Stratified fibre bundles

Hans-Joachim Baues and Davide L. Ferrario

Abstract

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category 𝔉 of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem.

AMS SC : 55R55 (Fiberings with singularities); 55R65 (Generalizations of fiber spaces and bundles); 55R70 (Fiberwise topology); 55R10 (Fibre bundles); 18F15 (Abstract manifolds and fibre bundles); 54H15 (Transformation groups and semigroups); 57S05 (Topological properties of groups of homeomorphisms or diffeomorphisms).

Keywords: Stratified fibre bundle, structure category, function space of admissible maps, principal bundle

1 Introduction

A stratified bundle is a filtered fibered space X = {X_{i}, i ≥ 0} for which the complements X_{i} ∖ X_{i−1}, termed strata, are fibre bundles. Moreover the attachment of strata is controlled by a structure category 𝔉 of fibers. For example the tangent bundle of a stratified manifold is a stratified vector bundle, see [4]. Moreover for a compact smooth G-manifold M the projection M → M/G to the orbit space is a stratified bundle by results of Davis [7]. In this paper we prove three basic properties: The pull back theorem shows that certain pull backs of stratified bundles are again stratified bundles, see (4.7). The bundle theorem (5.1) states that stratified bundles are the same as classical fibre bundles in case the structure category is a groupoid. Moreover the principal bundle theorem (6.12) shows that the function space of admissible maps, X^{V}, is a stratified bundle and the 𝔉^{op}-diagram V ↦ X^{V} plays the role of the associated principal bundle, see (6.17). In the proofs of the results we have to consider the intricate compatibility of quotient topology, product topology, and compact-open topology in function spaces. As application we study in [4] algebraic constructions on stratified vector bundles (like the direct sum, tensor product, exterior product) and we show that stratified vector bundles lead to a stratified K-theory generalizing the Atiyah-Hirzebruch K-theory. This was the main motivation for the proof of the basic properties in this paper.

2 Fibre families

Let 𝔉 be a small category together with a functor F: 𝔉 → Top to the category Top of topological spaces. We assume that the functor F satisfies the assumption:

(∗) For every object V in 𝔉 the space F (V) is locally compact, second-countable and Hausdorff.

Then 𝔉 is termed structure category and F is a fibre functor on 𝔉. If F is a faithful functor then 𝔉 is a topological enriched category in which morphism sets have the compact open topology. In many examples the functor F is actually the inclusion of a subcategory 𝔉 of Top so that in this case we need not to mention the fibre functor F.

A fibre family with fibres in 𝔉 (or a (𝔉, F)-family) is a topological space X, termed total space, together with a map p_{X}: X → X̄, termed projection to the base space X̄, and for every b ∈ X̄ a selected homeomorphism Φ_{b}: p_{X}^{−1}b ≈ F X_{b} where X_{b} is an object in 𝔉, called fibre, depending on b ∈ X̄. The homeomorphism Φ_{b} is termed chart at b. The family (p_{X}: X → X̄, X_{b}, Φ_{b}, b ∈ X̄) is denoted simply by X. Each object V in 𝔉 yields the point family also denoted by V given by the map p_{V} : F V → ∗ where ∗ is the singleton space.

Given two 𝔉-families X and Y a 𝔉-map from X to Y is a pair of maps (f, f̄) such that the following diagram

\xymatrix{X \ar[r]^{f} \ar[d]_{p_{X}} & Y \ar[d]^{p_{Y}} \\ X̄ \ar[r]^{f̄} & Ȳ ,}

commutes, and such that for every b ∈ X̄ the composition given by the dotted arrow of the diagram

\xymatrix{p_{X}^{−1}(b) \ar[r]^{f|p^{−1}(b)} & p_{Y}^{−1}(f̄b) \ar[d]^{Φ_{f̄b}} \\ F X_{b} \ar[u]^{Φ_{b}^{−1}} \ar@{.>}[r] & F Y_{f̄b}}

is a morphism in the image of the functor F. That is, there exists a morphism ϕ: X_{b} → Y_{f̄b} in 𝔉 such that the dotted arrow is equal to F (ϕ). We will often denote F X_{b} by X_{b} and it will be clear from the context whether X_{b} denotes an object in 𝔉 or a space in Top given by the functor F. If a 𝔉-map f = (f, f̄) is a 𝔉-isomorphism then f and f̄ are homeomorphisms but the converse need not be true.

If X is a 𝔉-family and Z is a topological space, then X × Z is a 𝔉-family with projection p_{X×Z} = p × 1_{Z}: X × Z → X̄ × Z. The fibre over a point (b, z) ∈ X̄ × Z is equal to p_{X}^{−1}(b) × {z}; using the chart Φ_{b} : p_{X}^{−1}(b) → X_{b} the chart Φ_{(b,z)} is defined by (x, z) ↦ Φ_{b}(x) ∈ X_{b} where of course we set (X × Z)_{(b,z)} = X_{b}. In particular, by taking Z = I the unit interval we obtain the cylinder object X × I and therefore the notion of homotopy: two 𝔉-maps f_{0}, f_{1} : X → Y are 𝔉-homotopic (in symbols f_{0} ∼ f_{1}) if there is a 𝔉-map F: X × I → Y such that f_{0} = F i_{0} and f_{1} = F i_{1}. Here i_{0} and i_{1} are the inclusions X → X × I at the levels 0 and 1 respectively.

Let 𝔉-Top be the category consisting of 𝔉-families p_{X} : X → X̄ in Top and 𝔉-maps. Homotopy of 𝔉-maps yields a natural equivalence relation ∼ on 𝔉-Top so that the homotopy category (𝔉-Top)/_{∼} is defined.

(2.1) Remark. Consider a structure category 𝔉 with fibre functor F. In general it is not assumed that F is faithful, but it is easy to see that the category 𝔉-Top is equivalent to 𝔉′-Top, where 𝔉′ = 𝔉/_{≡} is the category with the same objects as 𝔉 and morphisms given by equivalence classes of morphisms, with f ≡ f′ ⟺ F (f) = F (f′). We call 𝔉′ the faithful image category of F.

(2.2) Definition. If V is a fibre in 𝔉 and X̄ is a space in Top, then the projection onto the first factor p_{1} : X = X̄ ×V → X̄ yields the product family with fibre V; the charts Φ_{b}: {b} × V → V = X_{b} are given by projection and X_{b} = V for all b ∈ X̄. If X is a 𝔉-family 𝔉-isomorphic to a product family then X is said to be a trivial 𝔉-bundle. In general a 𝔉-bundle is a locally trivial family of fibres, i.e. a family X over X̄ such that every b ∈ X̄ admits a neighborhood U for which X|U is trivial. Here X|U is the restriction of the family X defined by U ⊂ X̄.

Given a family Y with projection p_{Y} : Y → Ȳ and a map f̄: X̄ → Ȳ, the pull-back X = f̄^{∗}Y is the total space of a family of fibres given by the vertical dotted arrow of the following pull-back diagram.

\xymatrix{X = f̄^{∗}Y \ar@{.>}[r] \ar@{.>}[d] & Y \ar[d]^{p} \\ X̄ \ar[r]^{f̄} & Ȳ .} (2.3)

The charts are defined as follows: For every b ∈ X̄ let X_{b} = Y_{f̄b}, and Φ_{b} : p_{X}^{−1}(b) → X_{b} the composition p_{X}^{−1}(b) → p_{Y}^{−1}(f̄b) ≈ Y_{f̄b} = X_{b} where the map p_{X}^{−1}(b) → p_{Y}^{−1}(f̄b) is a homeomorphism since X is a pull-back.

A 𝔉-map i: A → Y is termed a closed inclusion if ī: Ā → Ȳ is an inclusion, īĀ is closed in Ȳ and the following diagram is a pull-back:

\xymatrix{i^{∗}Y = A \ar[r]^{i} \ar[d]_{p_{A}} & Y \ar[d]^{p_{Y}} \\ Ā \ar[r]^{ī} & Y} (2.4)

Hence a closed inclusion i: A → Y induces homeomorphisms on fibres.

The push-out construction can be extended to the category 𝔉-Top, provided the push-out is defined via a closed inclusion.

(2.5) Lemma. Given 𝔉-families A, X, Y and 𝔉-maps f: A → X, i: A → Y with i a closed inclusion the push-out diagram in 𝔉-Top

\xymatrix{A \ar[r]^{f} \ar[d]_{i} & X \ar[d] \\ Y \ar[r] & Z} (2.6)

exists and X → Z is a closed inclusion.

Proof. The push-out is obtained by the following commutative diagram,

[figure: A; f; p_{A}; i; X; p_{X}; Ā; f̄; ī; X̄; Ȳ; Z̄; Y; p_{Y}; Z; p_{Z}]

in which the spaces Z and Z̄ are push-outs in Top of f, i and f̄, ī respectively; the map p_{Z} exists and is unique because of the push-out property. Moreover, the charts of fibres of Z are given as follows: Z̄ is the union of the two subsets X̄ and Ȳ ∖ Ā. For b = x̄ ∈ X̄ ⊂ Z̄ there is a canonical composition of homeomorphisms p_{Z}^{−1}(b) ≈ p_{X}^{−1}(x̄) ≈ Xx̄ that yields the chart

Φ_{b} : p_{Z}^{−1}(b) → Xx̄ = Z_{b}.

In fact, i : A → Y being a closed inclusion, for every ā ∈ Ā the fibre over ā in A is canonically isomorphic to the fibre over ā in Y and hence for every x̄ ∈ X the fibre over x̄ in X is canonically isomorphic to the fibre over x̄ in Z. Next, for b = ȳ ∈ Ȳ ∖ Ā ⊂ Z the homeomorphism is

Φ_{b} : p_{Z}^{−1}(b) ≈ p_{Y}^{−1}(ȳ) ≈ Yȳ = Z_{b}.

The 𝔉-map j : Y → Z induced by the push-out is a closed inclusion, since j̄ : Ȳ → Z̄ is the inclusion of a closed subspace of Z̄; furthermore, j is an isomorphism on fibres and an inclusion, and therefore Y = j^{∗}Z. q.e.d.

3 𝔉-complexes

For an object V in 𝔉 the family F (V) → ∗ with base space a singleton is termed a 𝔉-point and is also denoted by V. A disjoint union of 𝔉-points is called a 𝔉-set. This is a 𝔉-family for which the base space has the discrete topology. Let D^{n} be the unit disc in ℝ^{n} and S^{n−1} its boundary with base point ∗ ∈ S^{n−1}. The complement e^{n} = D^{n} ∖ S^{n−1} is the open cell in D^{n}. An 𝔉-cell is a product family V × e^{n} → e^{n} with V ∈ 𝔉.

We say that a 𝔉-family X is obtained from a 𝔉-family D by attaching n-cells if a 𝔉-set Z together with a 𝔉-map f is given, such that the following diagram

\xymatrix{Z × S^{n−1} \ar[r]^{f} \ar[d] & D \ar[d] \\ Z × D^{n} \ar[r]^{Φ} & X = D ∪_{f} (Z × D^{n})} (3.1)

is a push-out in 𝔉-Top. The inclusion Z × S^{n−1} → Z × D^{n} is a closed inclusion, therefore the push-out exists and the induced map D → X is a closed inclusion and X ∖ D = Z × e^{n} is a union of open 𝔉-cells. If Z is a 𝔉-point then we say that X is obtained from D by attaching a 𝔉-cell and Φ is the characteristic map of the 𝔉-cell.

(3.2) Definition. A relative 𝔉-complex (X, D) is a family X and a filtration

D = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ X_{n+1} ⊂ ⋯ ⊂ X

of 𝔉-families X_{n}, n ≥ −1, such that for every n ≥ 0 the 𝔉-family X_{n} is obtained from X_{n−1} by attaching n-cells and

X = lim_{n≥0} X_{n}.

The spaces X_{n} are termed n-skeleta of (X, D). If D is empty we call X a 𝔉-complex. Then X is a union of 𝔉-cells. It is not difficult to show that a relative 𝔉-complex (X, D) is Hausdorff and normal provided that D is Hausdorff and normal.

(3.3) Example. Let G be a compact Lie group and let 𝔉 be the category of orbits of G, that is, 𝔉 is the subcategory of Top consisting of spaces G/H, where H is a closed subgroup of G, and G-equivariant maps G/H → G/H′. Then each G-CW-complex (see [16]) is a 𝔉-complex.

4 Stratified bundles

Let A be a closed subset of a space X. We say that (X, A) is a CW-pair if there exists a homeomorphism (X, A) ≈ (X′, A′) of pairs where X′ is a CWcomplex and A′ a subcomplex of X′. A CW-space is a space homeomorphic to a CW-complex.

(4.1) Definition. We call a space X a stratified space if a filtration

X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ ⋯ ⊂ lim_{n→∞} X_{n} = X

is given and if for every i ≥ 1 there is given a CW-pair (M_{i}, A_{i}) and a map h_{i}: A_{i} → X_{i−1} with the following properties: The subspace X_{i} is obtained by attaching M_{i} to X_{i−1} via the attaching map h_{i}, i.e. there is a push-out diagram:

[figure: A_{i}; h_{i}; M_{i}; X_{i−1}; X_{i}.]

Moreover X_{0} is a CW-space. The complements X_{i} ∖ X_{i−1} are termed strata, while the filtration ∅ ⊂ X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ … is called stratification of X. The strata X_{i} ∖ X_{i−1} coincide with the complements M_{i} ∖ A_{i}. We call the pairs (M_{i}, A_{i}) and the space X_{0} the attached spaces of X. If all attached spaces (M_{i}, A_{i}) are CW-complexes with subcomplexes A_{i} and if all attaching maps are cellular then X is a CW-complex and X_{i} is a subcomplex of X. In this case we say that the stratified space X is a stratified CW-complex. In particular CW-complexes with the skeletal filtration are stratified CW-complexes. We say that a stratified space is finite if the number of non-empty strata is finite. We always assume that a stratified space which is not finite is a stratified CW-complex. The cellular approximation theorem shows that a stratified space is homotopy equivalent to a stratified CW-complex. We point out that a stratified space X is a Hausdorff and regular space, see Gray [9].

A map

f: X → X′ (4.2)

between stratified spaces is a filtration preserving map f = {f_{n}}_{n≥0} together with commutative diagrams

[figure: X_{i−1}; f_{i−1}; A_{i}; M_{i}; g_{i}; X′_{i−1}; A′_{i}; M′_{i}]

such that g_{i} ∪ f_{i−1} = f_{i} for i ≥ 1. A map is termed stratum-preserving if for every i

f_{i}(X_{i} ∖ X_{i−1}) ⊂ X′_{i} ∖ X′_{i−1}.

Let Stra be the category of stratified spaces and stratum-preserving maps.

Consider a manifold with boundary (M, ∂M), a manifold N, and a submersion h: ∂M → N. The push-out of h and the inclusion ∂M ⊂ M

[figure: ∂M; h; M; N; M ∪_{h} N.] (4.3)

yields a stratified space X = M ∪_{h} N with stratification X_{0} = N ⊂ X_{1} = X. For example if a diffeomorphism ∂M ≅ Z × P is given where Z and P are manifolds and if h : ∂M ≅ Z × P → Z is defined by the projection then X = M ∪_{h} N is a manifold with singularities, see Rudyak [13], Baas [2], Botvinnik [5], Sullivan [15], Vershinin [17]. Also the stratified manifolds (stratifolds) of Kreck [11] are stratified spaces. Manifolds with singularities and stratifolds have “tangent bundles” which are stratified vector bundles. These are important examples of stratified fibre bundles introduced in the next definition.

Recall that 𝔉 denotes a structure category together with a fibre functor F: 𝔉 → Top.

(4.4) Definition. A 𝔉-stratified fibre bundle is a stratified space X̄ together with a 𝔉-family

X → X̄ with the following properties. For i ≥ 1 the restriction X_{i} = X|X̄_{i} = X_{i−1}∪_{A_{i}} M_{i} is the push-out of 𝔉-maps

[figure: M_{i}; A_{i}; h_{i}; X_{i−1}; M̄_{i}; Ā_{i}; h̄_{i}; X̄_{i−1}]

where M_{i} → M̄_{i} is a 𝔉-bundle and A_{i} = M_{i}|Ā_{i}. Moreover, X_{0} → X̄_{0} is a 𝔉-bundle and X = lim_{i→∞} X_{i}. Hence the strata X_{i} ∖ X_{i−1} → X̄_{i} ∖ X̄_{i−1} are 𝔉-bundles.

(4.5) Example. By comparing definition (4.4) and definition (3.2) it is easy to see that a 𝔉-complex with the skeletal filtration is a 𝔉-stratified bundle since the spaces (V × D^{n}, V × S^{n−1}) are trivial 𝔉-bundles over the CW-pair (D^{n}, S^{n−1}). Moreover, if X is a 𝔉-stratified bundle such that all the attaching maps are cellular, then X is a 𝔉-complex. Here we use the bundle theorem in section 5 below. In general, a 𝔉-stratified bundle has the 𝔉-homotopy type of a 𝔉-complex (this is a consequence of a cellular approximation theorem for 𝔉-complexes, see [3]).

(4.6) Example. Let G be a compact Lie group and M a compact smooth G-manifold. Let 𝔉 be the orbit category of G. Then the augmented Gnormal system associated to M, as defined by Davis in [7], yields naturally a 𝔉-stratified bundle M → M̄ = M/G via the corresponding assembling functor. The strata are defined exactly as in the proof of theorem 4.9 of [7], by M_{n} = M(n − 1) and A_{n} = ∂M_{n}. Thus each open stratum contains the disjoint union of all the open strata given by the stratification of M (or the corresponding stratification on M̄) by normal orbit type of depth n in the poset of the normal orbit types.

A 𝔉-stratified map f: X → X′ between 𝔉-stratified bundles is is given by sequences {f_{n}}_{n≥0} and {g_{n}}_{n≥1} of 𝔉-maps such that, given the commutative diagrams

[figure: X_{i−1}; f_{i−1}; A_{i}; M_{i}; g_{i}; X′_{i−1}; A′_{i}; M′_{i}]

for every i, we have g_{i} ∪ f_{i−1} = f_{i} for i ≥ 1. If for every i the inclusion f(X_{i} ∖ X_{i−1}) ⊂ X′_{i} ∖ X′_{i−1} holds, f is termed stratum-preserving.

(4.7) Pull back theorem. Let X̄ and X̄′ be finite stratified spaces with attached spaces which are locally finite and countable CW-complexes. Let f̄: X̄ → X̄′ be a stratum-preserving map (in Stra) and X′ → X̄′ a 𝔉stratified bundle. Then the pull-back f̄^{∗}X′ → X̄ is a 𝔉-stratified bundle.

Proof. Let n be the number of strata of X̄ and X_{i} = f̄^{∗}X′_{i} for i = 1, … n. Clearly f̄_{0}^{∗}X′_{0} is a 𝔉-bundle on X̄_{0}. Since X′ is a 𝔉-stratified bundle there are 𝔉-bundles M′_{i} and 𝔉-maps h′_{i}: A′_{i} ⊂ M′_{i} → X′_{i−1} such that X′_{i} = X′_{i−1} ∪_{h′_{i}} M′_{i}. Let the attached spaces of X̄ be denoted by (M̄_{i}, Ā_{i}) and h̄_{i} the attaching maps. The stratified map f̄ yields maps ḡ_{i}: M̄_{i} → M̄′_{i} such that ḡ_{i}∪f̄_{i−1} = f̄_{i}. For every i ≥ 1 let M_{i} be the pull-back ḡ_{i}^{∗}M′_{i} → M̄_{i}. It is a 𝔉-bundle. Let A_{i} be its restriction to Ā_{i}. By assumption M̄_{i} and M̄′_{i} are locally finite and countable; furthermore since f̄ is stratum-preserving, for every i we have f̄(X̄_{i} ∖ X̄_{i−1}) ⊂ X̄′_{i} ∖ X̄′_{i−1}. Thus, by applying proposition (7.4) below, the pull-back family X_{i} is obtained by attaching the 𝔉-bundle M_{i} to X_{i} via a 𝔉-map h_{i}: A_{i} ⊂ M_{i} → X_{i}, where h_{i} is a suitable 𝔉-map induced by the construction. This is true for i = 1, …, n, hence X_{n} = X is a 𝔉-stratified bundle. q.e.d.

A similar theorem was proved by Davis [7] (theorems 1.1 and 1.3), in the case of the pull-back f^{∗}M of a smooth G-manifold M, where f is a (weakly) stratified map and G is a compact Lie group.

Let (𝔉, F) and (𝔉′, F′) be structure categories. Then we define the structure category (𝔉 × 𝔉′, F × F′) by the functor

F × F′: 𝔉 × 𝔉′ → Top.

Here 𝔉×𝔉′ is the product category consisting of pairs (V, V′) of objects V ∈ 𝔉 and V′ ∈ 𝔉′ and pairs of morphisms. The functor F × F′ carries (V, V′) to the product space F (V) × F′(V′). For a 𝔉-family X and a 𝔉′-family X′ with the base space X̄ = B = X̄′ we define the fiberwise product X ×_{B} X′ by the pull-back diagram in Top

\xymatrix{X ×_{B} X′ \ar[r] \ar[d] \ar@{.>}[dr]^{p} & X′ \ar[d] \\ X \ar[r] & B}

Then it is clear that X ×_{B} X′ → B is a (𝔉 × 𝔉′, F × F′)-family. Moreover we get compatibility with stratifications as follows.

(4.8) Proposition. Let X be a 𝔉-stratified bundle and let X′ be a 𝔉′- stratified bundle such that the stratified spaces X̄ = B = X̄′ coincide. Moreover assume that X and X′ are locally compact. Then X ×_{B} X′ → B is a 𝔉 × 𝔉′-stratified bundle.

Proof. By construction X is the quotient space of a projection

q: X_{0} ∪ ∐_{i≥0} M_{i} → X,

and the same holds for X′, with a projection q′. By the Whitehead theorem the map q × q′ is a quotient map onto X × X′, since X and X′ are locally compact. Now, X ×_{B} X ⊂ X × X′ is a closed subspace, hence the restricted projection

q × q′: (q × q′)^{−1}X ×_{B} X → X′ ×_{B} X′

is a quotient map. We can consider now the union

(X_{0} ×X̄_{0} X′_{0}) ∐_{i}(M_{i} ×M̄_{i} M′_{i}) ⊂ (q × q′)^{−1}X ×_{B} X → X′.

Then it is possible to see that the projection q × q′ yields a quotient map

(X_{0} ×X̄_{0} X′_{0}) ∐_{i}(M_{i} ×M̄_{i} M′_{i}) → X ×_{B} X′,

which gives the 𝔉 × 𝔉′-stratification of X ×_{B} X′. q.e.d.

5 The bundle theorem

A groupoid is a category in which all morphisms are isomorphisms. If the structure category 𝔉 is a groupoid then 𝔉-complexes and 𝔉-stratified bundles are actually 𝔉-bundles. More precisely we show the following result which we could not find in the literature though special cases are well known like the clutching construction of Atiyah [1].

(5.1) Bundle theorem. Let 𝔉 be a structure category which is a groupoid. Then a 𝔉-stratified bundle X → X̄ is a 𝔉-bundle over X̄. Conversely, given a stratified space X̄ and a 𝔉-bundle X over X̄ then X is a 𝔉-stratified bundle. In particular, a 𝔉-complex X → X̄ is a 𝔉-bundle over X̄ and, given a CWcomplex X̄ and a 𝔉-bundle X over X̄, then X is 𝔉-isomorphic to a 𝔉-complex.

Proof. If X is a 𝔉-complex then the theorem is a consequence of corollary (5.4) and lemma (8.5) below. Otherwise, assume that X is a 𝔉-stratified bundle. Since X is obtained by attaching 𝔉-bundles to 𝔉-bundles via 𝔉maps, we can apply a finite number of times corollary (5.4) below and obtain the result. On the other hand assume that X̄ is a stratified space and that X → X̄ is a 𝔉-bundle. By applying a finite number of times lemma (8.3) it is possible to show that X is a 𝔉-stratified bundle. q.e.d.

(5.2) Example. Let 𝔉 be the category consisting of finite sets {1, 2, …, n} and permutations of these sets. A 𝔉-family X is a finite-to-one map X → X̄. By theorem (5.1), if X is a 𝔉-complex, then p_{X} is locally trivial and hence a covering map.

(5.3) Lemma. Let 𝔉 be a groupoid. Let Y → Ȳ be a 𝔉-bundle, Z̄ a set and h: V × Z̄ × S^{n−1} → Y a 𝔉-map which is the attaching map of X = Y ∪_{h} (V × Z̄ × D^{n}). Given a set Ū ⊂ Ȳ open in Ȳ such that Y |Ū is trivial, there is an open set Ū′′ ⊂ X̄ such that Ū′′ ∩ Ȳ = Ū and such that X|Ū′′ is trivial. As a consequence, the push-out space X = Y ∪_{h} (V × Z̄ × D^{n}) is a 𝔉-bundle.

Proof. Let Φ denote the characteristic map Φ: V × Z̄ × D^{n} → X and Φ̄, h̄ the maps induced on the base spaces. The restriction of V × Z̄ × S^{n−1} to h̄^{−1}Ū is trivial, and with a suitable change of coordinates (which exists since Aut_{𝔉}(V) is a topological group) we can assume that the restriction

h_{0} : V × h̄^{−1}Ū → Y |Ū ≈ V × Ū

is of the form 1_{V} × h̄. Now, there is an open set Ū′ ⊂ Z̄ × D^{n} with the property that Ū′ ∩ Z̄ × S^{n−1} = h̄^{−1}Ū. Since X̄ = Ȳ ∪h̄ (Z̄ × D^{n}) the image Ū′′ = Φ̄(Ū′)∪Ū is open in X̄. Moreover, since V is locally compact Hausdorff the following diagram is a push-out.

\xymatrix{V × h̄^{−1}Ū \ar[r] \ar[d] & V × Ū \ar[d] \\ V × Ū′ \ar[r] & V × Ū′′}

But by assumption X is the push-out Y ∪_{h} (V × Z̄ × D^{n}), and the restriction of the push-out to the pre-images of Ū′′ yields a push-out

\xymatrix{V × h̄^{−1}Ū \ar[r] \ar[d] & V × Ū \ar[d] \\ V × Ū′ \ar[r] & X|Ū′′.}

Therefore X|Ū′′ ≈ V × Ū′′ by an isomorphism which is an extension of the chosen isomorphism Y |Ū ≈ V × Ū. This implies that every point in Ȳ has a neighborhood Ū′′ in X̄ such that X|Ū′′ is trivial. On the other hand, if x is a point in X̄ ∖ Ȳ, then there is a neighborhood Ū′′ of x contained in X̄ ∖ Ȳ and in this case X|Ū′′ is trivial because it is 𝔉-homeomorphic to V × Φ̄^{−1}Ū′′ ⊂ V × Z̄ × e^{n}. q.e.d.

(5.4) Corollary. Let (X, Y) a relative 𝔉-complex. Assume that 𝔉 is a groupoid and that Y → Ȳ is a 𝔉-bundle. Then X → X̄ is a 𝔉-bundle.

Proof. Consider the cellular filtration

Y = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ ⋯ ⊂ X = lim_{n} X_{n}.

By applying inductively lemma (5.3) we see that X_{n} is a 𝔉-bundle over X̄_{n} for every n ≥ 0. Moreover, there is a sequence of sets Ū_{n} ⊂ X̄_{n} open in X̄_{n} such that Ū_{n} ∩ X̄_{n−1} = Ū_{n−1} and 𝔉-isomorphisms α_{n}: X_{n}|Ū_{n} ≈ V × Ū_{n}. Moreover α_{n} can be chosen to be extension of α_{n−1} (see the proof of (5.3)). Since V is locally compact Hausdorff we can take limits, and obtain a 𝔉homeomorphism

X|Ū ≈ V × Ū,

where Ū = lim_{n} Ū_{n}. q.e.d.

6 The principal bundle theorem

It is well known that each fibre bundle X → X̄ with fibre V and structure group G yields the associated principal bundle X_{G} → X̄ with fibre G such that X_{G} is a right G-space for which there is an isomorphism of bundles.

\xymatrix{X \ar[rr]^{≅} \ar[dr] & & X_{G} ×_{G} V \ar[dl] \\ & X̄} (6.1)

Moreover Steenrod [14] p. 39 points out that in case G is a subgroup of the group of homeomorphisms of V with the compact open topology then the principal bundle

X_{G} = X^{V} (6.2)

is the function space X^{V} of “admissible” maps V → X with the compact open topology. The action of g ∈ G on α ∈ X^{V} is given by composition

α · g = \xymatrix{V \ar[r]^{g} & V \ar[r]^{α} & X} ∈ X^{V} .

In this section we generalize these results for 𝔉-complexes and 𝔉-stratified bundles.

(6.3) Definition. Given a faithful fibre functor F: 𝔉 → Top and an object V ∈ 𝔉 we obtain a new fibre functor

F^{V} : 𝔉 → Top

which carries an object W in 𝔉 to the space F^{V} (W) = hom_{𝔉}(V, W) with the compact open topology. The functor F^{V} need not be faithful, even though F is faithful.

(6.4) Remark. Since for every V the fibre F (V) is Hausdorff, we see that for each W ∈ 𝔉 the fibre F^{V} (W) = W^{V} of F^{V} is Hausdorff. Moreover, since W and V are 2^{∘} countable, also W^{V} is 2^{∘} countable (see e.g. [8]). We will show that if (𝔉, F) is a NKC structure category then W^{V} is locally compact so that F^{V} fulfills assumption (∗).

The definition of a NKC structure category is the following. First, a family K of compact sets of V is termed generating if for every 𝔉-family Y the subsets

N_{K,U} = {f ∈ Y^{V} , f(K) ⊂ U}

with K ∈ K and U open in Y yield a sub-basis for the topology of Y^{V}.

(6.5) Definition. A structure category 𝔉 with faithful fibre functor F has the NKC-property if for every V ∈ 𝔉 there is a generating family of compact sets K such that for every K ∈ K, every W ∈ 𝔉 and every compact subset C ⊂ W the subspace

N_{K,C} ⊂ hom_{𝔉}(V, W) = W^{V}

is compact. Examples are given in section 11 below.

(6.6) Remark. Since the spaces W in 𝔉 are assumed locally compact this implies that for a NKC category the function spaces W^{V} = hom_{𝔉}(V, W) are locally compact. Furthermore, it is not difficult to see that a closed subcategory of a NKC category is also a NKC subcategory.

(6.7) Example. Let 𝔉 denote the category consisting of a finite dimensional vector space V and morphisms given by elements in a closed subgroup G of GL(V). The fibre functor is the embedding F: 𝔉 → Top. Then, by (5.1), a 𝔉-complex X is a vector bundle with structure group G and fibre V. We will see in theorem (11.2) that (𝔉, F) has the NKC property. Therefore by theorem (6.12) the principal bundle X^{V} is a (𝔉, F^{V})-complex and hence a (𝔉, F)-bundle by (5.1). This is the result of Steenrod in (6.2).

(6.8) Example. Let G be a compact Lie group and 𝔉 its orbit category (see (3.3)). The fibre functor F: 𝔉 → Top is the embedding. Then a smooth Gmanifold X has a structure of a 𝔉-complex where X̄ = X/G and p_{X}: X → X̄ is the projection onto the orbit space. For every closed subgroup H ⊂ G the orbit V = G/H is an object of 𝔉 and the function space X^{V} is homeomorphic to the fixed subspace X^{H} ⊂ X via the evaluation map at 1 ∈ G. We will see as a consequence of proposition (11.1) that the orbit category 𝔉 with the embedding functor F: 𝔉 → Top has the NKC property. Hence by theorem (6.12) the space X^{H} is a 𝔉-complex with fibre functor F^{G/H}.

We now define for every (𝔉, F)-family Y the associated (𝔉, F^{V})-family Y^{V} as follows. Let Y^{V} denote the space of all 𝔉-maps V → Y with the compact-open topology. Then we obtain the projection

Y^{V} → Ȳ (6.9)

which sends a 𝔉-map a: V → Y to the point p_{Y} a(v_{0}) ∈ Ȳ, where v_{0} ∈ V is an arbitrary point of the fibre V. The pre-image of a point b ∈ Ȳ under this projection is equal to F hom(V, Y_{b}) = Y_{b}^{V}. Thus Y^{V} → Ȳ is a (𝔉, F^{V})-family.

We recall some properties of the function spaces X^{V} and W^{V} = hom_{𝔉}(V, W). By assumption (∗) the images under the fibre functor F of all the objects in 𝔉 are locally compact second-countable and Hausdorff. This implies that they are metrizable. Since every object V is metrizable, the automorphism group Aut(V) is a topological group with the compact-open topology induced by the inclusion Aut(V) ⊂ hom_{𝔉}(V, V).

Given a 𝔉-map f: X → Y, let f^{V} denote the map f^{V} : X^{V} → Y^{V} defined by composing with f the 𝔉-maps V → X. Moreover for φ: V → W in 𝔉 let X^{φ}: X^{W} → X^{V} be the map induced by φ. Let us recall that given a 𝔉-map f: Z × V → X the adjoint f̂ of f is the function f̂: Z → X^{V} defined by f̂(z)(v) = f(z, v) for every z ∈ Z and v ∈ V.

(6.10) Lemma. For every space Z a 𝔉-function f: Z×V → X is continuous if and only if the adjoint f̂: Z → X^{V} is continuous.

Proof. Because V is locally compact and Hausdorff, a function f: Z×V → X is continuous if and only if its adjoint f̂ : Z → Map(V, X) is continuous, where Map(V, X) denotes the space of all (not necessarily controlled by 𝔉) maps from V to X with the compact-open topology. But a function f: Z × V → X is a 𝔉-map if and only if its adjoint sends Z into the subspace X^{V} ⊂ Map(V, X). This implies the lemma. q.e.d.

(6.11) Lemma. The evaluation map X^{V} × V → X which sends (g, v) to g(v) is continuous. Moreover, if f: X → Y is continuous then the induced function f^{V} : X^{V} → Y^{V} is continuous. Also, given φ: V → W the induced map X^{φ}: X^{W} → X^{V} is continuous.

Proof. The evaluation map is the adjoint of the identity of X^{V}, hence continuous by lemma (6.10). The evaluation at v ∈ V is the restriction of the evaluation to the subspace X^{V} × {v} of X^{V} × V and hence continuous. The projection p: X^{V} → X̄ is the composition of the evaluation at any v_{0} ∈ V with the projection p_{X} and hence continuous. The induced function f^{V} is the adjoint of the composition X^{V} × V → X → Y where the first arrow is the evaluation and the second is f. To see that X^{φ} is continuous it is enough to see that it is the adjoint of the composition X^{W} × V → X^{W} × W → X, where the first arrow is 1 × φ while the second is the evaluation. q.e.d.

The next result is a crucial observation which shows that the function space X^{V} of a stratified bundle X is again a stratified bundle.

(6.12) Theorem. Let 𝔉 be a NKC category with fibre functor F. If X is a (𝔉, F)-stratified bundle and V an object in 𝔉 then the function space X^{V} is also a (𝔉, F^{V})-stratified bundle.

Proof. If X is a (𝔉, F)-complex then this follows from corollary (10.16) below. In the general case one can apply a finite number of times corollary (10.18) and lemma (9.4). q.e.d.

Let TopX̄ be the category of spaces over X̄; objects are maps X → X̄. Given a 𝔉-family X, the function spaces X^{V} for every object V in 𝔉 yield a functor X^{∘} which carries V to X^{∘}(V) = X^{V}. Moreover, given a 𝔉-map f: X → Y there is a natural transformation f^{∘} : X^{∘} → Y^{∘} defined by f^{V} : X^{V} → Y^{V}. Let 𝔉^{op}-DiagX̄ denote the diagram category in which the objects are functors 𝔉^{op} → TopX̄ and the morphisms are the natural transformations between functors. The operator (−)^{∘} sends a 𝔉-family X to the 𝔉^{op}-diagram X^{∘} and a 𝔉-map f to the natural transformation f^{∘}. Thus (−)^{∘} is a functor

(−)^{∘}: 𝔉-TopX̄ → 𝔉^{op}-DiagX̄.

The 𝔉-diagram X^{∘} is the generalization of the concept of principal bundle. In fact, if X → X̄ is a fibre bundle with structure group G and fibre V and 𝔉 = G → Top is given by the G-space V then the diagram X^{∘}: 𝔉^{op} → Top coincides with the G-space X_{G} given by the principal bundle X_{G} → X̄ associated to X, see (6.2).

For a topological enriched category 𝔉 there is a naturally associated functor ĥom: 𝔉 → 𝔉^{op}-Diag which sends an object W of 𝔉 to the 𝔉^{op}-diagram hom_{𝔉}(−, W) = W^{∘}. In case X is a 𝔉-stratified bundle and (𝔉, F) is a NKC structure category then by theorem (6.12) the function space X^{V} is not only a space over X̄ but a (𝔉, F^{V})-stratified bundle. Moreover this holds for every object V ∈ 𝔉 in a compatible way. Therefore the diagram X^{∘} given by stratified bundles X^{V} leads to the concept of a stratified bundle diagram. For this consider the definition of a stratified bundle in (4.4) and replace the category Top by the category of diagrams in Top. This yields a notion of a stratified bundle diagram. If the stratification is given by the skeletal filtration of the base space X̄ then a (𝔉, ĥom)-stratified bundle diagram is the same as a “free CW-complex” in the sense of Davis–Lück [6].

(6.13) Definition. If 𝔉 is a topologically enriched category, then a (𝔉, ĥom)- stratified bundle diagram in the category 𝔉^{op}-DiagX̄ is termed principal stratified bundle diagram. This is a 𝔉^{op}-diagram of stratified 𝔉-bundles over a stratified space X̄. For example X^{∘}: 𝔉^{op} → TopX̄ is such a principal stratified bundle diagram.

If 𝔉 has a single object V and hom_{𝔉}(V, V) is a topological group, then a (𝔉, ĥom)-stratified bundle diagram is nothing but a principal G-bundle with structure group G = hom_{𝔉}(V, V) over X̄, see theorem (5.1).

Now consider a principal stratified bundle diagram P: 𝔉^{op} → TopX̄ over X̄ and a fibre functor F: 𝔉 → Top. We can build a (𝔉, F)-stratified bundle X → X̄ in Top as the coend of the two functors P and F:

X = P ⊗_{𝔉} F = ∫ ^{V ∈𝔉} P (V) × F (V).

We recall the coend construction: the space P ⊗_{𝔉} F is the quotient space of the coproduct

∐_{V ∈𝔉}P (V) × F (V)

with the identification (P (α)(x), y) ∼ (x, F (α)(y)) for every morphism α ∈ hom_{𝔉}(V, W), x ∈ P (W) and y ∈ F (V). There is a well-defined map X → X̄ and it is not difficult to show:

(6.14) Lemma. P ⊗_{𝔉} F is a (𝔉, F)-family.

Proof. See e.g. [10], page 44, for the case of a groupoid 𝔉. It suffices to show it in the case P is a (𝔉, ĥom)-point, that is a diagram P over a point. Since P is principal, there is an object W of 𝔉 such that P (V) = hom_{𝔉}(V, W) for every V ∈ 𝔉, or, equivalently, P = W^{∘}. If g: V → V′ is a morphism in 𝔉, then P (g): W^{V′} → W^{V} is defined by P (g)(α) = αg. Thus the coend W^{∘} ⊗_{𝔉} F is defined as

(∐_{V ∈𝔉}W^{V} × F (V)) / ∼, (6.15)

where (αg, x) ∼ (g, F (α)(x)) for every g ∈ hom_{𝔉}(V, V′), α ∈ W^{V} and x ∈ F (V). Now it is easy to see that for every W

W^{∘} ⊗_{𝔉} F = F (W)

and that given a morphism α^{∘}: W^{∘} ⊗_{𝔉} F → W′^{∘} ⊗ F the following diagram commutes:

\xymatrix{W^{∘} ⊗_{𝔉} F \ar[r]^{α^{∘}} \ar[d]^{=} & W′^{∘} ⊗_{𝔉} F \ar[d]^{=} \\ F (W) \ar[r]^{F (α)} & F (W′)} (6.16)

q.e.d.

(6.17) Theorem. Let 𝔉 be a structure category with a faithful fibre functor F so that 𝔉 is topological enriched by the compact open topology. Let (𝔉, F) be a NKC structure category and X a (𝔉, F)-stratified bundle. Then the associated diagram X^{∘} is a principal stratified bundle diagram with fibre functor ĥom. Conversely, if P is a principal (𝔉, ĥom)-stratified bundle diagram and F a fibre functor, then the coend P ⊗_{𝔉} F is a stratified bundle in Top with structure category 𝔉 and fibre functor F. Moreover

X^{∘} ⊗_{𝔉} F = X.

This result generalizes the classical equation

X_{G} ×_{G} V = X

for the principal bundle X_{G} associated to X in (6.1). This construction can be generalized. Let φ: G → H be a continuous homomorphism between topological groups and let G (H) be a topological transformation group for V (W). Then a (G, V)-bundle X (i.e. a bundle with fibre V and structure group G) yields the principal bundle X_{G} which in turn yields the φ-associated bundle

φ_{#}(X) = X_{G} ×_{G} φ^{∗}W → X̄.

Here φ^{∗}W is the G-space with the G-action induced by φ, that is g · w = φ(g) · w for g ∈ G and w ∈ W. The associated bundle φ_{#}(X) has fibre W and structure group H. Hence φ induces the functor

(6.18) φ_{#} : (G, V)-BundlesX̄ → (H, W)-BundlesX̄.

We now generalize this functor for stratified bundles.

Let (𝔉, F) and (𝔊, G) be structure categories with faithful fibre functor F, G so that 𝔉 and 𝔊 have the compact open topology. Let

φ: 𝔉 → 𝔊

be a continuous functor. If (𝔉, F) is a NKC structure category and X a (𝔉, F)-stratified bundle then the principal stratified bundle diagram X^{∘} is defined and we obtain the φ-associated stratified bundle

φ_{#}(X) = X^{∘} ⊗_{𝔉} (Gφ) → X̄.

Here φ_{#}(X) is a (𝔊, G)-stratified bundle. Hence φ induces the functor

φ_{#} : (𝔉, F)-StraX̄ → (𝔊, G)-StraX̄.

Proof of theorem (6.17 ). We have seen that, as a consequence of theorem (6.12), for each V ∈ 𝔉 the function space X^{V} is obtained by gluing W^{V} - bundles M_{i}^{V} along with maps h_{i}^{V} : A_{i}^{V} ⊂ M_{i} → X_{i−1}^{V}, for suitable objects W ∈ 𝔉. In other words, the diagram X^{∘} is obtained by gluing the 𝔉-diagrams M_{i}^{∘} (which are (𝔉, ĥom)-bundles) along with 𝔉-diagram maps h_{i}^{∘}: A_{i}^{V} → X_{i−1}^{∘}. Thus X^{∘} is a (𝔉, ĥom)-stratified bundle, that is a 𝔉-stratified bundle with diagram-fibre functor ĥom. By definition this is a principal stratified bundle diagram.

Conversely, let P be a principal stratified bundle diagram and F a fibre functor. It is not assumed that F is faithful, but that the spaces F V are second-countable locally compact Hausdorff spaces, see assumption (∗). Hence, if the i-skeleton of P is obtained by attaching a (𝔉, ĥom)-bundle M̂_{i} to P_{i−1} via a 𝔉-diagram map ĥ_{i}: Â_{i} ⊂ M̂_{i} → P_{i−1}, i.e. P_{i} = M̂_{i} ∪ĥ_{i} P_{i−1}, the coends are corners of a push-out diagram in 𝔉-Top:

\xymatrix{Â_{i} ⊗_{𝔉} F \ar[r]^{ĥ_{i}×_{𝔉}F} \ar[d] & P_{i−1} ⊗_{𝔉} F \ar[d] \\ M̂_{i} ⊗_{𝔉} F \ar[r] & P_{i} ⊗_{𝔉} F}

Thus X has a filtration given by X_{i} = P_{i} ⊗_{𝔉} F, for all i ≥ 0, and each X_{i} is obtained by attaching M_{i} = M̂_{i} ⊗_{𝔉} F to X_{i−1} via the map

h_{i} = ĥ_{i} ⊗_{𝔉} F: A_{i} = Â_{i} ⊗_{𝔉} F → X_{i−1}.

Moreover, since

lim_{i≥0} (P_{i} ⊗_{𝔉} F) = (lim_{i≥0} P_{i}) ⊗_{𝔉} F,

X is equal to the colimit X = lim_{i≥0} X_{i}. To show that it is a (𝔉, F)-stratified bundle we need only to show that the attaching maps are 𝔉-maps. But ĥ_{i} are (𝔉, ĥom)-maps, and by diagram (6.16) the coends of such maps are (𝔉, F)-maps. The proof is hence complete. q.e.d.

7 Pull-back of a push-out 𝔉-family

In this section we prove the pull-back theorem (4.7).

Consider a 𝔉-family Y′, a 𝔉-CW-pair (M′, A′) and a 𝔉-map h′: A′ → Y′. Then we can glue M′ to Y′ and obtain a 𝔉-family X′, as in the following diagram.

\xymatrix{A′ \ar[rrr]^{h′} \ar[ddd]^{j_{A′}} \ar[dr]^{p_{A′}} & & & Y′ \ar[ddd]^{j_{Y′}} \ar[dl]^{p_{Y′}} \\ & Ā′ \ar[r]^{h̄′} \ar[d]^{jĀ′} & Ȳ′ \ar[d]^{jȲ′} \\ & M̄′ \ar[r]^{Φ̄′} & X̄′ \\ M′ \ar[rrr]^{Φ′} \ar[ur]^{p_{M′}} & & & X′ \ar[ul]^{p_{X′}}} (7.1)

Now consider a space Ȳ, a CW-pair (M̄, Ā) and a map h̄: Ā → Ȳ. Let X̄ be the push-out space X̄ = Ȳ ∪h̄ M̄. Assume that maps f̃_{A}, f̃, f̄Ȳ and f̄ are given, such that the following diagram commutes.

\xymatrix{Ā′ \ar[rrr]^{h̄′} \ar[ddd]^{j_{A′}} & & & Ȳ′ \ar[ddd]^{jȲ′} \\ & Ā \ar[ul]^{f̃_{A}} \ar[r]^{h̄} \ar[d]^{jĀ} & Ȳ \ar[d]^{jȲ} \ar[ur]^{f̄Ȳ} \\ & M̄ \ar[r]^{Φ̄} \ar[dl]^{f̃} & X̄ \ar[dr]^{f̄} \\ M̄′ \ar[rrr]_{Φ̄′} & & & X̄′} (7.2)

Then the pull-back families X = f̄^{∗}X′, Y = f̄ ∗ ; Ȳ Y′, A = f̃_{A}^{∗}A′ and M = f̃^{∗}M′ are defined. Moreover, we can define the maps h = (h̄, h′): A → Y, j_{Y} = (jȲ , j_{Y′}): Y → X, j_{A} = (jĀ, j_{A′}): A → M and Φ = (Φ̄, Φ′): M → X. They fit into the following diagram.

[figure: Ā × A′; A; h; j_{A}; Y; j_{Y}; Ȳ × Y′; M̄ × M′; M; Φ; X; X̄ × X′.] (7.3)

The aim of this section is to prove the following proposition.

(7.4) Proposition. Assume that M̄ and M̄′ are locally finite and countable CW-complexes and that f̄(X̄ ∖ Ȳ) ⊂ X̄′ ∖ Ȳ′. Then the pull-back family X = f̄^{∗}X′ is obtained by attaching the pull-back 𝔉-family M = f̃^{∗}M′ to Y = f̄ ∗ ; Ȳ Y′ via the 𝔉-map h: A ⊂ M → Y induced by (h̄, h′), i.e. the middle square in diagram (7.3 ) is a push-out.

The proof of proposition (7.4) will be the content of the rest of the section.

(7.5) Lemma. The maps in (7.3 ) are well-defined and the diagram commutes.

Proof. Consider (ā, a′) ∈ A. Then f̃_{A}(ā) = p_{A′}(a′), hence

f̄Ȳ h̄(ā) = h̄′f̃_{A}(ā) = h̄′p_{A′}(a′) = p_{Y′}h′(a′).

Hence (h̄(ā), h′(a′)) ∈ Y. On the other hand, if (ȳ, y′) ∈ Y, then f̄Ȳ (ȳ) = p_{Y′}(y′), hence

f̄jȲ (ȳ) = jȲ′ f̄Ȳ (ȳ) = jȲ′p_{Y′}(y′) = p_{X′}j_{Y′}(y′).

That is, j_{Y} (ȳ, y′) = (jȲ (ȳ), j_{Y′}(y′)) ∈ X. The same argument can be applied literally to j_{A} and Φ. The diagram commutes since it is the restriction of a commutative diagram on the Cartesian products. q.e.d.

(7.6) Lemma. The maps j_{A}: A → M and j_{Y} : Y → X are a closed inclusions of 𝔉-families.

Proof. Consider the following diagram. The left square is a pull-back (by definition) and the right square is a pull-back (the pair (M′, A′) is a 𝔉-CW-pair).

\xymatrix{A \ar[r] \ar[d] & A′ \ar[r] \ar[d] & M′ \ar[d] \\ Ā \ar[r] & Ā′ \ar[r] & M̄′} (7.7)

Now consider this diagram.

\xymatrix{A \ar[r] \ar[d] & M \ar[r] \ar[d] & M′ \ar[d] \\ Ā \ar[r] & M̄ \ar[r] & M̄′}

The composition of the two squares is equal to the composition of the two squares in diagram (7.7), hence it is a pull-back. The right square is a pullback by definition, hence the left square is a pull-back. (see e.g. exercise 8 of [12], page 72). Since Ā ⊂ M̄ is a closed inclusion, j_{A} is a closed inclusion. The proof is the same for j_{Y}. q.e.d.

Now consider the maps q = Φ ⊔ j_{Y}, q̄ = Φ̄ ⊔ jȲ, q̄′ = Φ̄′ ⊔ jȲ′ and q′ = Φ′ ⊔ j_{Y′}. They can be arranged in the following diagram.

\xymatrix{M ⊔ Y \ar[rrr] \ar[ddd]^{q} \ar[dr] & & & M′ ⊔ Y′ \ar[ddd]^{q′} \ar[dl] \\ & M̄ ⊔ Ȳ \ar[r] \ar[d]^{q̄} & M̄′ ⊔ Ȳ′ \ar[d]^{q̄′} \\ & X̄ \ar[r] & X̄′ \\ X \ar[rrr] \ar[ur] & & & X′ \ar[ul]} (7.8)

By definition q̄, q̄′ and q̄′ are quotient maps. We want to show that under suitable conditions q is a quotient map. If this is the case, then proposition (7.4) is proved, since q is exactly the projection defining the topology of the push-out.

(7.9) Lemma. Assume that f̄(X̄ ∖ Ȳ) ⊂ X̄′ ∖ Ȳ′. Then q is onto, q|Y and q|(M ∖ A) are mono.

Proof. Let x = (x̄, x′) ∈ X, so that then f̄(x̄) = p_{X′}(x′). If x̄ ∈ Ȳ, then f̄(x̄) ∈ Ȳ′, therefore x′ ∈ p_{X′}^{−1}Ȳ′ = Y′. Thus there is y = (ȳ, y′) ∈ Y such that q(y) = x. On the other hand, if x̄ ∉ Ȳ, since assumption f̄x̄ ∉ Ȳ′, necessarily x′ ∉ Y′ hence there is a unique m′ ∈ M′ ∖ A′ such that q′(m′) = x′. For the same reason there is a unique m̄ ∈ M̄ ∖ Ā such that q̄(m̄) = x̄. Now consider the since q̄′ is mono in M̄′ ∖ A′, the chain of equalities

q̄′f̃(m̄) = f̄q̄(m̄) = f̄(x̄) = p_{X′}(x′) = p_{X′}q′(m′) = q̄′p_{M′}(m′)

implies that f̃(m̄) = p_{M′}(m′), hence that (m̄, m′) ∈ M. We have shown that q is onto. The restriction of q to Y is mono since jȲ × j_{Y′} is mono. To see that the restriction of q to M ∖ A is mono, consider that if (m̄, m′) ∈ M then m̄ ∉ Ā implies m′ ∉ A′, hence M ∖ A ⊂ (M̄ ∖ Ā) × (M′ ∖ A′). Now, q̄ and q′ are mono when restricted to M̄ ∖ Ā and M′ ∖ A′ respectively, hence q|(M ∖ A) is mono. q.e.d.

From now on, we will assume that the condition of lemma (7.9) is fulfilled.

(7.10) Lemma. The restriction q|(M ∖ A): M ∖ A → X ∖ Y is a homeomorphism.

Proof. The map q|(M ∖A) is bijective and continuous. We need to show that it is open. Consider an open set U in M ∖ A, and a point x = (x̄, x′) ∈ U. There are neighborhood UM̄ ⊂ M̄ ∖ Ā and U_{M′} ⊂ M′ ∖ A′ such that

x ∈ (UM̄ × U_{M′}) ∩ M ⊂ U.

Since q̄ and q′ are homeomorphisms of M̄ ∖Ā and M′ ∖A′ onto their images, the sets UX̄ = q̄UM̄ and U_{X′} = q′U_{M′} are open subsets of X̄ ∖ Ȳ and X′ ∖ Y′. Hence UX̄ ×U_{X′} ∩X is open in X and contained in X ∖Y. Moreover, since q is mono in M ∖ A, we have that q [(UM̄ × U_{M′}) ∩ M] = UX̄ × U_{X′} ∩ X, hence UX̄ × U_{X′} ∩ X ⊂ q(U). But x is arbitrary, therefore q(U) is open and so the map q|M ∖ A is open. q.e.d.

(7.11) Lemma. A subset S ⊂ M ⊔ Y is saturated (i.e. q^{−1}q(S) = S ) if and only if S ∩ A = h^{−1}(S ∩ Y).

(7.12) Lemma. Assume that M̄ and M′ are metrizable. If UM̄ ⊂ M̄ and U_{A′} ⊂ A′ are open sets with compact closure such that

(\overline{UM̄} × \overline{U_{A′}}) ∩ M ⊂ q^{−1}U,

then there is an open subset U_{M′} ⊂ M′ with compact closure such that

U_{M′} ∩ A′ = U_{A′}
(\overline{UM̄} × \overline{U_{M′}}) ∩ M ⊂ q^{−1}U.

In the same way, if UĀ ⊂ Ā and U_{M′} ⊂ M′ are open sets with compact closure such that

(\overline{UĀ} × \overline{U_{M′}}) ∩ M ⊂ q^{−1}U,

then there is an open subset UM̄ ⊂ M̄ with compact closure such that

UM̄ ∩ Ā = UĀ
(\overline{UM̄} × \overline{U_{M′}}) ∩ M ⊂ q^{−1}U.

Proof. By assumption the set q^{−1}U ∩ M is open in M ⊂ M̄ × M′, therefore there exists an open set { \atop q^{−1}U} ⊂ M̄ × M′ such that { \atop q^{−1}U} ∩ M = q^{−1}U ∩ M. We need to find the open set U_{M′} ⊂ M′ with the desired properties. Consider the following function η: M̄ × M′ → ℝ defined by

η(x) = d(x, { \atop q^{−1}U}^{c}) + d(x, M), (7.13)

where { \atop q^{−1}U}^{c} is the complement of { \atop q^{−1}U} in M̄ × M′ and d is the distance given by a metric on M̄ × M′. Consider a point x in the compact closure \overline{UM̄} × \overline{U_{A′}}. If x ∈ M as well, then by the assumption x ∈ { \atop q^{−1}U}, and since { \atop q^{−1}U} is open this implies η(x) ≥ d(x, { \atop q^{−1}U}^{c}) > 0. On the other hand, since M is closed in M̄ ×M′, if x ∉ M then η(x) ≥ d(x, M) > 0. Thus η(x) > 0 for every x ∈ (\overline{UM̄} × \overline{U_{A′}}), and so there is an open set U_{M′} ⊂ M′ with compact closure such that U_{M′} ∩ A′ = U_{A′} and

x ∈ \overline{UM̄} × \overline{U_{M′}} ⟹ η(x) > 0.

This means that if x ∈ (\overline{UM̄} ×\overline{U_{M′}})∩M then d(x, { \atop q^{−1}U}^{c}) > 0, thus x ∈ { \atop q^{−1}U}, and this implies that

(\overline{UM̄} × \overline{U_{M′}}) ∩ M ⊂ q^{−1}U.

For the second part of the lemma, the proof is exactly the same, it is only necessary to exchange the roles of Ā and M′. q.e.d.

(7.14) Lemma. Let X be a 𝔉-bundle over a locally finite and countable CWcomplex X̄. Then X is metrizable. Moreover, every open subset O ⊂ X is the union of an ascending sequence of open subsets O_{n} ⊂ X with compact closure \overline{O_{n}} ⊂ O_{n+1}.

Proof. Since X̄ is locally finite and countable, it is metrizable and 2^{∘} countable. Since the fibres are 2^{∘} countable by assumption, X is 2^{∘} countable, since it is obtained by attaching a countable number of 𝔉-cells V × D^{n}, which are 2^{∘} countable. Now, X̄ and the 𝔉-cells V × D^{n} are completely regular, hence X is regular; therefore by Urysohn metrization theorem X is metrizable. Let d denote its metric. Now consider an ascending chain of finite subcomplexes X̄_{n} ⊂ X̄ such that X̄_{n} is contained in the interior of X̄_{n+1} and X̄ = ⋃_{n} X̄_{n} (such a sequence exists since X̄ is locally finite and countable). Using the fact that the fibres are locally compact and second countable, it is possible to show that X can be written as the union of an ascending sequence of subspaces X_{n} ⊂ X_{n+1} ⊂ … where each X_{n} is compact and is contained in the interior X̊_{n+1}. For every n define the following open set

O_{n} = \{x ∈ X : d(x, O^{c}) > \frac{1}{n}\} ⋂ X̊_{n},

where O^{c} denotes the complement of O. If x is an element of the closure \overline{O_{n}} (closure of O_{n} in X), then x ∈ O_{n+1} ⋂ X_{n} ⊂ X_{n}, hence \overline{O_{n}} is a closed subset of the compact space X_{n}, hence \overline{O_{n}} is compact. Moreover, since X_{n} ⊂ X̊_{n+1}, we have that \overline{O_{n}} ⊂ O_{n+1} as claimed. We need to show that O = ⋃_{n} O_{n}: if x ∈ O, then d(x, O^{c}) > 0, therefore there is n_{1} ≥ 1 such that d(x, O^{c}) > \frac{1}{n} for every n ≥ n_{1}. Since ⋃_{n} X_{n} = X, there is n_{2} ≥ n_{1} such that x ∈ X_{n_{2}} ⊂ X̊_{n_{2}+1}. But these conditions imply that x ∈ O_{n} for n = n_{2} + 1. Thus O = ⋃ O_{n}. q.e.d.

(7.15) Lemma. Assume that M̄ and M′ are metrizable. If UĀ ⊂ Ā and U_{A′} ⊂ A′ are open sets such that

(UĀ × U_{A′}) ∩ M ⊂ q^{−1}U,

then there are open sets UM̄ ⊂ M̄ and U_{M′} ⊂ M′ such that

UM̄ ∩ Ā = UĀ
U_{M′} ∩ A′ = U_{A′}
(UM̄ × U_{M′}) ∩ M ⊂ q^{−1}U.

Proof. By lemma (7.14), it is possible to find increasing sequences U k ; Ā ⊂ UĀ and U_{A′}^{k} ⊂ U_{A′} with compact closures and such that

⋃_{k≥1} U k ; Ā = UĀ

⋃_{k≥1} U_{A′}^{k} = U_{A′}.

Now, by applying lemma (7.12) twice it is possible to define compact subsets U 1 ; M̄ and U_{M′}^{1} such that

(\overline{U 1 ; M̄} × \overline{U_{M′}^{1}}) ∩ M ⊂ q^{−1}U.

By induction, we will show that it is possible to find two sequences of open sets U k ; M̄ and U_{M′}^{k} with compact closures, with the property that

U k ; M̄ ∩ Ā = U k ; Ā

U_{M′}^{k} ∩ A′ = U_{A′}^{k}

for every k, k′ ≥ 1 and

(\overline{U k ; M̄} × \overline{U_{M′}^{k′}}) ∩ M ⊂ q^{−1}U (7.16)

for every k, k′ ≥ 1. Assume that the sequences are defined for j ≤ k − 1. By applying k−1 times lemma (7.12) and taking the intersection of the resulting open sets, we can show that there is U k ; M̄ such that U k ; M̄ ∩ Ā = U k ; Ā and

(\overline{U k ; M̄} × \overline{U_{M′}^{k′}}) ∩ M ⊂ q^{−1}U

for every k′ ≤ k − 1. Now we can apply k times lemma (7.12) and take the intersection of the resulting open sets, to finally find and open set with compact closure U_{M′}^{k} such that U_{M′}^{k} ∩ A′ = U_{A′}^{k} and

(\overline{U j ; M̄} × \overline{U_{M′}^{k}}) ∩ M ⊂ q^{−1}U

for every j ≤ k. Thus equation (7.16) holds for every k, k′ ≥ 1. But this implies that the open sets

UM̄ = ⋃_{k≥1} U k ; M̄

U_{M′} = ⋃_{k≥1} U_{M′}^{k}

have the desired property. q.e.d.

(7.17) Lemma. If M̄ and M̄′ are locally finite and countable, then the map q is a quotient map.

Proof. Consider a subset U ⊂ X such that q^{−1}U is open in M ⊔ Y. We want to show that U is open. Let x ∈ U ∖ Y. Since q^{−1}U ∩ (M ∖ A) is open, by lemma (7.10) there is an open neighborhood of x in X ∖ Y contained in U.

If U ∩ Y = ∅ then we have proved that U is open. Otherwise, let x ∈ Y ∩ U. Since q^{−1}U is open, q^{−1}U ∩Y is open and contains y ∈ q^{−1}(x)∩Y. Therefore there are two open sets UȲ and U_{Y′} such that

y ∈ (UȲ × U_{Y′}) ∩ Y ⊂ q^{−1}Y ∩ Y.

Let

UĀ = h̄^{−1}UȲ
U_{A′} = h′^{−1}U_{Y′}

Since h̄ and h′ are continuous, they are open subsets of Ā and A′ respectively. By lemma (7.15) there are open sets UM̄ ⊂ M̄ and U_{M′} ⊂ M′ such that

UM̄ ∩ Ā = UĀ = h̄^{−1}UȲ
U_{M′} ∩ A′ = U_{A′} = h′^{−1}U_{Y′},

and

(UM̄ × U_{M′}) ∩ M ⊂ q^{−1}U.

Consider the sets

UX̄ = q̄(UM̄) ∪ UȲ ⊂ X̄
U_{X′} = q′(U_{M′}) ∪ U_{Y′} ⊂ X′.

Since q̄ and q′ are quotient maps (by definition), and since UM̄ ∪ UȲ and U_{M′} ∪U_{Y′} are saturated with respect to the maps q̄ and q′ (see lemma (7.11)), UX̄ and U_{X′} are open in X̄ and X′. Therefore U_{x} = UX̄ × U_{X′} ∩ X is open in X and contains x. It is left to show that U_{x} ⊂ U. Consider t ∈ U_{x} ∩ Y. Then q^{−1}t = h^{−1}t ⊔ {t} ⊂ M ⊔ Y. But since j_{Y} is mono, t ∈ (UȲ × U ′Y) ∩ Y and therefore t ∈ U. Otherwise, if t ∈ X ∖ Y then q^{−1}t = m, unique point belonging to (UM̄ × U_{M′}) ∩ M ⊂ q^{−1}U. Thus q^{−1}t ⊂ q^{−1}U and therefore t ∈ U. We have proved that U_{x} ⊂ U and the proof is now complete. q.e.d.

(7.18) Remark. Since a pull-back of a 𝔉-bundle is a 𝔉-bundle, if M′ is a 𝔉-bundle, then M is a 𝔉-bundle. Furthermore, by (7.6), A is the restriction of M to Ā. Thus, in this case, proposition (7.4) implies that if M̄ and M̄′ are locally finite countable CW-complexes and M′ a 𝔉-bundle, if X′ is obtained by attaching the 𝔉-bundle M′ to a 𝔉-family Y′, then the pull-back 𝔉-family f̄^{∗}X′ is obtained by attaching the pull-back 𝔉-bundle M to the pull-back 𝔉-family f̃^{∗}Y′.

8 Push-out of pull-back 𝔉-bundles

In this section we prove the bundle theorem (5.1). In case of the pull-back of 𝔉-bundles, the following properties hold.

(8.1) Lemma. Let Ȳ be a space, h̄: S^{n−1} → Ȳ a map and X̄ = Ȳ ∪h̄ D^{n}. Let Φ̄: D^{n} → X̄ denote the characteristic map. If X → X̄ is a 𝔉-bundle and Y is the restriction of X to Ȳ, then the following diagram is a push-out

\xymatrix{h̄^{∗}Y \ar[d] \ar[r] & Y \ar[d] \\ Φ̄^{∗}X \ar[r] & X} (8.2)

Proof. Since the diagram commutes, one gets the map t: P → X, where P denotes the push-out space Φ̄^{∗}X ∪ Y. It is easy to see that t is bijective and covers the identity of X̄. Now we show that t is a local homeomorphism when restricted to the space over Ū, where Ū ⊂ X̄ is an open set such that X|Ū is trivial. In fact, in this case the diagram (8.2) is reduced as

\xymatrix{h̄^{∗}(W × (Ū ∩ Ȳ)) \ar[d] \ar[r] & W × (Ū ∩ Ȳ) \ar[d] \\ W × Φ̄^{−1}Ū \ar[r] & W × Ū,}

which is a push-out since W is locally compact Hausdorff and X̄ = Ȳ ∪h̄ D^{n}. Thus P |Ū and X|Ū are homeomorphic. But P |Ū and X|Ū, when Ū ranges over all the trivializing neighborhoods of X̄, are open covers of P and X respectively, with t(P |Ū) = X|Ū for each Ū. Hence t is an open map. q.e.d.

Since exactly the same argument can be applied to the attaching of a set Z̄ of n-cells, we have the following generalization of lemma (8.1).

(8.3) Lemma. Let Ȳ be a space, Z̄ a set and h̄: Z̄ ×S^{n−1} → Ȳ an attaching map with X̄ = Ȳ ∪h̄ (Z̄ × D^{n}). Let Φ̄: Z̄ × D^{n} → X̄ denote the characteristic map. If X → X̄ is a 𝔉-bundle and Y is the restriction of X to Ȳ then the following diagram is a push-out

\xymatrix{h̄^{∗}Y \ar[d] \ar[r] & Y \ar[d] \\ Φ̄^{∗}X \ar[r] & X.}

We recall now the following important property. Let M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{n} ⊂ lim_{n≥0} M_{n} be a sequence of spaces, and M = lim_{n→∞} M_{n}. Then for every locally compact Hausdorff space W

lim_{n→∞} (W × M_{n}) = W × ( lim_{n→∞} M_{n}) = W × M. (8.4)

(8.5) Lemma. Let (M̄, Ā) be a CW-pair; let M → M̄ be a 𝔉-bundle and A the restriction of M to Ā. Then (M, A) is a 𝔉-CW-pair.

Proof. Consider the filtration on skeleta

Ā ⊂ M̄_{1} ⊂ M̄_{2} ⊂ ⋯ ⊂ M̄_{n} ⊂ ⋯ ⊂ M̄.

It induces a filtration on restrictions of the bundles M → M̄

A ⊂ M_{1} ⊂ M_{2} ⊂ ⋯ ⊂ M_{n} ⊂ ⋯ ⊂ M.

By lemma (8.3) for every n ≥ 0 the diagram

\xymatrix{h̄_{n}^{∗}M_{n−1} \ar[d] \ar[r] & M_{n−1} \ar[d] \\ Φ̄_{n}^{∗}M_{n} \ar[r] & M_{n},}

is a push-out, where h̄_{n}: Z̄_{n} × S^{n−1} → M̄_{n−1} is the attaching map on M̄_{n−1} and Φ̄_{n} is the corresponding characteristic map. Since D^{n} is contractible, there exists a 𝔉-isomorphism of (pair of ) bundles

(Φ̄_{n}^{∗}M_{n}, h̄_{n}^{∗}M_{n−1}) ≅ W × (Z̄_{n} × D^{n}, Z̄_{n} × S^{n−1}) (8.6)

for a suitable object W ∈ 𝔉. This means that M_{n} is obtained by attaching W × Z̄_{n} × D^{n} to M_{n−1} with as attaching map the composition of the map induced by pull-back h̄^{∗} and the isomorphism of (8.6). Thus (M_{n}, A) there are skeleta M_{n} ⊂ M such that

A ⊂ M_{0} ⊂ M_{1} ⊂ … M.

To show that (M, A) is a relative 𝔉-complex (and hence a 𝔉-CW-pair) we need to show that lim_{n→∞} M_{n} = M. Since there is a continuous bijection

t: lim_{n→∞} M_{n} → M covering the identity of M̄, we need to show that M has the topology of the limit. It suffices to restrict M to the trivializing neighborhoods Ū ⊂ M̄, and this is true as a consequence of (8.4). q.e.d.

Lemma (8.5) implies the following corollary, by taking A = ∅. Let M̄ be a CW-complex; let M → M̄ be a 𝔉-bundle. Then M is a 𝔉-complex. This is the second part of (5.1).

9 Push-out of function spaces

We now start the proof of the principal bundle theorem which is achieved in the next two sections.

Let V be an object of 𝔉. We recall that a family K of compact sets of V is termed generating if for every 𝔉-family Y the subsets N_{K,U} with K ∈ K and U open in Y yield a sub-basis for the topology of Y^{V}.

(9.1) Definition. Let Y ⊂ X a pair of 𝔉-families and a closed inclusion. We say that Y has the N-neighborhood extension property in X (NNEP) if for every V ∈ 𝔉 there exists a generating family K of compact subsets of V such that the following is true: let U be open in X^{V} and for i = 1, …, l let K_{i} ⊂ V be compact sets in K and U_{i} ⊂ Y be open such that

⋂_{i=1}^{l} N_{K_{i},U_{i}} ⊂ U ∩ Y^{V};

then there are l open subsets U′_{i} ⊂ X such that U′_{i} ∩ Y = U_{i} and

⋂_{i=1}^{l} N_{K_{i},U′_{i}} ⊂ U.

An important easy property of the sets N_{K,U} is the following: if U ⊂ Y and h: A → Y is a map, then

N_{K,h^{−1}U} = (h^{V})^{−1}N_{K,U}.

(9.2) Lemma. Let Z be a set and (M_{z}, A_{z}) be an NNEP-pair for every z ∈ Z. Then the coproduct of inclusions

∐_{z∈Z} A_{z} ⊂ ∐_{z∈Z} M_{z}

yields a NNEP-pair.

(9.3) Lemma. Assume that A has the NNEP in M and that h: A → Y is a 𝔉-map to a 𝔉-family Y. Then Y has the NNEP in the push-out space X = M ∪_{h} Y.

Proof. By lemma (2.5) Y → X is a closed inclusion. Let Φ: M → X denote the characteristic map of the push-out. Let U ⊂ X^{V} be open, and let K_{i} ⊂ V and U_{i} ⊂ Y be compact subsets of a generating family K of V and open subsets such that

⋂_{i=1}^{l} N_{K_{i},U_{i}} ⊂ U ∩ Y^{V}

as in definition (9.1). The map Φ^{V} is continuous, hence U′′ = (Φ^{V})^{−1}U ⊂ M^{V} is open in M^{V}. Moreover, the intersection

(h^{V})^{−1} ⋂_{i=1}^{l} N_{K_{i},U_{i}} = ⋂_{i=1}^{l} N_{K_{i},h^{−1}U_{i}}

is contained in U′′. Since A has the NNEP in X, there are l open sets U′′_{i} in M such that U′′_{i} ∩ A = h^{−1}U_{i} and

⋂_{i=1}^{l} N_{K_{i},U′′_{i}} ⊂ U′′.

Now consider the subsets U′_{i} = U_{i} ⋃ Φ(U′′_{i}) ⊂ X. Since U′_{i} ∩Y = U_{i}, Φ^{−1}U′_{i} = U′′_{i} and X is the push-out of M and Y, each U′_{i} is open in X. Moreover, consider

f ∈ ⋂_{i=1}^{l} N_{K_{i},U′_{i}}.

If f ∈ Y^{V} then

f ∈ ⋂_{i=1}^{l} N_{K_{i},U′_{i}} ∩ Y^{V} = ⋂_{i=1}^{l} N_{K_{i},U′_{i}∩Y} = ⋂_{i=1}^{l} N_{K_{i},U_{i}} ⊂ U ∩ Y^{V} ,

hence f ∈ U. On the other hand, if f ∉ Y^{V}, then there is a unique f′′: K → M ∖ A such that f = Φ^{V} (f′′). Since for every i fK_{i} = hf′′K_{i} ⊂ U′_{i}, we have

f′′K_{i} ⊂ Φ^{−1}U′_{i} = U′′_{i},

thus

f′′ ∈ ⋂_{i=1}^{l} N_{K_{i},U′′_{i}} ⊂ U′′.

Hence f = Φ^{V} (f′′) ∈ Φ^{V} U′′ = Φ^{V} (Φ^{V})^{−1}U = U. Therefore

⋂_{i=1}^{l} N_{K_{i},U′_{i}} ⊂ U,

and the proof is complete. q.e.d.

(9.4) Lemma. Assume that (M, A) is a NNEP-pair, Y is a 𝔉-family and h: A → Y is a 𝔉-map. Let X be the push-out X = M ∪_{h} Y. Then the following diagram

[figure: A^{V}; h^{V}; Y^{V}; M^{V}; Φ^{V}; X^{V}]

is a push-out.

Proof. By lemma (9.3) (X, Y) is a NNEP-pair. Let K denote the generating family of compact sets of V. Let P denote the push-out M^{V} ∪_{h^{V}} Y^{V}. By the push-out property there is a continuous map P → X^{V}. It is easy to see that it is bijective, so that, by identifying P and X^{V}, the only thing to prove is that if U is an open in the push-out topology, then it is open in the compact-open topology. This is true if and only if for every f_{0} ∈ U there is a subset U′ ⊂ X^{V} open in the compact-open topology such that f_{0} ⊂ U′ ⊂ U.

Consider first the case f_{0} ∉ Y^{V}. Then there exists a unique f′_{0} ∈ M^{V} ∖A^{V} such that Φ^{V} (f′_{0}) = f_{0}. Since A^{V} is closed in M^{V}, f′_{0} is contained in an open neighborhood U′′_{0} ⊂ M^{V} such that U′′_{0} ∩ A^{V} = ∅ and U′′_{0} is of type

⋂_{i=1}^{l} N_{K_{i},U′′_{i}}

for some K_{i} ⊂ V and some U′′_{i} in M ∖ A. For each i the image U′_{i} = ΦU′′_{i} is open in X, hence the set

⋂_{i=1}^{l} N_{K_{i},U′_{i}}

is open in X^{V}, contains f_{0} and is contained in U.

Now consider f_{0} ∈ Y^{V}. The intersection Y^{V} ∩ U is open in Y^{V} and hence there is a neighborhood of f_{0} in Y^{V} contained in Y^{V} ∩U. Because Y^{V} has the compact-open topology this means that there are l compact subsets K_{i} ⊂ V and open sets U_{i} ⊂ Y such that

f_{0} ∈ ⋂_{i=1}^{l} N_{K_{i},U_{i}} ⊂ U, (9.5)

where as above N_{K_{i},U_{i}} denotes the set of all the maps in Y^{V} such that fK_{i} ⊂ U_{i}. Without loss of generality we can assume that K_{i} belongs to the generating family K for every i = 1 … l.

Consider the sets

U′′ = (Φ^{V})^{−1}U ⊂ M^{V}

and

⋂_{i=1}^{l} N_{K_{i},h^{−1}U_{i}} ⊂ A^{V} ∩ U′′.

Since (M, A) is a NNEP-pair, by definition (9.1) there are open sets U′′_{i} ⊂ M such that U′′_{i} ∩ A = h^{−1}U_{i} and

⋂_{i=1}^{l} N_{K_{i},U′′_{i}} ⊂ U′′.

Let U′_{i} = U_{i} ⋃ ΦU′′_{i} for every i = 1, …, l. As in the proof of lemma (9.3), they are l open subsets of X with the property that U′_{i} ∩Y = U_{i}, hence f_{0} ∈ N_{K_{i},U′_{i}} for every i. And, again as in the proof of lemma (9.3),

U′ = ⋂_{i=1}^{l} N_{K_{i},U′_{i}} ⊂ U.

This concludes the proof. q.e.d.

(9.6) Lemma. Consider a sequence of 𝔉-families

X_{0} ⊂ X_{1} ⊂ ⋯ ⊂ X_{n} ⊂ ⋯ ⊂ X = lim_{n→∞} X_{n}

such that for every n ≥ 1 (X_{n}, X_{n−1}) is a NNEP-pair. Then (X, X_{0}) is a NNEP-pair and

X^{V} = lim_{n→∞} X_{n}^{V}

Proof. Let U ⊂ X^{V} be open, and ⋂_{i=1}^{l} N_{K_{i},U_{i}^{0}} ⊂ U ∩ X_{0}^{V} a subset of X_{0}^{V}, with K_{i} ⊂ V compact containing bases and U_{i}^{0} open in X_{0}. By induction, it is possible to define sequences of sets U_{i}^{n} open in X_{n} such that for every ≥ 1

U_{i}^{n} ∩ X_{n−1} = U_{i}^{n−1}

⋂_{i=1}^{l} N_{K_{i},U_{i}^{n}} ⊂ X_{n}^{V} ∩ U.

Now, because X = lim_{n→∞} X_{n}, the sets U_{i}^{∞} = ⋃_{n=0}^{∞} U_{i}^{n} are open in X, hence

⋂_{i=1}^{l} N_{K_{i},U_{i}^{∞}} ⊂ U.

is an open set. Moreover, by construction U_{i}^{∞}∩X_{0} = U_{i}^{0}, therefore X_{0} has the NNEP in X. The same construction shows that X^{V} = lim_{n→∞} X_{n}^{V}. q.e.d.

10 NKC-categories

The crucial property of a NKC-category as defined in (6.5) is the next result.

(10.1) Proposition. Assume that 𝔉 is NKC. Then for every W ∈ 𝔉 the pair (W × D^{n}, W × S^{n−1}) is a NNEP-pair.

The proof of proposition (10.1) will take the rest of the section.

(10.2) Lemma. Let R be a space and V and W objects of 𝔉. Then

(W × R)^{V} = W^{V} × R.

Proof. Let pr_{1} and pr_{2} denote the projections onto the first and the second factor of W × R. The map pr_{2}ϵ_{0} : (W × R)^{V} → R sending f: V → W × R to pr_{2}f(0) ∈ R is continuous. Furthermore, the map pr_{1}^{V} : (W × R)^{V} → W^{V} is continuous. Therefore the map F = (pr_{1}^{V} , pr_{2}ϵ_{0}): (W × R)^{V} → W^{V} × R is continuous. Consider now the function G: W^{V} × R → (W × R)^{V} defined by G(f, r)(v) = (f(v), r) for every v ∈ V and every r ∈ R. It is the adjoint of the map V × W^{V} × R → W × R defined by (v, f, r) ↦ (f(v), r), which is continuous since the evaluation map is continuous. Therefore G is continuous. It is readily seen that GF = 1 and F G = 1. q.e.d.

Let D_{ϵ}^{n} = {x ∈ D^{n} : |x| > 1 − ϵ}, for any ϵ small, and let ρ_{ϵ}: D_{ϵ}^{n} → S^{n−1} be the retraction defined by x ↦ \frac{x}{|x|}. With an abuse of notation we use the same symbol for the induced retraction ρ_{ϵ} = 1_{W} × ρ_{ϵ}: W × D_{ϵ}^{n} → W × S^{n−1}. For every W ∈ 𝔉 there is an induced retraction

ρ_{ϵ}^{V} : (W × D_{ϵ}^{n})^{V} → (W × S^{n−1})^{V} .

(10.3) Lemma. If C is a compact in (W × S^{n−1})^{V}, and C is contained in an open set A of (W × D^{n})^{V}, then there exists ϵ > 0 such that (ρ_{ϵ}^{V})^{−1}C ⊂ A.

Proof. If ϵ is small, we have D_{ϵ}^{n} = S^{n−1} × [0, ϵ), and hence (W × D_{ϵ}^{n})^{V} = (W × S^{n−1})^{V} × [0, ϵ) and ρ_{ϵ}^{V} is the projection onto the first factor. Each x ∈ C is a point of A, therefore there exists an open neighborhood of x of type O_{x} × [0, ϵ_{x}) contained in A. Being C compact there is a finite number of points x such that C is covered by O_{x} × [0, ϵ_{x}). Therefore there exists a ϵ > 0 such that ∪_{x}O_{x} × [0, ϵ) is an open subset of A containing C and thus (ρ_{ϵ}^{V})^{−1}C ⊂ A. q.e.d.

Now we can start the proof of proposition (10.1). Let U be an open subset U ⊂ (W × D^{n})^{V}. Let K be the generating family of compact subsets of V of the NKC property. Let K_{i} ⊂ V be compact subsets in K and U_{i} ⊂ W ×S^{n−1} open subsets, such that

⋂_{i=1}^{l} N_{K_{i},U_{i}} ⊂ U ∩ (W × S^{n−1})^{V} . (10.4)

By lemma (7.14) there is a sequence of open sets A_{i}^{k} ⊂ W × S^{n−1} with k ≥ 1 such that, for every k, A_{i}^{k} ⊂ A_{i}^{k+1},

⋃_{k≥1} A_{i}^{k} = U_{i} (10.5)

and the closure Ā_{i}^{k} is a compact subset of U_{i}.

(10.6) Lemma. For every i = 1, …, l the space N_{K_{i},Ā_{i}^{k}} is compact.

Proof. It is closed, because it is equal to the intersection of all the spaces N_{{x},Ā_{i}^{k}}, with x ∈ K_{i}, and they are closed because pre-images of the closed set Ā_{i}^{k} under the evaluation map (W × S^{n−1})^{V} × {x} → W × S^{n−1}, which is continuous because of lemma (6.11). Now we prove that it is compact. Consider the projection p: W × S^{n−1} → W onto the first factor. Since Ā_{i}^{k} is compact, its image pĀ_{i}^{k} is compact in W. Moreover, it is easy to see that

p^{V} (N_{K_{i},Ā_{i}^{k}}) ⊂ N_{K_{i},pĀ_{i}^{k}},

and hence that

N_{K_{i},Ā_{i}^{k}} ⊂ (p^{V})^{−1}N_{K_{i},pĀ_{i}^{k}} = N_{K_{i},pĀ_{i}^{k}} × S^{n−1}.

Now, since K_{i} and pĀ_{i}^{k} are compact and 𝔉 is an NKC-category, N_{K_{i},pĀ_{i}^{k}} is compact. Therefore N_{K_{i},Ā_{i}^{k}} is a closed subset of a compact space, and hence compact. q.e.d.

Consider for every k ≥ 1 the set

C_{k} = ⋂_{i=1}^{l} N_{K_{i},A_{i}^{k}} ⊂ (W × S^{n−1})^{V} . (10.7)

It is open, by definition of compact-open topology. Let C_{∞} denote the union ⋃_{k≥1} C_{k}.

(10.8) Lemma. We have that

C_{∞} = ⋂_{i=1}^{l} N_{K_{i},U_{i}} ⊂ (W × S^{n−1})^{V} ∩ U.

Proof. A 𝔉-map f ∈ (W ×S^{n−1})^{V} belongs to C_{∞} if and only if there is k ≥ 1 such that f ∈ C_{k}, that is if and only if there is k ≥ 1 such that fK_{i} ⊂ A_{i}^{k} for every i = 1, …, l. Of course this implies that fK_{i} ⊂ U_{i} for every i. On the other hand, if fK_{i} ⊂ U_{i}, then by equation (10.5) there is k_{i} ≥ 1 such that fK_{i} ⊂ A_{i}^{k} for every k ≥ k_{i}. This implies that there is k (take the maximum of all k_{i}) such that fK_{i} ⊂ A_{i}^{k} for every i = 1, …, l. q.e.d.

(10.9) Lemma. The closure C̄_{k} ⊂ (W × S^{n−1})^{V} is compact and contained in C_{∞}.

Proof. Because of its definition C̄_{k} ⊂ ∩_{i=1}^{l}\overline{N_{K_{i},A_{i}^{k}}}. Because for every i = 1, …, l the closure \overline{N_{K_{i},A_{i}^{k}}} is contained in N_{K_{i},Ā_{i}^{k}}, which is compact by lemma (10.6), we have that C̄_{k} is compact. Furthermore, by assumption Ā_{i}^{k} is contained in U_{i}, and hence N_{K_{i},Ā_{i}^{k}} is contained in N_{K_{i},U_{i}}. Therefore the closure of C_{k} is contained in C_{∞}. q.e.d.

Because of lemma (10.9) and lemma (10.8) the compact set C̄_{k} is contained in U in (W × D^{n})^{V}. By lemma (10.3) it is possible to find an ϵ_{k} > 0 such that

(ρ_{ϵ_{k}}^{V})^{−1}C_{k} ⊂ U. (10.10)

Without loss of generality we can assume that the sequence ϵ_{k}, with k ≥ 1, is decreasing.

Now consider for i = 1, …, l the following sets in (W × D^{n})^{V}:

U′_{i} = ⋃_{k≥1} ρ_{ϵ_{k}}^{−1}A_{i}^{k}. (10.11)

Since it is the union of open sets, for every i = 1, …, l, the set U′_{i} is open in W × D^{n}. Hence the set

U′ = ⋂_{i=1}^{l} N_{K_{i},U′_{i}} ⊂ (W × D^{n})^{V} , (10.12)

is open.

(10.13) Lemma. For every i = 1, …, l, U′_{i} ∩ (W × S^{n−1}) = U_{i}.

Proof. Because of (10.11), it suffices to prove that for every i and every k ≥ 1

ρ_{ϵ_{k}}^{−1}A_{i}^{k} ∩ (W × S^{n−1}) ⊂ U_{i},

and this is true because ρ_{ϵ_{k}}^{−1}A_{i}^{k} ∩ (W × S^{n−1}) = A_{i}^{k}. q.e.d.

(10.14) Lemma. U′ ⊂ U.

Proof. If f ∈ U′ ∩ (W × S^{n−1})^{V}, then fK_{i} ⊂ (W × S^{n−1}) ∩ U′_{i}, and by lemma (10.13) this implies fK_{i} ⊂ U_{i}. Thus, by (10.4), f_{0} ∈ U. On the other hand, assume that f ∈ (W × e^{n})^{V}. For every i = 1, …, l the image of the compact f(K_{i}) is in U′_{i} and hence, because of (10.11), for every i there exists an integer k_{i} ≥ 1 (depending on i) such f′(K_{i}) ⊂ ρ_{ϵ_{k_{i}}}^{−1}A_{i}^{k_{i}}. In particular, for every i = 1, …, l,

f′(K_{i}) ⊂ W × D_{ϵ_{k_{i}}}^{n},

and thus, because f is a 𝔉-map and ϵ_{k} is a decreasing sequence,

f′(V) ⊂ W × D_{ϵ_{m}}^{n},

where m denotes the maximum of k_{1}, …, k_{l}. Moreover, the sequence of spaces A_{i}^{k} is increasing with k ≥ 1, and therefore A_{i}^{k_{i}} ⊂ A_{i}^{m} for every i = 1, …, l. Thus for every i = 1, …, l,

f′(K_{i}) ⊂ ρ_{ϵ_{m}}^{−1}A_{i}^{m}.

But this means that for every i

f ∈ N_{K_{i},ρ_{ϵm}^{−1}A_{i}^{m}} = (ρ_{ϵ_{m}}^{V})^{−1}N_{K_{i},A_{i}^{m}},

hence that

f ∈ (ρ_{ϵ_{m}}^{V})^{−1}C_{m}.

Because of equation (10.10) this implies that f ∈ U. As claimed, we have proved that U′ ⊂ U. q.e.d.

This is the end of the proof of proposition (10.1). We can now draw the consequences which are needed.

(10.15) Lemma. Let 𝔉 be a NKC-category. Let Y be a 𝔉-family, Z a 𝔉-set and h: Z × S^{n−1} → Y a 𝔉-map, with n ≥ 1. If X is a family obtained as a push-out

[figure: Z × S^{n−1}; h; Y; Z × D^{n}; Φ; X]

then the following diagram

[figure: Z^{V} × S^{n−1}; (Z × S^{n−1})^{V}; h^{V}; Y^{V}; Z^{V} × D^{n}; (Z × D^{n})^{V}; Φ^{V}; X^{V}]

is a push-out, and Y has the N-neighborhood extension property in X.

Proof. By lemma (9.2) and proposition (10.1) if Z is a 𝔉-set the pair (Z × D^{n}, Z × S^{n−1}) is a NNEP-pair. Hence, by lemma (9.3) and lemma (9.4) X^{V} is the push-out of (Z × D^{n})^{V} and Y^{V} and Y has the N-neighborhood extension property in X. q.e.d.

(10.16) Corollary. Let (X, D) be a relative 𝔉-complex where 𝔉 is a NKCcategory. Let Φ : Z_{n} × D^{n} → X be its characteristic map of n-cells and let h_{n} denote the n-attaching map of X, i.e. the restriction of Φ to Z_{n} × S^{n−1}. For every n ≥ 1 and every V ∈ 𝔉 the diagram

[figure: Z_{n}^{V} × S^{n−1}; (Z_{n} × S^{n−1})^{V}; h_{n}^{V}; X_{n−1}^{V}; Z_{n}^{V} × D^{n}; (Z_{n} × D^{n})^{V}; Φ^{V}; X_{n}^{V}]

is a push-out (in Top). Moreover,

X^{V} = lim_{n→∞} X_{n}^{V} .

Proof. The first part is a direct consequence of lemma (10.15). Then, by applying (9.6) we obtain that (X, X_{n}) is a NNEP-pair for every n and that X^{V} = lim_{n→∞} X_{n}^{V}. q.e.d.

(10.17) Corollary. Let (X, D) be a relative 𝔉-complex (or a 𝔉-CW-pair) with 𝔉 NKC. Then (X, D) is a NNEP-pair.

(10.18) Corollary. Let 𝔉 be NKC. Let (M, A) be a 𝔉-CW-pair, Y a 𝔉-family and h: A → Y a 𝔉-map. If X is the push-out X = M ∪_{h} Y then (X, Y) is a NNEP-pair.

Proof. By corollary (10.17) (M, A) is a NNEP-pair, hence by lemma (9.3) and lemma (9.4) (X, Y) is a NNEP-pair. q.e.d.

11 Examples of NKC categories

(11.1) Proposition. If the fibre functor F is faithful and the Hom-sets hom_{𝔉}(V, W) are compact spaces then 𝔉 is a NKC category.

The proof of the previous proposition is simple. The interesting fact is that there are NKC structure categories with non-compact hom-sets, as a consequence of the following proposition.

(11.2) Proposition. If 𝔉 is a closed subcategory of the category Vect of finitely dimensional ℝ-vector spaces and linear maps, then 𝔉 has the NKC property.

Proof. See [4], lemma (4.2). q.e.d.

This result yields the principal stratified bundle associated to a 𝔉-stratified vector bundle.

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