O^{1}, v^{α}), −]_{∗}: dTop_{∗} → Set_{∗}.

3.5. Sequences of homotopy monoids. Applying the functor ↑π_{0} to the fibre sequence of a pointed d-map f: X → Y (see (51)), we obtain the fibre sequence of homotopy monoids (and pointed sets)

[figure: …; f_{∗2}; ↑π_{2}Y; d_{∗1}; ↑π_{1}(↑K^{−}f); p_{∗1}; ↑π_{1}X; f_{∗1}; ↑π_{1}Y; d_{∗0}; ↑π_{0}(↑K^{−}f); p_{∗0}; ↑π_{0}X; f_{∗0}; ↑π_{0}Y]  (55)

which is of order two but not exact, and probably much less interesting than its ordinary analogue, the fibre sequence of homotopy groups (which is exact).

Actually, each composition in (55) is null (since, up to d-homotopy equivalence, every map in the fibre sequence (51) is the h-kernel of the following one). On the other hand, take [x] ∈ ↑π_{0}X such that f_{∗0}[x] = [f(x)] = [∗_{Y}]; this only means that there is a finite sequence of paths in Y

f(x) = y_{0} → y_{1} ← y_{2} → … y_{n} = ∗_{Y} ,  (56)

but does not give, generally, a path ∗_{Y} → f(x′) with [x′] = [x].

If f is a bilateral fibration (1.7) we do get exactness at ↑π_{0}X (but nothing more). In fact, we can lift in X all the previous paths

x = x_{0} → x_{1} ← x_{2} → … x_{n},  f(x_{n}) = ∗_{Y} ,  (57)

so that p_{∗0}: ↑π_{0}(↑K^{−}f) → ↑π_{0}(X) takes [x_{n}, 0_{∗_{Y}}] to [x_{n}] = [x].

4. Based directed spaces

Generalising pointed objects, one can keep fixed a set of base-points, with the same advantage of controlling homotopy invariance at them.

4.1. Bipointed objects. Let us begin considering the ‘comma’ category dTop\S^{0} = (S^{0}↓dTop) of bipointed d-spaces. An object (X, x, x′) is a d-space equipped with a pair of points (possibly equal); bipointed maps preserve them and bipointed (directed) homotopies leave each of them fixed.

We already know (from I.3.3.2) that the homotopy functor

↑π_{1}: dTop\S^{0} → Set,

↑π_{1}(X, x, x′) = ↑Π_{1}(X)(x, x′) = [(↑I, 0, 1), (X, x, x′)]_{S^{0}}, (58)

is (strictly) invariant up to bipointed d-homotopy; this is also evident by its presentation as a representable homotopy functor, in the last formula above.

For instance, one can examine by such functors the following d-space X ⊂ ↑R^{2}

X = ([0, 9]×[0, 4]) \ (U ∪ V)

[figure: X; U; V; C; B; A; D; 0; 1; 9; 0; 1; 4; x; •; x′; •]  (59)

Keeping x′ fixed at (9, 4) and letting x vary in X, we get different results for the set ↑π_{1}(X, x, x′): two equivalence classes if x is in the rectangle A or in B, three if it is in C, none in D, one in the rest of X. As it happens for the fundamental monoid (3.4), each such information is invariant up to bipointed d-homotopy and can be computed via the fundamental category.

The ordered circle ↑O^{1}, bipointed at minimum and maximum, yields a representable homotopy functor [↑O^{1}, −]: dTop\S^{0} → Set, which counts the ‘linked pairs of paths’ in X, from x to x′, up to bipointed homotopy. But the obvious symmetry ↑O^{1} → ↑O^{1} produces a symmetry on [↑O^{1}, X], i.e. an action of the group Z_{2}, and we may prefer to count those pairs of paths up to this action

O_{1}: dTop\S^{0} → Set,  O_{1}(X, x, x′) = ([(↑O^{1}, v^{−}, v^{+}), (X, x, x′)]_{S_{0}})/Z_{2};  (60)

it might be interesting to develop methods of deducing this invariant from ↑Π_{1}X.

4.2. Based objects. More generally, let A be a fixed d-space, called the basis, and consider the comma category dTop\A = (A↓dTop) of d-spaces under A. An object is a pair (X, x), where x: A → X is a d-map; a map f: (X, x) → (Y, y) is defined by a d-map f: X → Y such that f.x = y. A based (directed) homotopy φ: f → g: (X, x) → (Y, y) is a homotopy φ: f → g in dTop which is ‘trivial on the basis’, i.e. φ^{∘}x = 0_{y} (each path φ(x(i), −): ↑I → Y is constant, for i ∈ A).

The category dTop\A acquires thus a d-homotopy structure similar to that of dTop, by suitable cylinder and cocylinder (as proved - in a more complex situation - in [8], Section 4; the crucial point is the fact that pushouts in dTop exist and are preserved by ↑I, a left adjoint).

We shall only recall how to construct the based cylinder and cocylinder. The latter is obvious

↑P (X, x) = (↑P X, x^{P}),  x^{P} = ↑P x.e: A → ↑P A → ↑P X,  (61)

while the cylinder ↑I(X, x) is obtained from a pushout in dTop, which collapses (independently) each fibre at a point x(i), for all i ∈ A

[figure: A; ∂^{α}; x; ↑IA; e; ↑Ix; A; x^{I}; X; ∂^{α}; ↑IX; ↑I(X, x)]  (62)

(to be precise, ↑I(X, x) is in fact this pushout object equipped with the d-map x^{I}). The fundamental category can then be viewed as a functor

↑Π_{1}: dTop\A → Cat\↑Π_{1}(A).  (63)

In particular, if A is a set (viewed as a discrete d-space and a discrete category), we have

↑Π_{1}: dTop\A → Cat\A,  ↑Π_{1}(X, x) = (↑Π_{1}X, x_{∗}),  (64)

a functor invariant up to A-based d-homotopies, of d-spaces and of categories (trivial on each point of A, in both domains). Again, the crucial fact is that this functor cannot destroy information at the base points x(i), for i ∈ A, (in contrast with the free ↑Π_{1} on dTop, cf. I.3.5).

This follows immediately from the invariance theorem I.3.2. First, if φ: f → g is Abased, then each path φ(x(i)): f(x(i)) → g(x(i)) is trivial. Second, if f: (X, x) → (Y, y) is an A-based d-homotopy equivalence, then

f_{∗1}: ↑Π_{1}(X)(x(i), x(j)) → ↑Π_{1}(Y)(fx(i), fx(j))  (65)

is bijective (for i, j ∈ A).

4.3. Variable basis. Finally, we want to be able to change bases, and to have morphisms between objects with different bases. This is solved by the category of morphisms dTop^{2}, already considered in 2.1.

An object is a d-map x: X′ → X′′, and a morphism is a commutative square in dTop

f = (f′, f′′): x → y, f′: X′ → Y′, f′′: X′′ → Y′′; yf′ = f′′x.  (66)

Again, the homotopy structure of this category of functors 2 → dTop has already been studied, in general, in [8], Section 4. It is of the same type as the one of dTop, with the obvious derived cylinder and cocylinder functors, whose values on objects are, respectively

\begin{matrix}↑I(x: X′ → X′′) = (↑Ix: ↑IX′ → ↑IX′′), \\ ↑P (x: X′ → X′′) = (↑P x: ↑P X′ → ↑P X′′). \end{matrix}  (67)

The fundamental category can then be viewed as a functor

↑Π_{1}: dTop^{2} → Cat^{2}.  (68)

Also here, it is interesting to restrict to the case where the domain of our d-maps x, y, … is a (variable) set, viewed as a discrete d-space via the functor D: Set → dTop (or a discrete category, via D: Set → Cat). We obtain then the comma category (D↓dTop) ⊂ dTop^{2}: an object is a d-map x: DA → X, a map (f_{0}, f) consists of a set-mapping f_{0} and a d-map f forming a commutative square (Df_{0} is written as f_{0})

\xymatrix{DA \ar[r]^{f_{0}} \ar[dd]_{x} & DB \ar[dd]^{y} \\  &  & (f_{0}, f): (x: DA → X) → (y: DB → Y). \\ X \ar[r]_{f} & Y}  (69)

A based d-homotopy φ: (f_{0}, f) → (g_{0}, g): x → y can only exist if f_{0} = g_{0}, and consists then of a d-homotopy φ: f → g such that φ^{∘}x = 0_{yf_{0}}. The category (D↓Cat) is similar.

The fundamental-category functor between such comma categories

↑Π_{1}: (D↓dTop) → (D↓Cat),  ↑Π_{1}(x: DA → X) = (x: DA → ↑Π_{1}X).  (70)

associates to the above homotopy φ of (D↓dTop), a natural transformation φ_{∗1}: f_{∗1} → g_{∗1} which is the identity on all objects x(i) (for i ∈ A): φ_{∗1}x(i) = id: y(f_{0}i) = y(g_{0}i). Therefore

f_{∗1} = g_{∗1}: ↑Π_{1}(X)(x(i), x(j)) → ↑Π_{1}(Y)(y(f_{0}i), y(f_{0}j))  (i, j ∈ A),  (71)

and, again, all sets ↑Π_{1}(X)(x(i), x(j)) are (bijectively) invariant up to based d-homotopy equivalence.

References

[1] F. Borceux, Handbook of categorical algebra 1-3, Cambridge Univ. Press, Cambridge 1994.

[2] L. Fajstrup - E. Goubault - M. Raussen, Algebraic topology and concurrency, Preprint 1999. http://www.di.ens.fr/∼goubault/

[3] P. Gaucher, Homotopy invariants of higher dimensional categories and concurrency in computer science, in: Geometry and concurrency, Math. Struct. in Comp. Science 10 (2000), no. 4, pp. 481-524.

[4] P. Gaucher - E. Goubault, Topological Deformation of Higher Dimensional Automata, Preprint 2001. http://www-irma.u-strasbg.fr/∼gaucher/diCW.ps

[5] E. Goubault, Geometry and concurrency: a user’s guide, in: Geometry and concurrency, Math. Structures Comput. Sci. 10 (2000), no. 4, pp. 411-425.

[6] M. Grandis, Homotopical algebra in homotopical categories, Appl. Categ. Structures 2 (1994), 351-406.

[7] M. Grandis, Cubical homotopical algebra and cochain algebras, Ann. Mat. Pura Appl. 170 (1996) 147-186.

[8] M. Grandis, Categorically algebraic foundations for homotopical algebra, Appl. Categ. Structures 5 (1997), 363-413.

[9] M. Grandis, Higher fundamental functors for simplicial sets, Cahiers Topologie Geom. Differentielle Categ. 42 (2001), 101-136.

[10] M. Grandis, Ordinary and directed combinatorial homotopy, applied to image analysis and concurrency, Homology Homotopy Appl., to appear. http://www.dima.unige.it/∼grandis/

[11] M. Grandis, Directed homotopy theory, I. The fundamental category, Cahiers Topologie Géom. Différentielle Catég., to appear. http://www.dima.unige.it/∼grandis/

[12] S. Mac Lane, Categories for the working mathematician, Springer, Berlin 1971.

Dipartimento di Matematica Università di Genova Via Dodecaneso 35 16146-Genova, Italy

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