Homology, Homotopy and Applications, vol.4(1), 2002, pp.71–85

METRIZABLE SHAPE AND STRONG SHAPE EQUIVALENCES

L. STRAMACCIA

(communicated by Walter Tholen)

Abstract In this paper we construct a functor Φ : pro𝒯op → pro𝒜𝒩ℛ which extends Mardešić correspondence that assigns to every metrizable space its canonical 𝒜𝒩ℛ-resolution. Such a functor allows one to define the strong shape category of prospaces and, moreover, to define a class of spaces, called strongly fibered, that play for strong shape equivalences the role that 𝒜𝒩ℛspaces play for ordinary shape equivalences. In the last section we characterize SSDR-promaps, as defined by Dydak and Nowak, in terms of the strong homotopy extension property considered by the author.

Introduction

In ordinary Shape Theory there is a canonical way of associating with every topological space X an inverse system X̌ of absolute neighborhood retracts, namely its Čech system [15]. It is an inverse system in the homotopy category ho𝒯op of topological spaces, whose bonding morphisms are homotopy classes of maps. This gives a functor ho𝒯op → pro(ho𝒜𝒩ℛ), where 𝒜𝒩ℛ is the category of absolute neighborhood retracts. In Strong Shape Theory [14] one associates with every space X an inverse system X in the category 𝒯op of topological spaces, bonded by continuous maps. In [19] S. Mardešić introduced the notion of 𝒜𝒩ℛ-resolution and proved that every topological space X admits a canonically associated 𝒜𝒩ℛ-resolution M(X) ∈ pro𝒜𝒩ℛ. However, the correspondence X ↦ M(X) does not give a functor 𝒯op → pro𝒜𝒩ℛ. In their 1991 paper [8], Dydak and Nowak tried to overcome such difficulties defining a Mardešić-like functor 𝒯op → pro𝒜𝒩ℛ but, due to some technical error, their construction there does not work (see [14], [9]). In another more recent paper [9], the same authors correct their errors adopting a different point of view. In this paper we undertake the program above and construct a functor Φ : pro𝒯op → pro𝒜𝒩ℛ, from the category of prospaces (inverse systems of topological spaces) to the category of inverse systems of absolute neighborhood retracts, which has the following properties :

Received January 25, 2002, revised August 23, 2002; published on September 9, 2002. 2000 Mathematics Subject Classification: 55P55, 55P05, 55P10, 18B30, 18G55, 55P60, 54B30. Key words and phrases: Metrizable proreflector, shape equivalence, strong shape equivalence, double mapping cylinder, SSDR-promaps, strong homotopy extension property. \begin{matrix}c \\ ◯ \end{matrix} 2002, L. Stramaccia. Permission to copy for private use granted.

i) the restriction of Φ to the category of metrizable spaces coincides with Mardešić’s correspondence which assigns to every space its canonical 𝒜𝒩ℛ-resolution [13],

ii) Φ has a reflective lifting ho(Φ) : ho(pro𝒯op) → ho(pro𝒜𝒩ℛ) to the Steenrod homotopy categories,

iii) the restriction of ho(Φ) to ho(𝒯op) is naturally equivalent to Cathey and Segal’s functor R : ho(𝒯op) → ho(pro𝒜𝒩ℛ) [5],

iv) one can define the strong shape category of prospaces sSh(pro𝒯op) as a natural extension of the category sSh(𝒯op), defined in [5], [19], considering the full image factorization of ho(Φ). It is shown that sSh(pro𝒯op) can be obtained localizing pro𝒯op at the class of strong shape equivalences (cf.[17]). Crucial for the definition of the functor Φ is the consideration of the metrizable proreflector 𝒯op → proℳet which gives a reflector ho(pro𝒯op) → ho(proℳet) and the fact that the strong shape theory of metrizable spaces is well settled in the literature. The existence of the functor Φ allows one to characterize strong shape equivalences as those maps inducing bijections f_{Z}^{∗} : [Y, Z] → [X, Z], between sets of homotopy classes, for every strongly fibered space Z (section 2). Hence, strongly fibered spaces play, for strong shape equivalences, the role that ANR-spaces play for ordinary shape equivalences. Such a result was already stated by Dydak and Nowak in [8] and corrected in [9], where SSDR𝒯op- fibrant spaces were introduced. We compare our results with those of [9] in last section. In particular we prove that SSDRpromaps of [9] coincide with the class of level cofibrations that are strong shape equivalences. As a fundamental tool we use a generalization of the SHEP (strong homotopy extension property), introduced in [21].

1. Procategories and Localizations.

Let 𝒞 be any category. The category pro𝒞 of inverse system in 𝒞 has objects the contravariant functors X : Λ → 𝒞, where Λ = (Λ, ⩽) is a directed set. An inverse system in 𝒞 will be explicitly denoted by X = (X_{λ}, x_{λλ′}, Λ), where X_{λ} = X(λ) and x_{λλ′} = X(λ ⩽ λ′). We refer to [15] for all details concerning the definition of pro𝒞, but it will be useful to recall the following facts :

- a morphism x : X → X, where X ∈ 𝒞, is a family x = {x_{λ} : X → X_{λ} | λ ∈ Λ} of morphisms of 𝒞, with the property that x_{λλ′} ∘ x_{λ′} = x_{λ}, for all λ ⩽ λ′.

- given a morphism f : X → Y in pro𝒞, it is always possible to assume, up to isomorphisms, that Λ is cofinite (that is: every λ ∈ Λ has only finitely many predecessors), that Y = (Y_{λ}, q_{λλ′}, Λ) is indexed over the same directed set as X and that f is a level morphism, that is given by a family {f_{λ} : X_{λ} → Y_{λ} | λ ∈ Λ} of morphisms of 𝒞, with y_{λλ′} ∘ f_{λ′} = f_{λ} ∘ x_{λλ′}, for λ ⩽ λ′ ([15], Thm.3.1). Note that a level morphism is actually a natural transformation of functors.

1.1. A full subcategory 𝒦 of 𝒞 is proreflective in 𝒞 ([16], [20], [22]) if, for every X ∈ 𝒞, there exists an inverse system X ∈ pro𝒦 and a morphism x : X → X in pro𝒞, which is universal (initial) with respect to every other morphism f : X → K, with K ∈ pro𝒦. In such a case x : X → X is called a 𝒦-expansion for X. It is clear that a 𝒦-expansion for X is uniquely determined up to isomorphisms in pro𝒦. This fact allows one to define a functor P : 𝒞 → pro𝒦, X ↦ X, which is is called the proreflector. Let ℬ be any category having inverse limits. Every functor F : 𝒞 → ℬ has an extension F^{∗} : pro𝒞 → ℬ which is defined by F^{∗} = lim · proF where proF : pro𝒞 → proℬ is the functor such that proF (X) = (F (X_{λ}), F (x_{λλ′}), Λ), while lim : proℬ → ℬ is the inverse limit functor. We give now a construction for the functor F^{∗} in the case ℬ = pro𝒦, for some category 𝒦.

Let X = (X_{λ}, x_{λλ′}, Λ) ∈ pro𝒞 and let F (X_{λ}) = (K_{i}^{λ}, k_{ii′}^{λ}, I_{λ}), for every λ ∈ Λ. Then, (F (X_{λ}), F (x_{λλ′}), Λ) is an inverse system in pro𝒦, whose inverse limit is the system

F^{∗}(X) = (K_{i}^{λ}, k_{ii′}^{λλ′}, Γ),

where Γ = ⋃{Λ × I_{λ} | λ ∈ Λ} is directed by the relation

(λ, i) ⩽ (λ′, i′) ⇔ \{ \begin{matrix}λ ⩽ λ′ in Λ, and \\ k_{ii′}^{λλ′} : K_{i′}^{λ′} → K_{i}^{λ} is part of F (x_{λλ′}). \end{matrix}

Let f : X → Y be a level morphism in pro𝒞, with Y = (Y_{λ}, y_{λλ′}, Λ) and f = {f_{λ} : X_{λ} → Y_{λ} | λ ∈ Λ}. If we assume, as it is possible, that each F (f_{λ}) : F (X_{λ}) → F (Y_{λ}), λ ∈ Λ, is a level morphism, then F (Y_{λ}) = (H_{i}^{λ}, h_{ii′}^{λ}, I_{λ}), hence it follows that

F^{∗}(Y) = (H_{i}^{λ}, h_{ii′}^{λλ′}, Γ),

while F^{∗}(f) is the level morphism given by

F^{∗}(f) = {F (f_{λ})_{i} : K_{i}^{λ} → H_{i}^{λ} | (λ, i) ∈ Γ}.

Note that, if P : 𝒞 → pro𝒦 is a proreflector, then P^{∗} : pro𝒞 → pro𝒦 is actually a reflector [20], [22] .

1.2. Recall that, given a class Σ of morphisms in a category 𝒞, the localization of 𝒞 at Σ is a pair (𝒞 [Σ^{−1}], L_{Σ}), where 𝒞 [Σ^{−1}] is a category (possibly in a larger universe) having the same objects as 𝒞 and L_{Σ} : 𝒞 → 𝒞 [Σ^{−1}] is a functor which is the identity on objects, having the following properties :

- L_{Σ} inverts all morphisms of Σ, that is L_{Σ}(s) is an isomorphism in 𝒞[Σ^{−1}], for all s ∈ Σ,

- L_{Σ} is universal (initial) among all functors F : 𝒞 → ℰ that invert all morphisms of Σ. Σ is usually called the class of weak equivalences of 𝒞. If 𝒟 is another category, endowed with a notion Δ of weak equivalences, then a functor F : 𝒞 → 𝒟 can be extended to a functor F̃ : 𝒞 [Σ^{−1}] → 𝒟 [Δ^{−1}] if and only if F preserves weak equivalences, that is F (s) ∈ Δ, for all s ∈ Σ. F̃ is the unique functor satisfying F̃ ∘ L_{Δ} = L_{Σ} ∘ F; it acts on objects as F does [18].

Let 𝒞 = 𝒯op be the category of topological spaces and let Σ be the class of homotopy equivalences. Then, 𝒯op [Σ^{−1}] = ho(𝒯op) is the usual homotopy category of spaces. In general, if 𝒞 has a Quillen model structure, with Σ the class of its weak equivalences, then ho𝒞 = 𝒞 [Σ^{−1}]. Moreover, pro𝒞 inherits a Quillen model structure and its (Steenrod) homotopy category is ho(pro𝒞) = pro𝒞 [Σ^{∗}^{−1}], where Σ^{∗} is the class of level weak equivalences, that is the class of those level morphisms which belong levelwise to Σ. Σ^{∗} will usually be considered as the class of weak equivalences of pro𝒞 [10], [16].

Theorem 1.3. (cf. [19]) Let 𝒞 and 𝒦 have classes of weak equivalences Σ and Π, respectively, and let P : 𝒞 → pro𝒦 be any functor. If P preserves weak equivalences, then also P^{∗} preserves weak equivalences. If, moreover, P is a proreflector, then P̃^{∗} : (pro𝒞)[Σ^{∗}^{−1}] → (pro𝒞)[Γ^{∗}^{−1}] is a reflector.

Proof. Let f ∈ Σ^{∗}, f = {f_{λ}}. P^{∗}(f) has level components of the form P (f_{λ})_{i}, (λ, i) ∈ Γ, which are all members of Π, by assumption. If P is a proreflector and Π = Σ ∩ {morphisms of 𝒦}, then P^{∗} is a reflector, hence left adjoint to the embedding E : pro𝒦 → pro𝒞. Since both P^{∗} and E preserve weak equivalences, the assertion follows from ([2], Thm.1.1).

1.4. The usual cylinder functor on 𝒯op can be extended naturally to a cylinder functor on pro𝒯op : for every X = (X_{λ}, x_{λλ′}, Λ) ∈ pro𝒯op, let X × I = (X_{λ} × I, x_{λλ′} × 1, Λ), where I is the unit interval. One obtains, as a consequence, a notion of (global) homotopy between promaps (that is: between morphisms of prospaces) and a corresponding notion of (global) homotopy equivalence. Two promaps f, g : X → Y are globally homotopic if there is a homotopy H : X × I → Y such that H ∘ e^{0} = f and H ∘ e^{1} = g, where e^{0}, e^{1} : X → X × I are the obvious promaps. The quotient category of pro𝒯op modulo global homotopy is denoted by π(pro𝒯op) and π : pro𝒯op → π(pro𝒯op) is the quotient functor. In general, the classes of global and level homotopy equivalences in pro𝒯op do not coincide, as shown in ([10], pp.55-56); however, every global homotopy equivalence X → Y is a level homotopy equivalence, whenever the bonding morphisms of X are epi ([19], Cor. 1.3).

1.5. Let F : 𝒞 → 𝒦 be any functor and let 𝒞_{F} be the category having the same objects as 𝒞 while a morphism in 𝒞_{F} (X, Y) is a triple (1_{X}, u, 1_{Y}), where u ∈ 𝒦(F (X), F (Y)). 𝒞_{F} is called the full image of F. There are functors F^{0} : 𝒞 → 𝒞_{F} and F^{1} : 𝒞_{F} → 𝒦, defined by F^{0}(X) = X and F^{0}(f) = (1_{X}, F (f), 1_{Y}), for f : X → Y in 𝒞, and F^{1}(X) = F (X), F^{1}(1_{X}, u, 1_{Y}) = u. They give a factorization F = F^{1} ∘ F^{0} of F which is uniquely determined, up to an isomorphism, among all factorizations F = H′′ ∘H′, where H′ is bijective on objects and H′′ is fully faithful. F = F^{1} ∘ F^{0} is called the full image factorization of F [18]. Recall that, when F is a reflector and Σ_{F} is the class of morphisms of 𝒞 inverted by F, then there is an isomorphism 𝒦 ≅ 𝒞[Σ_{F}^{−1}] ([18], 19.3.1). Moreover, from the uniqueness of the full image factorization and since L_{Σ_{F}} is the identity on objects, one also obtains an isomorphism 𝒞[Σ_{F}^{−1}] ≅ 𝒞_{F} . Let Σ_{F^{0}} denote the class of morphisms in 𝒞 that are inverted by F^{0}. Then clearly Σ_{F} = Σ_{F^{0}} holds.

In what follows we give a brief account of the construction of the Steenrod homotopy category ho(pro𝒯op) of pro𝒯op, following the point of view of [5].

Definition 1.6. Let f : X → Y and p : E → B be promaps. f has the left lifting property with respect to p (and p has the right lifting property with respect to f) if every commutative square

                       [figure: X; E; Y; B; a; b; f; p]

has a filler h : Y → E, such that h ∘ f = a and p ∘ h = b. Let Σ be a a class of morphisms in pro𝒯op, then

- a promap p : E → B is a Σ-fibration if it has the right lifting property with respect to all f ∈ Σ,

- a prospace Z = (Z_{μ}, z_{μμ′}, M) is Σ-fibrant iff the unique morphism Z → ∗ is a Σ-fibration, where ∗ denotes the final object in 𝒯op,

- a Σ-fibrant prospace Z is said to be strongly Σ-fibrant if, moreover, for every μ^{∗} ∈ M, the unique map z_{μ^{∗}} : Z_{μ^{∗}} → lim_{μ<μ^{∗}} Z_{μ}, induced by the bonding maps of the system, is a Σ-fibration,

- a topological space Z is Σ-strongly fibered if it is the inverse limit of a strongly Σ-fibrant prospace Z ∈ pro𝒜𝒩ℛ.

In the homotopy theory of pro𝒯op, as defined in [10], a promap f is a trivial cofibration if it has the left lifting property with respect to every Hurewicz fibration p : E → B in 𝒯op. This notion is a natural extension of that of trivial cofibration in 𝒯op. On the other hand, it is clear that a map p having the right lifting property with respect to all trivial cofibrations f in pro𝒯op, has to be a Hurewicz fibration. In the sequel, for Σ the class of trivial cofibrations in pro𝒯op, we shall speak of (strongly) fibrant prospaces and strongly fibered spaces, omitting the reference to the class Σ.

1.7. There is a reflective functor F : π(pro𝒯op) → π(pro𝒯op)_{f} onto the full subcategory of fibrant prospaces [5], with unit of adjunction [i_{X}] : X → X̂, where i_{X} is a trivial cofibration. By ([5], Prop. 3.3) F has a reflective restriction F : π(pro𝒜𝒩ℛ) → π(pro𝒜𝒩ℛ)_{sf}, where π(pro𝒜𝒩ℛ)_{sf} is the full subcategory of strongly fibrant prospaces. For Z ∈ pro𝒜𝒩ℛ, i_{Z} : Z → Ẑ is called the strongly fibrant modification of Z. ho(pro𝒯op) is the full image of the functor F above and is equipped with the canonical functors F^{0} : π(pro𝒯op) → ho(pro𝒯op) and F^{1} : ho(pro𝒯op) → π(pro𝒯op)_{f}. The functor L = F^{0} ∘ π : pro𝒯op → ho(pro𝒯op) is known to localize pro𝒯op at the class of trivial cofibrations and also at the class of level homotopy equivalences [10], [16].

Remark 1.8. For X, Y ∈ pro𝒯op, with Y fibrant, there is a natural bijection

ho(pro𝒯op)(X, Y) ≅ [X, Y],

where [X, Y] is the set of global homotopy classes of morphisms X → Y. This is because every prospace X is in fact cofibrant in pro𝒯op ([10], Prop.3.4.1, pag. 95).

2. The functor Φ : pro𝒯op → pro𝒜𝒩ℛ and the category sSh(pro𝒯op).

The category ℳet of metrizable spaces is proreflective in 𝒯op. In order to obtain the metrizable expansion x : X → X of a topological space (X, τ), let us consider the set Λ of all continuous pseudometrics on X, directed by the relation

λ ⩽ λ′ ⟺ τ_{λ} ⊂ τ_{λ′},

Here τ_{λ} denotes the topology induced on X by the pseudometric λ, while the continuity of λ means that τ_{λ} ⊂ τ [1]. Let X_{λ} denote the metric identification of (X, τ_{λ}). For every λ ∈ Λ, let x_{λ} : X → X_{λ} be the identity map (X, τ) → (X, τ_{λ}) followed by the quotient map (X, τ_{λ}) → X_{λ}. Moreover, for λ ⩽ λ′, let x_{λλ′} : X_{λ′} → X_{λ} be the unique map induced on the quotients by the identity (X, τ_{λ′}) → (X, τ_{λ}). We note explicitly that in the inverse system X = (X_{λ}, x_{λλ′}, Λ), the bonding morphisms x_{λλ′} are all surjective maps. We shall denote by P_{M} : 𝒯op → proℳet, X ↦ X, the metrizable proreflector.

Theorem 2.1. (cf. [19]) The metrizable proreflector P_{M} : 𝒯op → proℳet induces a reflector ho(P_{M}^{∗}) : ho(pro𝒯op) → ho(proℳet).

Proof. In view of Thm.1.3, it suffices to prove that P_{M} preserves weak equivalences. Let us note that P_{M} respects the cylinders, in the sense that, if x : X → X = (X_{λ}, x_{λλ′}, Λ) is the metrizable expansion of the space X, then x × 1 : X × I → X×I = (X_{λ}×I, x_{λλ′} ×1, Λ) is the metrizable expansion of X ×I, see ([19], Thm.2.3). It follows that P_{M} takes homotopy equivalences to global homotopy equivalences. Since X has epi bonding morphisms, the proof is complete. The reflection morphism χ : X → P_{M}^{∗}(X), for the prospace X, is induced by the family {x_{λ} : X_{λ} → X_{λ} | λ ∈ Λ} of the metrizable expansions of each X_{λ}, following the construction given in the previous section (1.1).

In [13] S. Mardešić introduced the notion of 𝒜𝒩ℛ-resolution for topological spaces and proved that every space X has a canonically associated 𝒜𝒩ℛ-resolution m_{X} : X → M(X). Although the correspondence 𝒯op → pro𝒜𝒩ℛ, X ↦ M(X), is not functorial in general, Cathey and Segal [5] proved that it induces a reflective functor between the Steenrod homotopy categories R : ho(𝒯op) → ho(pro𝒜𝒩ℛ). Moreover, they obtained the strong shape category sSh(𝒯op) and the strong shape functor sS𝒯op by taking the full image factorization of R :

[figure: ho(𝒯op); ho(pro𝒜𝒩ℛ); sSh(𝒯op); R; sS𝒯op; R^{1}]

where sS𝒯op = R^{0} is the identity on objects, while R^{1} is fully faithful.

The fact that R is a reflective functor means that, for every X ∈ 𝒯op and for every K ∈ pro𝒜𝒩ℛ, Mardešić’s 𝒜𝒩ℛ-resolution m : X → M(X) induces a bijection ho(pro𝒯op)(X, K) ≅ ho(pro𝒜𝒩ℛ)(M(X), K).

Another feature of Mardešić’s correspondence is that it becomes a functor

M : ℳet → pro𝒜𝒩ℛ

when restricted to the category ℳet of metrizable spaces. This fact was pointed out in [19] and used to give an alternative description of the strong shape category of topological spaces. The same paper (Thm. 2.4) also gave a particularly simple construction for the 𝒜𝒩ℛ-resolution m_{X} : X → M(X) of a metrizable space X, which is actually an 𝒜𝒩ℛ-expansion. In such a case M(X) is the inverse system of all open neighborhoods of X in its convex hull H(X) in the Banach space C(X) of all real, bounded, continuous functions on X, while m_{X} is formed by all the inclusions. Let us note that, lifting the functor M : ℳet → pro𝒜𝒩ℛ to the Steenrod homotopy categories, amounts to taking the restriction ho(M) : ho(ℳet) → ho(pro𝒜𝒩ℛ) of Cathey and Segal’s functor R, to the homotopy subcategory of metrizable spaces. It follows that M : ℳet → pro𝒜𝒩ℛ has to preserve weak equivalences. By Thm.1.1, the functor M^{∗} : proℳet → pro𝒜𝒩ℛ also preserves weak equivalences and has a lifting

ho(M^{∗}) : ho(proℳet) → ho(pro𝒜𝒩ℛ).

Theorem 2.2. ho(M^{∗}) is a reflector.

Proof. Let X = (X_{λ}, x_{λλ′}, Λ) be an inverse system of metrizable spaces. The family {m_{λ} : X_{λ} → M(X_{λ})} of the 𝒜𝒩ℛ-resolutions constructed above, gives a morphism m_{X} : X → M^{∗}(X) in proℳet. We have to prove that, for every Z ∈ proℳet, it induces a bijection

ho(proℳet)(X, Z) ≅ ho(pro𝒜𝒩ℛ)(M^{∗}(X), Z).

Let i_{Z} : Z → Ẑ be the strongly fibrant modification of Z. By the preceding remarks, one has ho(proℳet)(X, Z) ≅ [X, Ẑ] and ho(pro𝒜𝒩ℛ)(M^{∗}(X), Z) ≅ [M^{∗}(X), Ẑ]. It follows that proving the formula above amounts to proving that m_{X} induces a bijection

[M^{∗}(X), Ẑ] ≅ [X, Ẑ].

This is a consequence of the fact that ho(M) is reflective and of the construction of M^{∗}(X), as recalled in the first section.

Let us now define the functor

Φ : pro𝒯op → pro𝒜𝒩ℛ

as follows: for every X ∈ pro𝒯op, let Φ(X) = M^{∗}(P_{M}^{∗}(X)). It is clear that ho(Φ) : ho(pro𝒯op) → ho(pro𝒜𝒩ℛ) exists and can be written as ho(Φ) = ho(M^{∗})∘ho(P_{M}^{∗}). By (1.7) we may assume, without restriction of generality, that Φ(X) is strongly fibrant in pro𝒜𝒩ℛ. Moreover, by the results above, it follows that ho(Φ) is a reflector. If X = (X_{λ}, x_{λλ′}, Λ), the reflection morphism μ : X → Φ(X) is the promap obtained as the composition of χ : X → P_{M}^{∗}(X), m_{P_{M}^{∗} (X)} : P_{M}^{∗}(X) → M^{∗}(P_{M}^{∗}(X)) and the strongly fibrant modification of M^{∗}(P_{M}^{∗}(X)). The restriction of ho(Φ) to ho(𝒯op) coincides with the functor R of Cathey and Segal [19] and, consequently, it defines the same strong shape category for the class of topological spaces. The functor Φ is an extension of Mardešić ’s functor defined on the subcategory of metrizable spaces. Let us define the strong shape category sSh(pro𝒯op) for inverse systems of topological spaces and the related strong shape functor sS, by taking the full image factorization of ho(Φ), as illustrated in the commutative diagram

[figure: ho(pro𝒯op); ho(pro𝒜𝒩ℛ); sSh(pro𝒯op); ho(Φ); sS; ho(Φ)^{1}]

3. Shape and Strong Shape Equivalences.

A continuous map f : X → Y is said to be a (strong) shape equivalence if it becomes an isomorphism in the (strong) shape category of topological spaces, that is sS(L(f)) is an isomorphism in sSh(pro𝒯op). We refer to [15] and [14] for basic facts concerning shape and strong shape theory. In particular we recall that:

- f is a shape equivalence iff it induces a bijection f_{K}^{∗} : [Y, A] → [X, A] between sets of homotopy classes, for all A ∈ 𝒜𝒩ℛ,

- a shape equivalence f is a strong shape equivalence iff, given maps g, h : Y → A, A ∈ 𝒜𝒩ℛ, and a homotopy F : X × I → A connecting g ∘ f and h ∘ f, there exists a homotopy G : Y × I → A connecting g and h, such that G ∘ (f × 1) is homotopic to F w.r.t. end maps.

The notion of strong shape equivalence in pro𝒯op is the obvious generalization of the notion given previously [8], [14] : f : X → Y is a strong shape equivalence whenever the following two conditions hold :

(SSE1) for every A ∈ 𝒜𝒩ℛ and for every h : X → A, there is a morphism g : Y → A, such that g ∘ f ≃ h,

(SSE2) given morphisms g, h : Y → A, A ∈ 𝒜𝒩ℛ, and a global homotopy F : X× I → A joining f ∘ g and f ∘ h, there exists a global homotopy G : Y × I → A joining g and h, such that F is homotopic to G ∘ (f × 1) w.r.t. end morphisms.

Notice that, if a composition g∘f satisfies (SSE1), then f satisfies (SSE1). In fact, that f satisfies (SSE1) amounts to saying that the induced map f^{∗} : [Y, A] → [X, A] is onto. On the other hand one has (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

Theorem 3.1. The morphism μ : X → Φ(X) is a strong shape equivalence.

Proof. This is almost obvious. ho(Φ)(μ) must be an isomorphism in ho(pro𝒜𝒩ℛ), because of the reflectivity. Since ho(Φ) = ho(Φ)^{1} ∘ sS and ho(Φ)^{1} is fully faithful, it follows that sS(μ) is an isomorphism in the strong shape category sSh(pro𝒯op), hence μ is a strong shape equivalence.

Corollary 3.2. For prospaces X, Y, the following relation holds

ho(pro𝒯op)(X, Y) ≅ [Φ(X), Φ(Y)].

We need to consider now the following facts. Let f : X → Y, f = {f_{λ} | λ ∈ Λ}, be a level promap.

3.3. For every λ ∈ Λ, let M(f_{λ}) be the mapping cylinder of f_{λ} [12], with canonical maps Π_{λ} : X_{λ}×I → M(f_{λ}) and j_{λ} : Y_{λ} → M(f_{λ}), such that Π_{λ}∘e_{0,λ} = j_{λ}∘f_{λ}. Note that j_{λ} has a left inverse p_{λ}, such that p_{λ} ∘Π_{λ} = f_{λ} ∘σ_{λ}, where σ_{λ} : X_{λ} ×I → X_{λ} is the usual map. Then f_{λ} has a decomposition f_{λ} = f_{λ}^{1} ∘ f_{λ}^{0}, where f_{λ}^{0} : X_{λ} → M(f_{λ}) is a cofibration and f_{λ}^{1} : M(f_{λ}) → Y_{λ} is a homotopy equivalence. Since such a decomposition is functorial [11], one can define (levelwise) the mapping cylinder decompositon of the promap f, given by

                [figure: X; Y =; X; Y; M(f); f; f^{0}; f^{1}]

where f^{0} = {f_{λ}^{0} | λ ∈ Λ} is a level cofibration, f^{1} = {f_{λ}^{1} | λ ∈ Λ} is a level homotopy equivalence and M(f) = (M(f_{λ}), m_{λλ′}, Λ). The maps m_{λλ′} are obtained from the universal properties of the various mapping cylinders.

3.4. DM(f_{λ}) denote the double mapping cylinder of f_{λ} [12], [14], [21], that is the adjunction space (X_{λ}×I)∪_{f_{λ}}(Y_{λ}×∂I), equipped with canonical maps K_{λ} : X_{λ}×I → DM(f_{λ}) and j_{i,λ} : Y_{λ} → DM(f_{λ}), i = 0, 1, such that K_{λ} ∘ e_{i} = j_{i,λ} ∘ f_{λ}, i = 0, 1. Since DM(f_{λ}) is a colimit object, there is a unique map V_{λ} : DM(f_{λ}) → Y_{λ}×I, with the property that V_{λ}∘K_{λ} = f_{λ}×1 and V_{λ}∘j_{i,λ} = e_{i}, i = 0, 1. For every λ ⩽ λ′, there is a unique map n_{λλ′} : DM(f′_{λ}) → DM(f_{λ}), such that n_{λλ′} ∘ K_{λ′} = K_{λ} ∘ (x_{λλ′} × 1) and n_{λλ′} ∘ j_{i,λ′} = j_{i,λ} ∘ y_{λλ′}, i = 0, 1. The situation is better illustrated by the following commutative diagram

[figure: X_{λ′}; X_{λ}; Y_{λ′}; f_{λ′}; f_{λ}; Y_{λ}; DM(f_{λ′}); V_{λ′}; V_{λ}; DM(f_{λ}); K_{λ′}; f_{λ′} × 1; f_{λ} × 1; K_{λ}; X_{λ′} × I; X_{λ} × I; x_{λλ′}; x_{λλ′} × 1; y_{λλ′}; n_{λλ′}; y_{λλ′} × 1; Y_{λ′} × I; Y_{λ} × I; e_{0}; e_{1}; e_{0}; e_{1}; j_{0}; j_{1}; j_{0}; j_{1}]

with the obvious meaning of the maps involved. It follows that there is an inverse system DM(f) = (DM(f_{λ}), n_{λλ′}, Λ) and level maps K : X × I → DM(f), j_{0}, j_{1} : Y → DM(f) and V : DM(f) → Y × I, with K ∘ e_{i} = j_{i} ∘ f, i = 0, 1, and such that f × 1 = K ∘ V and V ∘ j_{i} = e_{i}, i = 0, 1. We point out that, if f is a level cofibration, then V is one too [12].

Theorem 3.5. The class of strong shape equivalences of pro𝒯op has the following properties:

1. contains all level homotopy equivalences,

2. if two of f, g, g ∘ f are strong shape equivalences, so is the third,

3. a level promap f is a strong shape equivalence iff f^{0} is,

4. if f is a strong shape equivalence and a level cofibration, then for every g : X → A, A ∈ 𝒜𝒩ℛ, there is an h : Y → A such that h ∘ f = g.

Proof. (1) is clear (see also [8], 4.1, 4.2). (2) depends on the fact that Ss(g ∘ f) = Ss(g) ∘ Ss(f). (3) follows from (2). (4) Since f is a level cofibration, there is a weak pushout diagram in pro𝒯op, with respect to 𝒜𝒩ℛ

                     [figure: X; X × I; Y; Y × I; e_{0}; e_{0}; f; f × 1]

Given now promaps ϕ : Y → A, A ∈ 𝒜𝒩ℛ, and F : X × I → A such that F ∘ e_{0}^{X} = ϕ ∘ f, there exists a λ ∈ Λ, such that the relative λ-diagram commutes. Therefore there is a homotopy G_{λ} : Y_{λ} × I → A with G_{λ} ∘ (f_{λ} × 1) = F_{λ} and G_{λ} ∘ e_{0,λ} = ϕ_{λ}. Such data define a homotopy G : Y × I → A with G ∘ (f × 1) = F and G ∘ e_{0}^{Y} = ϕ. At this point the assertion follows from ([12], 2.2.4).

In view of the theorem above, one can restrict the study of strong shape equivalences to those promaps that are level cofibrations.

In [21] the strong homotopy extension property (SHEP) for maps has been introduced, with respect to 𝒜𝒩ℛ. This can be generalized to promaps in the following way: a promap f : X → Y has the SHEP, w.r.t. 𝒜𝒩ℛ, iff the following diagram

                     [figure: X; X × I; Y; Y × I; e_{0}; e_{1}; e_{0}; e_{1}; f; f × 1]

is a weak colimit in pro𝒯op, w.r.t. 𝒜𝒩ℛ. This means that, for given promaps u, v : Y → A and homotopy H : X × I → A, A ∈ 𝒜𝒩ℛ, connecting u ∘ f and v ∘ f, there exists a homotopy G : Y × I → A, connecting u and v and such that H = G ∘ (f × 1).

Theorem 3.6. (cf. [21], sec.2) Let f : X → Y be a level cofibration in pro𝒯op having property (SSE1). The following are equivalent :

1. f is a strong shape equivalence,

2. f has the SHEP w.r.t. 𝒜𝒩ℛ,

3. V has property (SSE1).

Proof. (1) implies (2) : since f is a level cofibration, this follows from ([3], 7.2.5).

(2) implies (3) : Let α : DM(f) → A, A ∈ 𝒜𝒩ℛ, and consider α ∘ K : X × I → A. It is a homotopy connecting α ∘ j_{0} ∘ f to α ∘ j_{1} ∘ f, then there is a homotopy T : Y × I → A such that T ∘ (f × 1) = α ∘ K and T ∘ e_{i} = α ∘ j_{i}. It follows that T ∘ V ∘ K = T ∘ (f × 1) = α ∘ K and T ∘ V ∘ j_{i} = T ∘ e_{i} = α ∘ j_{i}, i = 0, 1. From the universal property of the double mapping cylinder, one obtains that T ∘ V = α.

(3) implies (1) : Let h_{0}, h_{1} : Y → A, A ∈ 𝒜𝒩ℛ, be given together with a homotopy H : X × I → A connecting h_{0} ∘ f to h_{1} ∘ f. There is a unique γ : DM(f) → A such that γ ∘ j_{0} = h_{0}, γ ∘ j_{1} = h_{1} and γ ∘ K = Ȟ. Since we may write V = p_{V} ∘ Π_{V} ∘ e_{0} (see (3.3)), it follows that Π_{V} ∘ e_{0} satisfies (SSE1) too and is a level cofibration. Then there is a G : M(V) → A such that G ∘ Π_{V} ∘ e_{0} = γ. It turns out that G ∘ j_{V} is a (global) homotopy connecting h^{0} and h^{1}. Moreover, one has G ∘ j_{V} ∘ (f × 1) = G ∘ j_{V} ∘ V ∘ K = γ ∘ K = H.

Corollary 3.7. V is a shape equivalence whenever f is a strong shape equivalence and a level cofibration.

Proof. We only have to show that V induces, for all A ∈ 𝒜𝒩ℛ, an onto map V_{A}^{∗} : [Y×I, A] → [DM(f), A]. Let α : DM(f) → A, then α ∘ K : X×I → A is a homotopy connecting α ∘ j_{0} to α ∘ j_{1}. Since α ∘ K ∘ e_{i} = α ∘ j_{i} ∘ f, there exists a homotopy T : Y × I → A, such that T ∘ (f × 1) = α ∘ K and T ∘ e_{i} = α ∘ j_{i}. It follows T ∘ V ∘ K = T ∘ (f × 1) = α ∘ K and T ∘ e_{i} = α ∘ j_{i}. From the universal property of the double mapping cylinder, one has α = T ∘ V.

We need to state the following technical result.

Lemma 3.8. Let f : X → Y be a strong shape equivalence and a level cofibration in pro𝒯op. f induces a bijection f_{𝒟}^{∗} : [Y, lim 𝒟] → [X, lim 𝒟], for every finite diagram 𝒟 [18] in 𝒜𝒩ℛ, having at most one arrow connecting every two vertices.

Proof. Let 𝒟 have vertices D_{i}, i ∈ I, and morphisms D_{u} : D_{i} → D_{j}, for u : i → j in I. Assume that α : X → lim 𝒟 is given and let p_{i} : lim 𝒟 → D_{i}, i ∈ I, be the projections of the limit. By 3.5(4), for every i ∈ I, there is a promap β_{i} : Y → D_{i}, such that β_{i} ∘ f = p_{i} ∘ α. If I(i, j) = ∅, for all i ∈ I, i ≠ j, put h_{j} = β_{j}. If u ∈ I(i, j), define h_{i} = D_{u} ∘ j. In this way one obtains a natural cone from Y to the vertices of the diagram, which induces a unique promap h : Y → lim 𝒟, with h ∘ f = α.

The proof of the following theorem is partially inspired by Thm. 4.4 of [8].

Theorem 3.9. Let f : X → Y be a strong shape equivalence in pro𝒯op. Then f induces a bijection f^{∗} : [Y, Z] → [X, Z], for every strongly fibrant prospace Z ∈ pro𝒜𝒩ℛ.

Proof. First of all we may assume, as usual, that f is a level promap with cofinite index set. Moreover, using the mapping cylinder decomposition of f, we may also assume that f is a level cofibration. Let g : X → Z be a given promap, with Z = (Z_{μ}, z_{μμ′}, M) ∈ pro𝒜𝒩ℛ strongly fibrant. The fact that f is a shape equivalence, by 3.5(4), implies that, for every μ ∈ M, there is a k_{μ} : Y → Z_{μ} such that k_{μ} ∘ f = g_{μ}. By induction on the number #(μ) of the predecessors of μ, let us define h_{μ} = k_{μ} if #(μ) = 0, and assume to have defined h_{μ}, for every μ ∈ M with 1 ⩽ #(μ) < n, in such a way that z_{μμ′} ∘ h_{μ′} = h_{μ}, for μ ⩽ μ′. Let μ^{∗} ∈ M having #(μ^{∗}) = n. The promaps h_{μ}, for μ < μ^{∗}, define a map z_{μ^{∗}} : Z_{μ^{∗}} → lim_{μ⩽μ^{∗}} Z_{μ^{∗}}, and one has z_{μ^{∗}} ∘ k_{μ^{∗}} ∘ f = z_{μ^{∗}} ∘ g_{μ^{∗}}. By Lemma 3.8, there is a promap γ : Y → lim_{μ⩽μ^{∗}} Z_{μ^{∗}}, with the property that γ ∘ f = z_{μ^{∗}} ∘ k_{μ^{∗}}. In diagram

[figure: X; Y; lim_{μ⩽μ^{∗}} Z_{μ^{∗}}; Z_{μ^{∗}}; f; z_{μ^{∗}}; g_{μ^{∗}}; k_{μ^{∗}}; γ]

Then, z_{μ^{∗}}∘k_{μ^{∗}}∘f ≃ γ ∘ f. Again by Lemma 3.8, since f is a strong shape equivalence, there is a homotopy H : Y × I → lim_{μ⩽μ^{∗}} Z_{μ^{∗}}, with H : γ ≃ z_{μ^{∗}} ∘ k_{μ^{∗}}. Since z_{μ^{∗}} is a fibration, there is a homotopy H^{∗} : Y × I → Z_{μ^{∗}}, such that H^{∗} ∘ e_{0} = k_{μ^{∗}} and z_{μ^{∗}} ∘ H^{∗} = H. If we put h_{μ^{∗}} = H^{∗} ∘ e_{1}, the the definition of h : Y → Z is complete and one has h ∘ f = g. Let now h, h′ : Y → Z be such that h ∘ f ≃ h′ ∘ f, by means of a homotopy F : X × I → Z. For every λ ∈ Λ, DM(f_{λ}) = X_{λ} × I ∪ Y_{λ} × {0, 1} and the inclusion V_{λ} : DM(f_{λ}) → Y_{λ} × I is a cofibration. If F̃_{λ} : X_{λ} × I ∪ Y_{λ} × {0, 1} → Z_{λ} is defined by

F̃_{λ}(x, t) = \{ \begin{matrix}F_{λ}(x, y), & for (x, t) ∈ X_{λ} × I \\ h_{λ}(x), & t = 0 \\ h′_{λ}(x), & t = 1 \end{matrix}

Then F̃ : DM(f) → Z, F̃ = {F̃_{λ} | λ ∈ Λ}, is a level promap. Since the promap V : DM(f) → Y × I is a shape equivalence and a level cofibration, it follows that F̃ has an extension G : Y × I → Z, which turns out to be a homotopy connecting h to h′.

Theorem 3.10. A continuous map f : X → Y is a strong shape equivalence iff it induces a bijection f^{∗} : [Y, Z] → [X, Z], for every strongly fibered space Z.

Proof. Let f be a strong shape equivalence, then by Thm 3.9 it induces a bijection f^{∗} : [Y, Z] → [X, Z], for every strongly fibrant prospace Z ∈ pro𝒜𝒩ℛ. Let Z = lim Z. Since the projection of the limit p : Z → Z induces bijections [X, Z] → [X, Z] and [Y, Z] → [Y, Z], the first part of the theorem easily follows. Conversely, let f induce bijections f^{∗} : [Y, Z] → [X, Z], for every strongly fibered space Z. Since every ANRspace is strongly fibered, it follows at once that f is a shape equivalence. Taking Z = Φ(X), there is a g : Y → Φ(X) such that [g ∘ f] = [μ]. Since μ : X → Φ(X) is a strong shape equivalence, it follows that f is such.

Recently, Prasolov [17] has defined the strong shape category of prospaces sSh(pro𝒯op) by localizing pro𝒯op at the class of strong shape equivalences as defined by the properties (SSE1) and (SSE2) above. The two categories coincide. In fact, from the construction of sSh(pro𝒯op), since ho(Φ) is reflective, it follows that

sSh(pro𝒯op) ≅ (ho(pro𝒯op))^{ho(Φ)} ≅ ho(pro𝒯op)[Σ{−1 \atop ho(Φ)}] ≅

≅ ho(pro𝒯op)[Σ_{sS}^{−1}] ≅ pro𝒯op[𝒮𝒮ℰ^{−1}],

where 𝒮𝒮ℰ is the class of strong shape equivalences in pro𝒯op, that is those promaps f ∈ pro𝒯op such that L(f) ∈ Σ_{sS}.

4. SSDR-promaps.

In this section we discuss some points from [9] in connection with the results obtained in the previous section. We need some preliminary results before to go on.

Let us recall the following definition from [9] :

4.1. a promap f : X → Y is called an SSDR-promap provided that any commutative diagram in pro𝒯op

[figure: X; Map(K, A); Y; Map(L, A); a; b; f; i^{∗}]

has a filler Y → Map(K, A), whenever K is a finite CW complex, L is a finite subcomplex, i : L → K is the inclusion and A ∈ 𝒜𝒩ℛ. This notion is a generalization of that of SSDR-map introduced in [4]. Map(K, A) denotes the space of mappings with the compact-open topology.

In the sequel we shall denote by SSDR the class of SSDR-promaps while SSDR𝒯op will be the subclass of SSDR whose elements are of the form f : X → Y, Y ∈ 𝒯op.

Thm. 3.5 of [9] states that f : X → Y is an SSDR promap iff it satisfies the following two conditions :

(SSDR1) for every A ∈ 𝒜𝒩ℛ and for every h : X → A, there is a g : Y → A, such that g ∘ f = h,

(SSDR2) given morphisms g, h : Y → A, A ∈ 𝒜𝒩ℛ, and a global homotopy F : X × I → A joining f ∘ g and f ∘ h, there exists a global homotopy G : Y × I → A joining g and h, such that F = G ∘ (f × 1).

Since (SSDR2) says exactly that f has the SHEP w.r.t. 𝒜𝒩ℛ, from theorems 3.5(4) and 3.6, one obtains the

Theorem 4.2. Theorem 4.2 Let f : X → Y be a level cofibration in pro𝒯op. f is an SSDR promap iff it is a strong shape equivalence.

Remark 4.3. As a consequence of the theorem, it follows that every trivial cofibration is an SSDR-promap. In fact, by ([10], 3.3.36), one may assume, up to isomorphisms, that f is a level trivial cofibration. Then, it is clear that every (strongly) SSDR-fibrant prospace is also (strongly) fibrant. Moreover, if Z is a (strongly) SSDR-fibered prospace, then its inverse limit lim Z is SSDR_{𝒯op}-fibrant: let f : X → Y ∈ SSDR_{𝒯op} and a : X → lim Z, be given. If p : lim Z → Z is the limiting cone, there is a y : Y → Z, such that y ∘ f = p ∘ a and, by the universal property of the limit, there is also a t : Y → lim Z, with p ∘ t = y. It follows that t ∘ f = a.

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L. Stramaccia stra@dipmat.unipg.it

Dipartimento di Matematica e Informatica Universitá di Perugia via Vanvitelli, I-06123 Perugia