Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 26, 2007, 107–123 \begin{matrix}c \\ ◯ \end{matrix} Vadim V. Shurygin, jr.

Vadim V. Shurygin, jr.

PRODUCT PRESERVING BUNDLE FUNCTORS ON MULTIFIBERED AND MULTIFOLIATE MANIFOLDS

(submitted by M. A. Malakhaltsev)

Abstract. We show that the set of the equivalence classes of multifoliate structures is in one-to-one correspondence with the set of equivalence classes of finite complete projective systems of vector space epimorphisms. After that we give the complete description of all product preserving bundle functors on the categories of multifibered and multifoliate manifolds.

In the middle 1980s Eck [2], Kainz and Michor [3], Luciano [8] described all product preserving bundle functors on the category of smooth manifolds in terms of Weil bundles [14] (see also [5]). In 1996 Mikulski [9] classified all product preserving bundle functors on fibered manifolds. In the recent years Weil functors and product preserving functors are of great interest, see e.g. Kolář and Mikulski [6], Kriegl and Michor [7], Muños, Rodrigues, and Muriel [11], Mikulski and Tomáš [10, 13].

Kodaira and Spencer in [4] introduced the notion of a multifoliate structure on a smooth manifold. In the present paper, we introduce the category of multifibered manifolds which is a subcategory of the category of multifoliate manifolds and, following the lines of Mikulski [9], describe all product preserving bundle functors on these categories.

2000 Mathematical Subject Classification. 58A05, 58A32.

Key words and phrases. product preserving bundle functors, multifoliate structures, projective systems.

We denote the category of smooth manifolds by Mf and that of fibered manifolds by FM [5]. All manifolds and maps between manifolds under consideration are assumed to be of class C^{∞}.

1. Projective systems of vector spaces

Let (Λ = {α, β, … }, ≤) be a partially ordered set. A projective system (an inverse system) over Λ [1] is a collection (S_{α}, ζ_{α}^{β}, Λ) consisting of sets S_{α}, α ∈ Λ, and maps ζ_{α}^{β} : S_{β} → S_{α}, α ≤ β, called projections, such that ζ_{α}^{α} = id_{S_{α}} for all α ∈ Λ and ζ_{α}^{β} ∘ζ_{β}^{γ} = ζ_{α}^{γ} when α ≤ β ≤ γ. The projective limit of a projective system (S_{α}, ζ_{α}^{β}, Λ) is the subset

S = lim_{⟵} S_{α} ⊂ ∏_{α∈Λ} S_{α}

consisting of all elements x = (x_{α}) such that ζ_{α}^{β}(x_{β}) = x_{α}. If the set S is not empty, then by ζ_{β} : S → S_{β} we denote the map which sends x = (x_{α}) into x_{β}. These maps are called canonical projections.

It will be convenient to denote projective systems under consideration as follows: ζ = (S_{α}, ζ_{α}^{β}, Λ, S).

In this section, we will consider projective systems of vector spaces ξ = (L_{α}, ξ_{α}^{β}, Λ, L) satisfying the following conditions:

i) L_{α}, α ∈ Λ, and L are finite-dimensional vector spaces over ℝ; ii) all the maps ξ_{α}^{β} and ξ_{α} are linear epimorphisms.

By an isomorphism between two projective systems ξ = (L_{α}, ξ_{α}^{β}, Λ, L) and ξ′ = (L′_{α′}, ξ_{α′}^{β′}, Λ′, L′) we mean a collection (ω, {ψ_{α}}_{α∈Λ}) consisting of an isomorphism ω : Λ → Λ′ of partially ordered sets and linear isomorphisms ψ_{α} : L_{α} → L′_{ω(α)} such that ξ_{ω(α)}^{ω(β)} ∘ ψ_{β} = ψ_{α} ∘ ξ_{α}^{β} for α ≤ β. An isomorphism (ω, {ψ_{α}}_{α∈Λ}) gives rise to the isomorphism ψ := lim_{⟵} ψ_{α} : L → L′ defined by ψ((x_{α})) = (ψ_{α}(x_{α})). The map ψ is the unique isomorphism between L and L′ such that ξ_{ω(α)} ∘ ψ = ψ_{α} ∘ ξ_{α}. Projective systems ξ = (L_{α}, ξ_{α}^{β}, Λ, L) and ξ′ = (L′_{α′}, ξ_{α′}^{β′}, Λ′, L′) are said to be isomorphic if there exists an isomorphism between them.

An isomorphism from ξ to itself of the form (id, {f_{α}}_{α∈Λ}) is said to be an automorphism of ξ. Denote by GL(ξ) the group of all linear automorphisms of L of the form f = lim_{⟵} f_{α} where (id, {f_{α}}) is an automorphism of ξ.

Definition. A vector subspace K ⊂ L is said to be invariant if every f ∈ GL(ξ) maps K into itself.

One can easily see that the sum and the intersection of two invariant subspaces are invariant subspaces. For any α ∈ Λ, the subspace K_{α} := ker ξ_{α} ⊂ L is invariant and L_{α} ≅ L/K_{α}. In what follows we will identify L_{α} and L/K_{α}.

Definition. A projective system ξ = (L_{α}, ξ_{α}^{β}, Λ, L) is said to be complete if any finite-codimensional invariant subspace of L is of the form ker ξ_{α} for some α ∈ Λ.

Let ξ = (L_{α}, ξ_{α}^{β}, Λ, L) be a projective system (not necessarily complete). Consider the set {K_{a}}_{a∈Λ̃} of all finite-codimensional invariant subspaces K_{a} of L. For any two invariant subspaces K_{a}, K_{b} such that K_{a} ⊃ K_{b}, denote by ξ_{a}^{b} : L_{b} = L/K_{b} → L_{a} = L/K_{a} the canonical epimorphism. Let us endow Λ̃ with the partial order defined as follows: a ≤ b if and only if K_{a} ⊇ K_{b}. One can easily see that the collection ξ̃ = (L_{a}, ξ_{a}^{b}, Λ̃, L̃ = lim_{⟵} L_{a}) is a complete projective system. We call it

the completion of ξ. Obviously, ξ̃ is complete. Since, for any α ∈ Λ, the subspace K_{α} = ker ξ_{α} is invariant, one can consider Λ as a subset of Λ̃.

Definition. A projective system ξ = (L_{α}, ξ_{α}^{β}, Λ, L) is called finite if Λ is finite.

Obviously, when ξ is finite, its limit L is a finite-dimensional vector space.

Proposition 1.1. If ξ = (L_{α}, ξ_{α}^{β}, Λ, L) is a finite complete projective system then

(1) Λ contains the greatest element ε;

(2) L is isomorphic to L_{ε}.

Proof. Indeed, the zero subspace is invariant and of finite codimension.

Definition. Two projective systems

ξ = (L_{α}, ξ_{α}^{β}, Λ, L) and ξ′ = (L′_{α}, ξ_{α′}^{β′}, Λ′, L′)

are said to be equivalent if there exists an isomorphism φ : L → L′ such that φ ∘ f ∘ φ^{−1} ∈ GL(ξ′) for any f ∈ GL(ξ) and

Φ : f ∈ GL(ξ) ↦ φ ∘ f ∘ φ^{−1} ∈ GL(ξ′)

is a group isomorphism.

One can easily see that isomorphic projective systems are equivalent.

Proposition 1.2. Let ξ = (L_{α}, ξ_{α}^{β}, Λ, L) be a projective system and ξ̃= (L_{a}, ξ_{a}^{b}, Λ̃, L̃) its completion. Then ξ and ξ̃ are equivalent.

Proof. In fact, the maps

φ = (φ_{α}) : L̃ → L, φ_{α} : L̃ ∋ x = (x_{a})_{a∈Λ̃} ↦ x_{α} ∈ L_{α}

and

ψ = (ψ_{a}) : L → L̃, ψ_{a} : L ∋ x ↦ x + K_{a} ∈ L_{a} = L/K_{a}

are mutually inverse isomorphisms which induce an isomorphism of the groups GL(ξ) and GL(ξ̃). □

The proof of the following proposition is immediate.

Proposition 1.3. If complete projective systems ξ = (L_{α}, ξ_{α}^{β}, Λ, L) and ξ′ = (L′_{α}, ξ_{α′}^{β′}, Λ′, L′) are equivalent, then ξ is isomorphic to ξ′.

Definition. Let ξ = (L_{α}, ξ_{α}^{β}, Λ, L) be a projective system. A local diffeomorphism φ : U ⊂ L → V ⊂ L between two open subsets of L is called a ξ-diffeomorphism if for any x ∈ U there exist an open subset W (x) ⊂ U and a system of diffeomorphisms {φ_{α} : ξ_{α}(W) → ξ_{α}(φ(W))}_{α∈Λ} such that φ_{α} ∘ ξ_{α} = ξ_{α} ∘ φ for any α ∈ Λ.

Denote the pseudogroup of all ξ-diffeomorphisms by Γ(ξ). The tangent map dφ_{x} of any ξ-diffeomorphism φ : U → V at every point x ∈ U can be viewed as an element of GL(ξ).

Definition. A ξ-structure on an n-dimensional smooth manifold (n = dim L) is a maximal atlas compatible with the pseudogroup Γ(ξ). A smooth manifold endowed with a ξ-structure is called a ξ-manifold.

Definition. Let ξ = (L_{α}, ξ_{α}^{β}, Λ, L) and ξ′ = (L′_{α}, ξ′_{α}^{β}, Λ, L′) be two projective systems over the same partially ordered set Λ. A smooth map g : U ⊂ L → V ⊂ L′ is called a Λ-smooth map if for any x ∈ U there exist an open subset W (x) ⊂ U and a system of smooth maps {g_{α} : ξ_{α}(W) → ξ′_{α}(g(W))}_{α∈Λ} such that g_{α} ∘ ξ_{α} = ξ′_{α} ∘ g for any α ∈ Λ.

Definition. Let M be a ξ-manifold and M′ a ξ′-manifold. A smooth map f : M → M′ between a ξ-manifold M and a ξ′-manifold M′ is called a Λ-smooth map if it is Λ-smooth in terms of the atlases defining ξ- and ξ′-structures on these manifolds.

For a fixed finite partially ordered set Λ, all ξ-manifolds for all projective systems ξ over Λ together with Λ-smooth maps as morphisms form a subcategory Mf_{proj}(Λ) of the category Mf.

Proposition 1.4. The category Mf_{proj}(Λ) admits products.

Proof. Let M be a ξ-manifold and M′ a ξ′-manifold, where

ξ = (L_{α}, ξ_{α}^{β}, Λ, L) and ξ′ = (L′_{α}, ξ′_{α}^{β}, Λ, L′).

Then M × M′ is a (ξ × ξ′)-manifold, where ξ × ξ′ denotes the projective system (L_{α} × L′_{α}, ξ_{α}^{β} × ξ′_{α}^{β}, Λ, L × L′). □

2. Multifoliate manifolds

Multifoliate structures on smooth manifolds were introduced by K. Kodaira and D.C. Spencer [4] as follows.

Definition. A pair (Λ, p) consisting of a finite partially ordered set Λ and a surjective map

p : {1, …, n} ∋ i ↦ p(i) ∈ Λ

is called a multifoliate structure on the set {1, …, n}.

Denote by GL(Λ, p) the group of all linear isomorphisms

f : ℝ^{n} ∋ (x^{i}) ↦ (f_{j}^{i}x^{j}) ∈ ℝ^{n}

satisfying the condition

f_{j}^{i} = 0 if p(i) ≱ p(j),

and by Γ(Λ, p) the pseudogroup of all local diffeomorphisms g : U ⊂ ℝ^{n} → V ⊂ ℝ^{n} such that dg_{x} ∈ GL(Λ, p) for all x ∈ U.

Definition. A (Λ, p)-multifoliate structure on an n-dimensional smooth manifold is a maximal atlas compatible with the pseudogroup Γ(Λ, p). We call the local coordinates determined by a chart of this atlas adapted coordinates. A smooth manifold endowed with a (Λ, p)-multifoliate structure is called a (Λ, p)-multifoliate manifold.

Definition. Let M be a (Λ, p)-multifoliate manifold and N be a (Λ, p′)- multifoliate manifold. A Λ-multifoliate map f : M → N is a smooth map, satisfying the condition

\frac{∂f^{a}}{∂x^{i}} = 0 if p′(a) ≱ p(i)

in adapted coordinates. Clearly, this definition does not depend on the choice of a local coordinate system.

For a fixed finite partially ordered set Λ, all (Λ, p)-multifoliate manifolds for all surjective maps p : {1, …, n} → Λ and for all natural numbers n ≥ card Λ together with Λ-multifoliate maps as morphisms form a subcategory Mf_{Λ} of the category Mf. We call it the category of multifoliate manifolds over Λ.

Proposition 2.1. The category Mf_{Λ} admits products.

Proof. If M_{a} is a (Λ, p_{a})-multifoliate manifold, p_{a} : {1, …, n_{a}} → Λ, a = 1, 2, then the product M_{1} × M_{2} is a (Λ, p)-multifoliate manifold, where p : {1, …, n_{1} + n_{2}} → Λ is defined by

p(i) = \{ \begin{matrix}p_{1}(i), & i ≤ n_{1}; \\ p_{2}(i − n_{1}), & i > n_{1}. \end{matrix}

□

Definition. We say that two multifoliate structures (Λ, p) and (Λ′, p′) on the same set {1, …, n} are equivalent if there exists a linear automorphism φ : ℝ^{n} → ℝ^{n} such that φ ∘ f ∘ φ^{−1} ∈ GL(Λ′, p′) for any f ∈ GL(Λ, p) and

Φ : f ∈ GL(Λ, p) ↦ φ ∘ f ∘ φ^{−1} ∈ GL(Λ′, p′)

is a group isomorphism.

Clearly, if (Λ, p) is a multifoliate structure on {1, …, n}, then, for each permutation σ on {1, …, n}, the multifoliate structure (Λ, p ∘ σ) is equivalent to (Λ, p).

For a multifoliate structure (Λ, p) on {1, …, n}, define the sets H_{α} = {i | p(i) ≤ α}, α ∈ Λ, and let k(α) = card H_{α}. The vector spaces

L_{α} = {(x^{i_{1}}, …, x^{i_{k(α)}}) | x^{i_{s}} ∈ ℝ, i_{s} ∈ H_{α}, s = 1, …, k(α)}

and the natural epimorphisms pr_{α}^{β} : L_{β} → L_{α}, α ≤ β, form a projective system whose limit can be naturally identified with ℝ^{n}. Denote this system and its completion, respectively, by ξ(Λ, p) and ξ̃(Λ, p).

Theorem 2.1. [12] The correspondence (Λ, p) ↦ ξ̃(Λ, p) induces a bijection between the equivalence classes of multifoliate structures (Λ, p) and the equivalence classes of finite complete projective systems of vector space epimorphisms.

Proof. We give here a sketch of the proof and refer for details to [12].

Show first that the correspondence (Λ, p) ↦ ξ̃(Λ, p) induces a map from the set of equivalence classes of multifoliate structures to the set of equivalence classes of finite complete projective systems of vector space epimorphisms. By Propositions 1.2 and 1.3, it suffices to show that the groups GL(Λ, p) and GL(ξ(Λ, p)) are isomorphic. In fact, there is a natural isomorphism GL(Λ, p) → GL(ξ(Λ, p)) which assigns to g ∈ GL(Λ, p) a collection of maps {g_{α} : L_{α} → L_{α}} defined as follows: g_{α}(y_{α}) = pr_{α}(g(y)) where y ∈ ℝ^{n} is such that pr_{α}(y) = y_{α}.

To prove that the correspondence indicated in the theorem is one-to-one, we need to pass to the dual inductive system [1].

Let ξ = (L_{α}, ξ_{α}^{β}, Λ, L) be a finite complete projective system and let ε ∈ Λ be the greatest element. The dual spaces L_{α}^{∗} together with the dual maps (ξ_{α}^{β})^{∗} form an inductive system ξ^{∗} = (L_{α}^{∗}, (ξ_{α}^{β})^{∗}, Λ). The existence of the greatest element implies that the inductive limit of ξ^{∗} exists and can be identified with the dual space L^{∗}. Under this identification the dual maps ξ_{α}^{∗} : L_{α}^{∗} → L^{∗} are the canonical maps of ξ^{∗}. Obviously, all the maps (ξ_{α}^{β})^{∗} and ξ_{α}^{∗} are monomorhisms. We will call the inductive system ξ^{∗} = (L_{α}^{∗}, (ξ_{α}^{β})^{∗}, Λ, L^{∗}) the dual of ξ.

For any f ∈ GL(ξ) and for each α ∈ Λ, we have ξ_{α}^{∗} ∘ f_{α}^{∗} = f^{∗} ∘ ξ_{α}^{∗}. Denote by GL(ξ^{∗}) the group of all linear automorphisms h : L^{∗} → L^{∗}

which are the limits of inductive systems of linear automorphisms h_{α} : L_{α}^{∗} → L_{α}^{∗}. Since the maps ξ_{α}^{∗} are monomorphisms, it will be convenient to consider each L_{α}^{∗} as a subspace of L^{∗}. Then h_{α} = h|L_{α}^{∗} or, in other words, h maps L_{α}^{∗} into itself. The correspondence f ↦ f^{∗} is an isomorphism of the groups GL(ξ) and GL(ξ^{∗}).

The dual system ξ^{∗} is complete in the sense that it contains all subspaces which are invariant with respect to each f^{∗} ∈ GL(ξ^{∗}).

By a chain in ξ^{∗} we mean a sequence of embeddings

L_{α_{k}}^{∗} ⟵^{(ξ_{α_{k−1}}^{α_{k}})^{∗}} L_{α_{k−1}}^{∗} ⟵^{(ξ_{α_{k−2}}^{α_{k−1}})^{∗}} ⋯ ⟵^{(ξ_{α_{1}}^{α_{2}})^{∗}} L_{α_{1}}^{∗}

such that α_{1} < α_{2} < ⋯ < α_{k} and α_{i} is the successor of α_{i−1} in Λ, i = 2, …, k, (that is, α_{i−1} ≤ β ≤ α_{i} implies that either β = α_{i−1} or β = α_{i}). The space L_{α_{k}}^{∗} is called the end of the chain.

L_{α}^{∗} is said to be a subspace of the first floor if α is a minimal element of Λ. L_{α}^{∗} is said to be a subspace of the k-th floor if each chain with end L_{α}^{∗} is of length no greater than k and among all such chains there is at least one of length k.

If L_{α}^{∗} is a subspace of the first floor, we take a basis B_{α} = {e_{α}^{1}, …, e_{α}^{s(α)}} in L_{α}^{∗} and call the index α distinguished. Let C_{1} be the union of B_{α} for all subspaces of the first floor. One can verify that the system C_{1} is linearly independent. In fact, the assumption that the system is linearly dependent contradicts the completeness of ξ^{∗} (see [12] for details).

Let now L_{β}^{∗} be a space of the second floor. Then either L_{β}^{∗} ⊂ L{C_{1}}, where L{C_{1}} is the linear span of the system C_{1}, or one can choose a system of linearly independent elements B_{β} = {e_{β}^{1}, …, e_{β}^{s(β)}} in L_{β}^{∗} such that L_{β}^{∗} = L{e_{β}^{1}, …, e_{β}^{s(β)}} ⊕ (L_{β}^{∗} ∩ L{C_{1}}). In the latter case the index β is also called distinguished. Let C_{2} be the union of B_{β} for all subspaces of the second floor. The system C_{1} ∪ C_{2} is linearly independent (see [12] for details).

Suppose that we have chosen systems C_{ℓ} for every ℓ ≤ k. If L_{γ}^{∗} is a space of (k + 1)-th floor, then either L_{γ}^{∗} ⊂ L_{k} := L{C_{1} ∪ ⋯ ∪ C_{k}} or there exists a system of linearly independent elements B_{γ} = {e_{γ}^{1}, …, e_{γ}^{s(γ)}} such that L_{γ}^{∗} = (L_{γ}^{∗} ∩ L_{k}) ⊕ L{e_{γ}^{1}, …, e_{γ}^{s(γ)}}. In the latter case the index γ is called distinguished. Let C_{k+1} be the union of B_{γ} for all subspaces of the (k+1)-st floor. As above, the system C_{1}∪C_{2}∪⋯∪C_{k+1} is linearly independent.

This process stops when we reach L^{∗}. As a result, we obtain a subset Λ′ ⊂ Λ consisting of distinguished elements and the corresponding basis {e^{1}, …, e^{n}} in L^{∗}. Let p : {1, …, n} → Λ′ be the map defined as follows: p(m) = α where α is the minimal distinguished element such that e^{m} ∈ L_{α}^{∗}. The pair (Λ′, p) is a multifoliate structure on {1, …, n} and the group GL(ξ) is isomorphic to GL(Λ′, p). □

Corollary 2.1. For any finite partially ordered set Λ, the categories Mf_{proj}(Λ) and Mf_{Λ} are isomorphic.

Corollary 2.2. Let (Λ, p) be a multifoliate structure on {1, …, n} and ξ(Λ, p) the corresponding projective system. Let ξ̃(Λ, p) = (L̃_{a}, ξ_{a}^{b}, Λ̃, L̃) be the completion of ξ(Λ, p) and (Λ′, p′) the multifoliate structure on {1, …, n} determined by ξ̃(Λ, p). Then

(1) the partially ordered sets Λ and Λ′ are canonically isomorphic;

(2) the multifoliate structures (Λ, p) and (Λ′, p′) are equivalent.

Proof. (1) Every invariant subspace of ξ̃(Λ, p) is of the form

L̃_{(α_{1},...,α_{k})} = L/(ker ξ_{α_{1}} ∩ ⋯ ∩ ker ξ_{α_{k}}),

where α_{1}, …, α_{k} ∈ Λ are pairwise incomparable. Thus, Λ̃ is isomorphic to the set of all finite collections of pairwise incomparable elements (α_{1}, …, α_{k}) endowed with the partial order defined as follows: (α_{1}, …, α_{k}) ≤ (β_{1}, …, β_{ℓ}) if and only if ker ξ_{β_{1}} ∩ ⋯ ∩ ker ξ_{β_{ℓ}} ⊆ ker ξ_{α_{1}} ∩ ⋯ ∩ ker ξ_{α_{k}}.

The index (α_{1}, …, α_{k}) ∈ Λ̃ is distinguished if and only if k = 1, and so the set of all distinguished elements is naturally isomorphic to Λ.

(2) From Theorem 2.1 it follows that GL(Λ, p) ≅ GL(ξ(Λ, p)) and GL(ξ̃(Λ, p)) ≅ GL(Λ′, p′). The rest of the proof follows from Proposition 1.2. □

Corollary 2.3. If (Λ, p) and (Ω, q) are equivalent multifoliate structures on {1, …, n}, then (1) the partially ordered sets Λ and Ω are isomorphic;

(2) there exists a permutation σ on {1, …, n} such that q = p ∘ σ.

Proof. (1) Let (Λ′, p′) and (Ω′, q′) be the multifoliate structures corresponding to the complete projective systems ξ̃(Λ, p) and ξ̃(Ω, q) respectively. The systems ξ̃(Λ, p) and ξ̃(Ω, q) are equivalent. By Proposition 1.3, these systems are isomorphic. Hence the sets Λ′ and Ω′ of their distinguished elements are isomorphic. By Corollary 2.2, Λ and Ω are isomorphic.

(2) Let ω : Λ ∋ α ↦ ω(α) ∈ Ω be an isomorphism. Recall that, for any distinguished index β ∈ Λ′ ≅ Λ, s(β) denotes the number of linearly independent elements in the system B_{β} defined in the proof of Theorem 2.1. One can easily see that the cardinality of the subset p^{−1}(β) ⊂ {1, …, n} coincides with s(β). Since the projective systems ξ̃(Λ, p) and ξ̃(Ω, q) are isomorphic, for every distinguished index β ∈ Λ′ ≅ Λ, the numbers s(β) and s(ω(β)) coincide. This means that p^{−1}(α) and q^{−1}(ω(α)) have the same cardinality for any α ∈ Λ. From this observation it follows that one can find a permutation σ on {1, …, n} such that q = p ∘ σ. In general, such a permutation is not unique. □

3. Multifibered manifolds. The classification theorem

Definition. Let ξ = (L_{α}, ξ_{α}^{β}, Λ, L) be a projective system of vector spaces and let π = (M_{α}, π_{α}^{β}, Λ, M) be a projective system such that all M_{α} and M = lim_{⟵} M_{α} are smooth manifolds and all maps π_{α}^{β} : M_{β} → M_{α} and π_{α} : M → M_{α} are surjective submersions. Let A be a ξ-structure on M. We call π = (M_{α}, π_{α}^{β}, Λ, M) a multifibered manifold if the ξ-structure A on M is compatible with all projections π_{α} in the following sense: for any point x = (x_{α}) ∈ M, there are charts (U, h) centered at x on M and (U_{α}, h_{α}) centered at x_{α} on M_{α}, α ∈ Λ, such that the following diagram commutes

U \underrightarrow{h} L
π_{α}↓  ↓ξ_{α}
U_{α} \underrightarrow{h_{α}} L_{α} .

It follows from Corollary 2.1 that M carries a structure of multifoliate manifold. For any point x = (x_{α}) ∈ M the projective system of tangent spaces ξ_{x} = (T_{x_{α}}M_{α}, (dπ_{α}^{β})_{x_{β}}, Λ, T_{x}M) is isomorphic to ξ.

Definition. A multifibered map f : π → \overline{π} between two multifibered manifolds π = (M_{α}, π_{α}^{β}, Λ, M) and \overline{π} = (\overline{M}_{α}, \overline{π}_{α}^{β}, Λ, \overline{M}) is a collection of maps \{f_{α} : M_{α} → \overline{M}_{α}\}_{α∈Λ} such that for all α ≤ β the diagram

M_{β} \underrightarrow{f_{β}} \overline{M}_{β}
π_{α}^{β}↓  ↓\overline{π}_{α}^{β}
M_{α} \underrightarrow{f_{α}} \overline{M}_{α}

commutes. Each multifibered map determines a unique smooth map f : M → \overline{M}.

Multifibered manifolds over Λ together with multifibered maps form a subcategory of the category Mf_{Λ} of multifoliate manifolds over Λ. We denote it by FM_{Λ}.

Proposition 3.1. The category FM_{Λ} admits products.

Proof. If π = (M_{α}, π_{α}^{β}, Λ, M) and \overline{π} = (\overline{M}_{α}, \overline{π}_{α}^{β}, Λ, \overline{M}) are two multifibered manifolds, then their product is π × \overline{π} := (M_{α} × \overline{M}_{α}, π_{α}^{β} × \overline{π}_{α}^{β}, Λ, M × \overline{M}). □

The categories FM_{Λ} and Mf_{Λ} are local categories over manifolds.

Definition. An inductive system of Weil algebra homomorphisms over Λ is a collection μ = (A_{α}, μ_{α}^{β}, Λ) consisting of Weil algebras A_{α}, α ∈ Λ, and Weil algebra homomorphisms μ_{β}^{α} : A_{α} → A_{β}, α ≤ β, such that μ_{α}^{α} = id_{A_{α}} for all α ∈ Λ and μ_{γ}^{β} ∘ μ_{β}^{α} = μ_{γ}^{α} when α ≤ β ≤ γ. Let μ = (A_{α}, μ_{α}^{β}, Λ) and \overline{μ} = (\overline{A}_{α}, \overline{μ}_{α}^{β}, Λ) be two inductive systems of Weil algebra homomorphisms. By a morphism ν : μ → \overline{μ} we mean a family ν = (ν_{α})_{α∈Λ} of Weil algebra homomorphisms \{ν_{α} : A_{α} → \overline{A}_{α}\} such that for all α ≤ β the diagram

A_{α} \underrightarrow{ν_{α}} \overline{A}_{α}
μ_{β}^{α}↓  ↓\overline{μ}_{β}^{α}
A_{β} \underrightarrow{ν_{β}} \overline{A}_{β}

commutes.

Theorem 3.1. Any product preserving bundle functor F on the category FM_{Λ} or Mf_{Λ} is uniquely determined by the inductive system μ = (μ_{β}^{α} : A_{α} → A_{β}) of Weil algebra homomorphisms. Any natural transformation η : F → \overline{F} is uniquely determined by the morphism ν : μ → \overline{μ} of inductive systems of Weil algebra homomorphisms.

Since, by Theorem 2.1, any multifoliate manifold is locally a multifibered manifold, it is enough to consider the case of a bundle functor F : FM_{Λ} → FM.

The proof of the Theorem 3.1 is essentially the same as the Mikulski’s proof [9] for the case of a bundle functor FM → FM. We will reproduce the main scheme of the proof.

Let μ = (G_{α}, μ_{β}^{α}, Λ) be an inductive system of natural transformations of bundle functors, i.e., for any α ∈ Λ, there is given a bundle functor G_{α} : Mf → FM and for any α, β ∈ Λ such that α ≤ β, there is given a natural transformation μ_{β}^{α} : G_{α} → G_{β} with the properties μ_{β}^{α} ∘ μ_{γ}^{β} = μ_{γ}^{α} and μ_{α}^{α} = id. We define a bundle functor ∏_{μ} G_{α} : FM_{Λ} → FM as follows.

Consider a multifibered manifold π = (M_{α}, π_{α}^{β}, Λ, M). We let

∏_{μ} G_{α}(π) := {(x_{α}) | G_{β}(π_{α}^{β})(x_{β}) = μ_{β}^{α}(M_{α})(x_{α})} ⊂ ∏_{α∈Λ} G_{α}(M_{α}).

The set ∏_{μ} G_{α}(π) is a submanifold in ∏_{α∈Λ} G_{α}(M_{α}). We define the map

p_{μ}(π) : ∏_{μ} G_{α}(π) → M

as follows. Consider the bundle projection

∏_{α∈Λ} G_{α}(M_{α}) → ∏_{α∈Λ} M_{α}.

The image of its restriction to ∏_{μ} G_{α}(π) coincides with M = lim_{⟵} M_{α}, thus defining the map p_{μ}(π) : ∏_{μ} G_{α}(π) → M which is a surjective submersion.

Let f = (f_{α}) : π → \overline{π} be a multifibered map. We set

∏_{μ} G_{α}(f) := the restriction of ∏_{α∈Λ} G_{α}(f_{α}).

The map

∏_{μ} G_{α}(f) : ∏_{μ} G_{α}(π) → ∏_{μ} G_{α}(\overline{π})

is well-defined since all μ_{β}^{α} are natural transformations.

The correspondence

∏_{μ} G_{α} : FM_{Λ} → FM

is a bundle functor.

Now let \overline{μ} = (\overline{μ}_{β}^{α} : \overline{G}_{α} → \overline{G}_{β}) be another inductive system of natural transformations of bundle functors. Suppose that there is given a family ν = (ν_{α} : G_{α} → \overline{G}_{α}) of natural transformations such that, for any manifold X and α ≤ β, the diagram

\begin{matrix}G_{α}(X) & \underrightarrow{ν_{α}(X)} & \overline{G}_{α}(X) \\ μ_{β}^{α}(X)↓ & & ↓\overline{μ}_{β}^{α}(X) \\ G_{β}(X) & \underrightarrow{ν_{β}(X)} & \overline{G}_{β}(X) \end{matrix}  (1)

commutes. Then we define the natural transformation

∏_{μ,\overline{μ}} ν_{α} : ∏_{μ} G_{α} → ∏\overline{μ} \overline{G}_{α}

as follows.

For a multifibered manifold π = (M_{α}, π_{α}^{β}, Λ, M), we define the map

∏_{μ,\overline{μ}} ν_{α}(π) : ∏_{μ} G_{α}(π) → ∏\overline{μ} \overline{G}_{α}(π)

to be the restriction of ∏_{α∈Λ} ν_{α}(M_{α}). Since each ν_{α} is a natural transformation, the map ∏_{μ,\overline{μ}} ν_{α}(π) is well-defined. The family

∏_{μ,\overline{μ}} ν_{α} = \{∏_{μ,\overline{μ}} ν_{α}(π)\} : ∏_{μ} G_{α} → ∏\overline{μ} \overline{G}_{α}

is a natural transformation.

Let us denote by pt a one-point manifold. Consider a smooth manifold X. For any α ∈ Λ, we construct a multifibered manifold i_{α}(X) = (X_{γ}, r_{δ}^{γ}, Λ, X) in the following way. We let X_{γ} = X if γ ≥ α, and X_{γ} = pt otherwise. Each projection r_{δ}^{γ} is either the identity map id_{X} : X → X if γ ≥ α, δ ≥ α, or the unique map pt_{X} : X → pt if γ ≥ α, δ ≱ α, or the unique map pt → pt if γ ≱ α, δ ≱ α. Clearly, lim_{⟵} X_{α} = X. We can consider any map f : X → Y as a multifibered map f : i_{α}(X) → i_{β}(Y) for α ≤ β. Thus we obtain the bundle functors

i_{α} : Mf → FM_{Λ}

and the natural transformations

id_{β}^{α} : i_{α} → i_{β}, α ≤ β,

consisting of FM_{Λ}-morphisms id_{X} : i_{α}(X) → i_{β}(X). Obviously, the functors i_{α} preserve products.

Let F : FM_{Λ} → FM be a bundle functor. Consider the bundle functors

G_{α}^{F} : = F ∘ i_{α} : Mf → FM.  (2)

If F preserves products, then the functors G_{α}^{F} also preserve products.

We define an inductive system μ^{F} = ((μ^{F})_{β}^{α}) of natural transformations as follows:

(μ^{F})_{β}^{α} := F (id_{β}^{α}) : G_{α}^{F} → G_{β}^{F} .  (3)

Let \overline{F} : FM_{Λ} → FM be another bundle functor, and let η = \{η_{π}\} : F → \overline{F} be a natural transformation. We define the family of natural transformations

ν^{η} = (ν_{α}^{η} : G_{α}^{F} → G\overline{F} ; α)

by

ν_{α}^{η}(X) := η_{i_{α}(X)} : G_{α}^{F} (X) → G\overline{F} ; α (X)  (4)

for any manifold X. The diagram

G_{α}^{F} (X) \underrightarrow{ν_{α}^{η}(X)} G\overline{F} ; α (X)
(μ^{F})_{β}^{α}(X)↓  ↓(μ\overline{F})_{β}^{α}(X)
G_{β}^{F} (X) \underrightarrow{ν_{β}^{η}(X)} G\overline{F} ; β (X)

commutes for any manifold X and any α ≤ β.

Let F : FM_{Λ} → FM be a bundle functor. Following Mikulski, we construct a natural transformation

Θ = {Θ_{π}} : F → ∏_{μ^{F}} G_{α}^{F} .

Let π = (M_{α}, π_{α}^{β}, Λ, M) be a multifibered manifold. For any α ∈ Λ we define a multifibered map j_{α} : π → i_{α}(M_{α}) as follows: we let (j_{α})_{γ} := π_{α}^{γ} if α ≤ γ and (j_{α})_{γ} := pt_{M_{γ}} otherwise.

The image of the map

∏_{α∈Λ} F (j_{α}) : F (π) → ∏_{α∈Λ} F (i_{α}(M_{α})) = ∏_{α∈Λ} G_{α}^{F} (M_{α})

is contained in ∏_{μ^{F}} G_{α}^{F} (π). Therefore, the map

Θ_{π} := ∏_{α∈Λ} F (j_{α}) : F (π) → ∏_{μ^{F}} G_{α}^{F} (π) ⊂ ∏_{α∈Λ} G_{α}^{F} (M_{α}).

is well-defined.

The family Θ = {Θ_{π}} : F → ∏_{μ^{F}} G_{α}^{F} is a natural transformation. Let now μ = (μ_{β}^{α} : G_{α} → G_{β}) be an inductive system of natural

transformations of bundle functors G_{α} : Mf → FM. Consider the corresponding bundle functor F := ∏_{μ} G_{α} : FM_{Λ} → FM. Denote by μ^{∘} = (μ^{∘}_{β}^{α} : G^{∘}_{α} → G^{∘}_{β}) the corresponding inductive system of natural transformations (3). Then

G^{∘}_{γ}(X) = {x_{α} ∈ G_{α}(X_{α}) | (μ^{F})_{β}^{α}(M_{α})(x_{α}) = G_{β}^{F} (π_{α}^{β})(x_{β})} ⊂ ∏_{α∈Λ} G_{α}(X_{α}),

where X_{α} = X for α ≥ γ, otherwise X_{α} = pt.

For any manifold X and for any α ∈ Λ, we define the map

O_{α}(X) : G^{∘}_{α}(X) → G_{α}(X)  (5)

as the restriction of the standard projection ∏_{β∈Λ} G_{β}(X_{β}) → G_{α}(X).

The families

O_{α} = {O_{α}(X)} : G^{∘}_{α} → G_{α}

are natural transformations. They all are natural equivalences if and only if every map μ_{β}^{α}(pt) : G_{α}(pt) → G_{β}(pt) is a diffeomorphism. The diagram

G^{∘}_{α}(X) \underrightarrow{O_{α}(X)} G_{α}(X)
μ^{∘}_{β}^{α}(X)↓  ↓μ_{β}^{α}(X)
G^{∘}_{β}(X) \underrightarrow{O_{β}(X)} G_{β}(X)

is commutative for any manifold X and any α ≤ β.

Suppose now that the inductive system μ = (μ_{β}^{α} : G_{α} → G_{β}) satisfies the condition that all the maps μ_{β}^{α}(pt) : G_{α}(pt) → G_{β}(pt) are diffeomorphisms.

For any multifibered manifold π = (M_{α}, π_{α}^{β}, Λ, M) we let

T^{μ}(π) = \{ \begin{matrix}G_{α}(X) & if π = i_{α}(X) for some X ∈ Mf, α ∈ Λ, \\ ∏_{μ} G_{α} (π) & otherwise. \end{matrix}

Then T^{μ}(π) is a fibered manifold over M = lim_{⟵} M_{α}. We also define the map

I_{π} : T^{μ}(π) → ∏_{μ} G_{α} (π)

as follows:

I_{π} = \{ \begin{matrix}O_{α}^{−1}(X) & if π = i_{α}(X), \\ id_{Π_{μ} G_{α} (π)} & otherwise, \end{matrix}

where O_{α}(X) : G^{∘}_{α}(X) = ∏_{μ} G_{α} (i_{α}(X)) → G_{α}(X) are defined by (5). We let

T^{μ}(f) := I−1 ; \overline{π} ∘ ∏_{α∈Λ} G_{α}(f_{α}) ∘ I_{π}.

The correspondence T^{μ} thus defined is a bundle functor FM_{Λ} → FM, and the family

I = {I_{π}} : T^{μ} → ∏_{μ} G_{α}

is a natural transformation.

If all G_{α} preserve products, then G_{α}(pt) = pt, hence the maps μ_{β}^{α}(pt) are diffeomorphisms. In this case, I is a natural equivalence and the functor T^{μ} also preserves products.

Let now \overline{μ} = (\overline{μ}_{β}^{α} : \overline{G}_{α} → \overline{G}_{β}) be another inductive system of natural transformations such that all \overline{μ}_{β}^{α}(pt) are diffeomorphisms, and let ν = (ν_{α} : G_{α} → \overline{G}_{α}) be a family of natural transformations such that the diagram (1) is commutative. Following Mikulski, we define a natural transformation ν̃ = \{ν̃_{π}\} : T^{μ} → T \overline{μ} to be the composition

ν̃_{π} : T^{μ}(π) ⟶^{I_{π}} ∏_{μ} G_{α} (π) ⟶^{Π_{μ,\overline{μ}} ν_{α}(π)} ∏\overline{μ} \overline{G}_{α} (π) ⟶^{\overline{I}_{π}^{−1}} T \overline{μ}(π)

for any multifibered manifold π.

In the case F = T^{μ} the natural transformations (μ^{F})_{β}^{α} : G_{α}^{F} → G_{β}^{F} coincide with μ_{β}^{α}, i.e.,

μ^{F} = μ if F = T^{μ}.

Let F : FM_{Λ} → FM be a bundle functor such that (μ^{F})_{β}^{α}(pt), α ≤ β, are diffeomorphisms.

Then we define a natural transformation κ = {κ_{π}} : F → T^{μ^{F}} to be the composition

κ_{π} : F (π) ⟶^{Θ_{π}} ∏_{μ^{F}} G_{α}^{F} (π) ⟶^{I_{π}^{−1}} T^{μ^{F}} (π)  (6)

for any multifibered manifold π.

The proofs of the following propositions are similar to the proofs of Theorems 2.1 and 2.2 in [9].

Proposition 3.2. (1) Let F : FM_{Λ} → FM be a product preserving bundle functor. Then the natural transformation κ : F → T^{μ^{F}} is a natural equivalence.

(2) If μ = (μ_{β}^{α} : G_{α} → G_{β}) is an inductive system of natural transformations between product preserving bundle functors G_{α} : Mf → FM and κ is the natural transformation (6) for F = T^{μ}, then κ : T^{μ} → T^{μ} and κ_{π} = id_{T^{μ}(π)} for any multifibered manifold π.

(3) For μ = (μ_{β}^{α} : G_{α} → G_{β}) the functor T^{μ} is a product preserving bundle functor on the category FM_{Λ} unique up to a natural equivalence such that the natural transformation μ^{F} corresponding to F = T^{μ} coincides with μ.

Proposition 3.3. Let F, \overline{F} : FM_{Λ} → FM be two product preserving bundle functors. Let μ^{F} = ((μ^{F})_{β}^{α} : G_{α}^{F} → G_{β}^{F}) and μ\overline{F} = ((μ\overline{F})_{β}^{α} : G\overline{F} ; α → G\overline{F} ; β) be the corresponding inductive systems of natural transformations. Let ν = (ν_{α} : G_{α}^{F} → G\overline{F} ; α) be the family of natural transformations such that the diagram

G_{α}^{F} (X) \underrightarrow{ν_{α}(X)} G\overline{F} ; α (X)
(μ^{F})_{β}^{α}(X)↓  ↓(μ\overline{F})_{β}^{α}(X)
G_{β}^{F} (X) \underrightarrow{ν_{β}(X)} G\overline{F} ; β (X)

is commutative for any manifold X. Then the natural transformation η = \{η_{π}\} : F → \overline{F} given by the compositions

η_{π} : F (π) ⟶^{κ_{π}} T^{μ^{F}} (π) \underrightarrow{ν̃_{π}} T^{μ\overline{F}} (π) ⟶^{\overline{κ}_{π}^{−1}} \overline{F} (π)

is the unique natural transformation F → \overline{F} such that ν_{α}^{η} = ν_{α}, where ν_{α}^{η} is defined by (4).

Definition. We say that two bundle functors F and \overline{F} are equivalent if there exists a natural equivalence η : F → \overline{F}. We say that two inductive systems of natural transformations μ and \overline{μ} are equivalent if there exists a family ν = (ν_{α}) of natural transformations such that the diagram (1) is commutative for any manifold X.

The following proposition completes the proof of Theorem 3.1. It is proved just the same as Corollary 2.3 in [9].

Proposition 3.4. The correspondence F → μ^{F} induces a bijection between the equivalence classes of product preserving bundle functors on the category FM_{Λ} and the equivalence classes of inductive systems of natural transformations of product preserving bundle functors on the category Mf. The inverse bijection is induced by the correspondence μ → T^{μ}.

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Geometry Department, Branch of Mathematics, Chebotarev Research Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya, 17, Kazan, 420008, Russia E-mail address: vadimjr@ksu.ru

Received May 24, 2007