The holonomy groupoids of singularly foliated bundles

We define a notion of connection in a fibre bundle that is compatible with a singular foliation of the base. Fibre bundles equipped with such connections are shown to simultaneously generalise regularly foliated bundles in the sense of Kamber-Tondeur, bundles that are equivariant under the actions Lie groupoids with simply connected source fibres, and singular foliations. We define hierarchies of diffeological holonomy groupoids associated to such bundles, and in so doing recover the holonomy groupoids of singular foliations defined by Androulidakis and Skandalis as special cases. Finally we prove functoriality of all our constructions under appropriate morphisms.


Introduction
In this paper, we extend the notion of a partial connection in a fibre bundle to the singular setting, obtaining singular partial connections. Fibre bundles with singular partial connections, which we refer to as singularly foliated bundles, simultaneously generalise singular foliations, regularly foliated bundles, and bundles that are equivariant under the actions of Lie groupoids with simply connected source fibres. We use certain diffeological pseudo-bundles consisting of germs of sections to systematically construct hierarchies of holonomy groupoids for singularly foliated bundles as diffeological quotients of path spaces, and show that our constructions are functorial under suitably defined morphisms of singularly foliated bundles. In particular, we recover the well-known holonomy groupoid of any singular foliation [2] as a special case.
Singular foliations are involutive, locally finitely generated families of vector fields on manifolds. As famously proved by Stefan [54] and Sussmann [55], such objects integrate to give decompositions of their ambient manifolds into immersed submanifolds, possibly of differing dimension, called leaves. Singular foliations are ubiquitous in mathematics and its applications.
For instance, every Poisson manifold has a singular foliation by symplectic leaves, and conversely a singular foliation of a manifold by symplectic leaves suffices to determine a Poisson structure [27]. More generally, any integrable Dirac manifold admits a singular foliation by presymplectic leaves [25]. Singular foliations also generalise regular foliations, which are among the primary instances of Connes' noncommutative geometries [18,20].
An essential construction for the noncommutative perspective is the holonomy groupoid of a regular foliation, which was introduced by Winkelnkemper [57] as a model for the leaf space. As described by Phillips [51], the holonomy groupoid is in a precise sense the smallest desingularisation of the naive "space of leaves" obtained as the quotient by leaves that admits a (locally Hausdorff) manifold structure. It is upon the locally Hausdorff holonomy groupoid of a foliation (orétale versions thereof) that a great deal of progress has been made in index theory [24,15,39,9,35,10,13,16,11,8,12,47,45] and equivariant cohomology [19,23,33,21,34,28,49,22,48,44]. An alternative toolbox for the study of regular foliations that has been developing since the nineteen-nineties is diffeology [37,38], which provides a way of doing differential topology on conventionally badly behaved spaces X by declaring which maps from Euclidean domains into X are smooth. Recent progress by the author in this area [46] shows that the holonomy groupoid of a regular foliation is just the largest of an infinite family of diffeological holonomy groupoids constructed using solutions of parallel transport differential equations in diffeological bundles. Thus, while the Winkelnkemper holonomy groupoid is the smallest Lie groupoid that integrates a regular foliation, it is far from being the smallest diffeological groupoid that does so.
Defining holonomy groupoids for singular foliations dates back to the mid nineteen-eighties with work of Pradines and Bigonnet [14,52]. Significant further progress was made by Debord in [29,30] in the study of holonomy groupoids for singular foliations arising from Lie algebroids whose anchor maps are injective on a dense set (these types of foliations are now known as Debord foliations [43,Definition 3.6]). Such foliations are special in that their holonomy groupoids are Lie groupoids. At present, the most general family of singular foliations for which holonomy groupoids can be defined are those associated to locally finitely generated, involutive families of vector fields, in the spirit of those originally studied by Stefan and Sussmann. The holonomy groupoids of such general foliations were formulated by Androulidakis and Skandalis in [2].
Holonomy groupoids at this level of generality are topologically pathological, but, as is evident in the recent preprint [56] of Garmendia and Villatoro, are diffeologically quite well-behaved. The years since the Androulidakis-Skandalis construction have seen a great deal of further research conducted into singular foliations and their holonomy, see for instance [4,3,6,31,5,32,56].
The present paper constitutes a generalisation of the holonomy groupoid constructions in [46] to singular foliations, and is inspired in part by the recent work of Garmendia and Villatoro [56], who showed how to recover the Androulidakis-Skandalis holonomy groupoid as a quotient of a diffeological path space. In the author's view, the primary contribution of this paper is a novel perspective on the holonomy of singular foliations that closely resembles the classical theory of holonomy in fibre bundles defined via solutions to parallel transport differential equations.
In particular, this places the holonomy of singular foliations in the same realm of differential geometry that deals with symmetries of differential equations in the sense of [1,50]. In addition, the diffeological pseudo-bundles of germs that we introduce in this paper are shown to be extensions of jet bundles, which are closely related to (but distinct from) classical objects in sheaf theory such asétale spaces of sheaves. We believe that these pseudo-bundles may be of independent interest and utility. Let us now outline the content of the paper in more detail.
Section 2 consists of a recollection of the well-known definitions and results from singular foliations, jet bundle theory and diffeology that will be required for our constructions later in the paper. We remark here that our notation Γ k for the k th order jet bundles differs from the J k that is usually seen in the literature -this is to ensure compatibility with the pseudo-bundles of germs that we introduce in the following section. In particular, the k-jet at a point x ∈ M of a particular section σ of a fibre bundle over M is denoted rather than the usual j k x σ. Section 3 is where we introduce the key diffeological constructions with which the holonomy groupoids of singularly foliated bundles can be systematically constructed. In particular, we associate to any sheaf S of smooth sections of a fibre bundle π B : B → M over a manifold M a canonical diffeological pseudo-bundle Γ g (S ) over M , whose fibre over x ∈ M consists of all the germs [σ] g x at x of elements σ of S defined around x. When S is the sheaf of all sections of π B , the "pseudo-bundle of germs" Γ g (π B ) ought to be thought of as a "completion" of the usual tower of jet bundles Γ k (π B ) associated to π B , which is sufficiently rich to capture the behaviour of non-analytic smooth sections. The concept of jet prolongation of a vector field to a jet bundle is extended to germinal prolongation of a vector field to a bundle of germs, which is a crucial component in the definition of our holonomy groupoids.
We include in Section 3 a discussion of the relationship between pseudo-bundles of germs and classical sheaf theoretic concepts. In particular, we show in Proposition 3.9 that any suitably smooth morphism of sheaves of sections induces a morphism of the corresponding pseudo-bundles of germs, following which we give a counterexample to the converse being true.
Finally, Proposition 3.12 and Example 3.13 show that while the pseudo-bundle of germs of a sheaf is isomorphic as a set to the well-knownétale space associated to the sheaf, the topology it inherits from its diffeology is (often strictly) contained in the usualétale topology. These considerations regarding the topology of pseudo-bundles of germs are not required anywhere in our constructions, and are included out of independent interest. Section 3 concludes by recalling the diffeological path categories P(X ) of diffeological spaces X introduced in [46], and generalises the leafwise path category of a regular foliation introduced therein to the singular case. Elements of the leafwise path category P(F ) of a singular foliation F are triples where X is some locally-defined vector field in F, d > 0 a real number, and γ : M an integral curve of X such that X vanishes in a neighbourhood of γ(0) and of γ([d, ∞)).
This definition draws from the analogous definition used by Garmendia and Villatoro in [56].
We also define an abstract notion of holonomy groupoid associated to a lifting map into paths of a pseudo-bundle, which serves as the framework for the constructions of Section 4.
In Section 4 we generalise the notion of a singular foliation to a singularly foliated bundle.
In the same way that regularly foliated bundles in the sense of Kamber and Tondeur [41] are defined in terms of a partial connection on the total space of the bundle, our singularly foliated bundles are defined in terms of what we call singular partial connections. Roughly speaking, a singular partial connection ℓ in a fibre bundle over a singular foliation allows us to lift vector fields from the foliation of the base to fields on the total space. We show that singularly foliated bundles simultaneously generalise singular foliations, regularly foliated bundles, and bundles which are equivariant under the actions of Lie groupoids with simply connected source fibres. Now associated to any singularly foliated bundle π B : B → M with foliation F of the base are pseudo-bundles π k,F B : Γ k (π B ) F → M of germs/jets of sections which are locally invariant under flows of F. These pseudo-bundles generalise the "bundles of distinguished sections" considered in [46]. We prove the following. Theorem 1.1 (Theorem 4.10). Let π B : B → M be a singularly foliated bundle. Let k denote any of the symbols 0, . . . , ∞, g. Then for each (γ, [X] g , d) ∈ P(F) and for each (x, is the unique solution to the initial value problem is smooth. Here Fl denotes the flow and p k (X) denotes the k-prolongation of the vector field X, while (γ, [X] g , d) and [σ] k x are as in Equations (2) and (1) respectively.
In words, Theorem 4.10 simply says that conjugation by the flows of an element of F solves a parallel transport differential equation, and defines a lifting map from leafwise paths to paths in Γ g (π B ) F . Each of the lifting maps L(π k,F B ) of a singularly foliated bundle π B : B → M induces a transport functor from the leafwise path category P(F) to the diffeological groupoid of diffeomorphisms between the fibres of Γ k (π B ) F . The fibres of this functor determine an equivalence relation on P(F ), and the quotient of P(F) by this equivalence relation is the holonomy groupoid H(π k,F B ). The arguments of [46,Theorem 5.15] then apply to show that we have a hierarchy of diffeological holonomy groupoids associated to the tower of germ/jet bundles.
Following this, we prove in Theorem 4.15 that in the case of a singular foliation (M, F ), with M of dimension n, the holonomy groupoid H(π g,F M ×R n ) associated to the trivial singularly foliated bundle M ×R n → M recovers the holonomy groupoid of Garmendia-Villatoro. Thus, by [56,Theorem 5.5], our construction also generalises the holonomy groupoid of Androulidakis-Skandalis [2]. Section 4 is concluded by defining a class of morphisms of singularly foliated bundles, which generalises the morphisms described by Definition 6.11], and we prove in Theorem 4.18 that the hierarchy (3) of holonomy groupoids is functorial under such morphisms. The paper is concluded in Section 5 with a discussion of some open questions.

Acknowledgements
defined by the vector field X.

Singular foliations
We begin by recalling the standard sheaf-theoretic definition of a singular foliation.
One can alternatively describe a singular foliation of a manifold M as a locally finitely generated submodule of the compactly supported vector fields on M which is closed under Lie brackets, as in [2]. By [32, Remark 1.8] these definitions are equivalent. One of the most important facts regarding singular foliations is the Stefan-Sussmann integration theorem [54,55].
Theorem 2.2. Let M be a manifold with a singular foliation F . Then F integrates to give a decomposition of M into smoothly immersed submanifolds called leaves.
The following important theorem, due to Androulidakis and Skandalis [2, Theorem 0.1], says that the leaves of a singular foliation always arise as the orbits of a certain topological groupoid, called the holonomy groupoid of the foliation.  The groupoid H(F ) is called the holonomy groupoid of F .
Androulidakis and Skandalis built their groupoid using "bisubmersions" defined by iterated flows of the vector fields defining the foliation. Recent work by Garmendia and Villatoro [56] recovers the Androulidakis-Skandalis holonomy groupoid as a diffeological quotient of a certain path space. It is the Garmendia-Villatoro construction that most closely resembles the construction we give in this article.  3. Debord foliations, associated to algebroids whose anchor maps are injective on a dense set. The integration problem for these foliations was solved by Debord in [29,30] We will not be making use of Lie algebroids in this article. Like the constructions of Androulidakis-Skandalis and Garmendia-Villatoro, our constructions will be founded on the more general Definition 2.1.

Jet bundles and prolongation
We recall in this subsection some well-known theory of jet bundles, drawn primarily from [53,1].
Although the reader is likely familiar with this theory already, we include the following outline both to introduce our rather unconventional notation (which we choose for consistency with the bundles of germs to be introduced in Subsection 3.1) and to point out the structures that will be most relevant in our constructions. Definition 2.5. Let π B : B → M be a fibre bundle, and let k ≥ 0. We say that two local sections σ 1 and σ 2 of π B defined in a neighbourhood of x ∈ M have the same k-jet at x if σ 1 (x) = b = σ 2 (x), and for any local for all i = 1, . . . , dim(F ), and all multi-indices I with |I| ≤ k. Having the same k-jet at a point x is an equivalence relation on the set of local sections defined about x, and we denote the k-jet equivalence class of any such local section by [σ] k x .
The k-jets of local sections fit into a fibre bundle in a natural way.
Definition 2.6. Let π B : B → M be a fibre bundle, and let k ≥ 0. For each x ∈ M , denote the set of all k-jets of local sections defined near x by Γ k (π B ) x , and define and observe that any choice of local coordinate trivialisation ( on the set (π 0,k B ) −1 (O). These coordinates give the projection π k : Γ k (π B ) → M the structure of a fibre bundle, called the k th order jet bundle of π B .
The jet bundles of a fibre bundle π B : B → M admit projections π k,l B : Γ l (π B ) → Γ k (π B ) defined by for any l ≥ k, and these projections form a projective system. The projective limit of this system, denoted Γ ∞ (π B ), inherits a natural smooth structure as a projective limit of manifolds, which may be equivalently thought of as arising from the projective limit diffeology [40, Section 1.39]. The space Γ ∞ (π B ) is usually identified with the set of ∞-jets of local sections of π B , and admits a canonical projection π ∞ B : Γ ∞ (π B ) → M given by One therefore obtains a hierarchy of jet bundles Since we will be concerned primarily with singular foliations, which arise from families of vector fields, we will need to know about vector fields on jet bundles. A particularly important class of vector fields on fibre bundles, in which we will be primarily interested, is those that are projectable in the following sense.
Definition 2.7. Let π B : B → M be a fibre bundle. A vector field X on B is said to be projectable if there is a vector field (π B ) * (X) on M for which on all of B. We denote by X proj (B) the set of projectable vector fields on B.
Projectable vector fields on a bundle prolong in a natural way to vector fields on the associated jet bundles.
Definition 2.8. Let π B : B → M be a fibre bundle, and let X ∈ X proj (B) be a projectable vector field. For 1 ≤ k ≤ ∞, the k-jet prolongation of X is the vector field p k (X) ∈ X(Γ k (π B )) defined by Suppose now that π B : B → M is a fibre bundle and X is a projectable vector field on B which, in coordinates (4), is given by X = a i ∂ ∂x i + b α ∂ ∂f α , where a i are smooth functions depending only on the x i while the b α depend on both the x i and the f α . Then for any 1 ≤ k < ∞, the k-jet prolongation p k (X) of X is given in coordinates over a point ( where the sum over J ⊂ I is a sum over all strict subsets of the multi-index J, and where we have absorbed the constants arising from the symmetry of mixed partial derivatives into our notation as in [1,Equation 1.15]. For k = ∞, the formula simplifies to The prolongation formula (6) can be used to deduce the following fact, which justifies our choice of notation in denoting the image of p k in X(Γ k (π B )) by X proj (Γ k (π B )).
Proposition 2.9. Let π B : B → M be a fibre bundle, and let X be a projectable vector field on B. Then the p k (X) are a projectable family, in the sense that Via the jet prolongation operators, therefore, we can think of projectable vector fields on a fibre bundle π B : B → M as defining a tower of projectable vector fields associated to the tower of jet bundles for π B .

Diffeology
We recall in this subsection some basic objects of study in diffeology that will be relevant for our constructions. The most comprehensive reference on diffeology is the wonderful book [40] by P. Iglesias-Zemmour.
Definition 2.11. A function ϕ : U → X from an open subset U of some finite-dimensional Euclidean space to a set X is called a parametrisation. A diffeology on a set X is a family D of parametrisations satisfying the following axioms.
1. The family D contains all constant parametrisations.
2. If ϕ : U → X is a parametrisation such that every point u ∈ U has an open neighbourhood V ⊂ U for which ϕ| V is an element of D, then ϕ itself is an element of D.
3. For every element ϕ : U → X of D, every open set V of any finite-dimensional Euclidean space, and for every smooth function f : A set with a diffeology is called a diffeological space, and the elements of the diffeology are called its plots. If X and Y are two diffeological spaces, then a function f : X → Y is said to be smooth if for every plot ϕ : A smooth bijection of diffeological spaces is said to be a diffeomorphism if it is smooth with smooth inverse.
Every manifold is a diffeological space, with diffeology constituted by the set of all parametrisations that are smooth in the usual sense. Moreover a map between manifolds is smooth in the manifold sense if and only if it is smooth in the diffeological sense. Thus the category of manifolds and smooth maps is a full and faithful subcategory of the category of diffeological spaces and smooth maps.
Diffeologies can be pushed forward and pulled back by functions of sets. This fact will be invoked frequently for our constructions.
Definition 2.12. Let X and Y be sets, and let f : X → Y be a function. The following special cases are of particular importance. Let X be a diffeological space.
1. If ∼ is any equivalence relation on X , then the quotient diffeology X / ∼ is the pushfoward diffeology arising from the quotient X → X / ∼.

If
S is any subset of X , then the subspace diffeology on S is the pullback diffeology arising from the inclusion S ֒→ X .
3. If Y is any other diffeological space, then the product diffeology on X × Y is the smallest diffeology for which the projections onto the factors are subductions.
Quotients, subspaces and products will always be assumed to be equipped with the respective diffeologies defined above unless otherwise stated.
One of the features of the category of diffeological spaces is that the set of all morphisms between any two objects in the category is itself an object.
Definition 2.13. Let X and Y be diffeological spaces, and denote by C ∞ (X , Y) the set of all smooth maps f : to be a plot if and only if for every plot ϕ : V → X of X , The familiar notion of a fibre bundle over a manifold has a far reaching generalisation to diffeological spaces. It is the flexibility afforded by this generalisation that permits most of the constructions in this paper.
Definition 2.14. A diffeological pseudo-bundle is a subduction π B : B → X of diffeological spaces. Such a pseudo-bundle is in particular called a diffeological vector pseudo-bundle if each fibre of π B is a vector space for which the vector space operations are smooth with respect to the subspace diffeology, and such that the zero section is smooth. Definition 2.15. Let C be a small category, with object set identified as a subset of the morphism set via the map which sends each object to its associated identity morphism. We say that C is a diffeological category if its set of morphisms is equipped with a diffeology for which the range, source, and composition are all smooth. If C is in addition a groupoid, whose inversion map is smooth, we call C a diffeological groupoid.
Let now π B : B → X be a smooth surjection of diffeological spaces. Denote by Aut(π B ) the groupoid with object set X , and with morphisms from x to y constituted by the set Diff(B x , B y ) of all diffeomorphisms from the fibre B x over x to the fibre B y over y, with the obvious range, source, inversion and composition. The groupoid Aut(π B ) admits a smallest diffeology, called the functional diffeology, under which the evaluation map ev : is smooth, and under which Aut(π B ) is a diffeological groupoid (see [40, 8.7] for details).
Definition 2.16. Let π B : B → X be a surjection of diffeological spaces. Equipped with the functional diffeology, we refer to Aut(π B ) as the structure groupoid of the surjection π B . We say that π B is a diffeological bundle if the characteristic map (r, s) : Aut(π B ) → X × X is a subduction.
Diffeological bundles, unlike general diffeological pseudo-bundles, have a typical fibre to which all other fibres are diffeomorphic, and the pullback of a diffeological bundle along any plot is locally trivial [40, p. 240].
By definition, singular foliations arise from certain families of sections of tangent bundles.
To use diffeology to study singular foliations, therefore, we need a notion of tangent bundle for a diffeological space. A number of definitions have been proposed for this purpose, which, while coincident for manifolds, do not coincide for general diffeological spaces (see [17] for a detailed discussion). The point of view that we find useful here, as in [46], is that of internal tangent spaces and bundles. The paper [17] provides a categorical definition of internal tangent spaces based on work of Hector [37], which may be summarised as follows.
Definition 2.17. Let X be a diffeological space and let x ∈ X . Denote by p x the set of all plots centered at x, that is, plots ϕ : U → X such that 0 ∈ U and ϕ(0) = x. Denote by T 0 dom(ϕ) the tangent space at zero of the domain of any such plot, and by v ϕ the image of v ∈ T 0 dom(ϕ) by the subspace generated by all vectors of the form is any smooth function for which the germs of ϕ and ϕ ′ • f at zero are equal. The class of an element v ϕ , v ∈ T 0 dom(ϕ), will be denoted ϕ * ( v).
Let X be a diffeological space, and consider the set T X := x∈X T x X . For any plot ϕ : U → X and for any u ∈ U , denote by These maps were first considered by Hector [37]. Then there exists a smallest diffeology on T X , called the dvs diffeology [17], for which the natural projection π T X : T X → X is a diffeological vector pseudo-bundle, and which contains the parametrisations dϕ : T U → T X as plots.
The internal tangent bundle is functorial under smooth maps of diffeological spaces.
x ∈ X and extend by linearity to a map df : T X → T Y. Then df is a smooth map [17, Proposition 4.8] called the pushforward or differential of f .
Finally, we recall that every diffeological space admits a natural topology with respect to which all plots are continuous.
Definition 2.20. Let X be a diffeological space. The D-topology on X is the topology whose open sets are precisely those sets A ⊂ X for which ϕ −1 (A) is open for all plots ϕ of X .
For manifolds, the D-topology coincides with the usual topology. Although the D-topology will not play a central role in any of our constructions, we will see in Subsection 3.2 that it gives theétale space of any sheaf of sections of a fibre bundle a natural topology which is distinct from the usualétale topology.
) for all u ∈ V and for which the Then the collection of parametrisations with the property funct defines a diffeology on S loc .
Proof. Observe first that ifσ :  Let us now consider the diffeological subspace There is then clearly a surjective, smooth map π Γ loc (S ) : which is moreover a subduction. Indeed, for any plotx : U → M and for any u ∈ U we can we have that π Γ loc (S ) • ρ =x, making π Γ loc (S ) a subduction as claimed. The fibre Γ loc (S ) x over any x ∈ M is the nonempty space consisting of sections σ of S defined on some open neighbourhood of x, equipped with the functional diffeology of Definition 3.2. The subduction π Γ loc (S ) is the first step on the way to defining a genuinely useful object. Our next example shows why π Γ loc (S ) is too large to be of much use in its own right. We define an equivalence relation ∼ g on Γ loc (S ) by declaring (x, σ) ∼ g (y, η) if and only if x = y and We denote by Γ g (S ) the diffeological quotient of Γ(S ) by the equivalence relation ∼ g , and denote by π g S : Γ g (S ) → M the obvious surjection Since both the quotient map q : Γ loc (S ) → Γ g (S ) and the projection Γ loc (S ) → M are subductions, so too is the projection π g S : Γ g (S ) → M . Thus π g S : Γ g (S ) → M is a diffeological pseudo-bundle. For each k ≤ ∞ we have a canonical projection π k,g B : Γ g (π B ) → Γ k (π B ) onto the k th order jet bundle of π B defined by The arguments of [46,Proposition 5.14] show that these projections are smooth, and are compatible with the jet projections π l,k B : Γ k (π B ) → Γ l (π B ) in the sense that π l,g B = π l,k B • π k,g B for all l ≤ k. We therefore have a tower Definition 3.5. Let π B : B → M be a fibre bundle. For X ∈ X proj (B), the vector field p g (X) on Γ g (π B ) defined by the formula is called the germinal prolongation of X. The associated linear map p g : X proj (B) → X(Γ g (π B )) is called the germinal prolongation operator, and its image is denoted X proj (Γ g (π B )).
Our next result relates the germinal prolongation operator to the jet prolongations of projectable vector fields, and can be seen as a justification of the nomenclature "germinal prolongation".
Proposition 3.6. Let π B : B → M be a fibre bundle. Then for each k ≤ ∞, we have for all X ∈ X proj (B). Consequently the tower of prolongations of Proposition 2.10 completes to a tower . . .
Proof. For any k ≤ ∞ and X ∈ X proj (B), we have The injectivity of the jet prolongation operators apparent from Equation (6) together with Proposition 3.6 implies that the germinal prolongation operator p g : X proj (B) → X(Γ g (π B )) of a bundle π B : B → M is injective. Consequently, on X proj (Γ g (π B )) we have a Lie bracket that is well-defined by the formula with respect to which each (π k,g B ) * is a homomorphism of Lie algebras. While we will not be making use of this feature in this article, we remark that it distinguishes X proj (Γ g (π B )) as a rather special subspace of X(Γ g (π B )), which, like the vector fields of many other diffeological spaces [17], does not carry a natural Lie bracket in general.
It is in examples arising from singular foliations that one sees the justification for the terminology "pseudo-bundle" in that the fibres of a pseudo-bundle of germs need not be isomorphic in general.
for X, Y ∈ Γ(F ) x and α ∈ R. Then these Lie algebra operations are well-defined and smooth with respect to the subspace diffeology on Γ g (F) x ⊂ Γ g (F ), and π g F : Γ g (F ) → M is a diffeological vector pseudo-bundle of Lie algebras.
Proof. Well-definedness of the operations follows from the comments made in Example 3.3. Let us therefore check only smoothness. LetX : U → Γ g (F ) x andỸ : V → Γ g (F ) x be plots. Then we may assume U and V to be sufficiently small so as to be equal to composites follows from a coordinate calculation, and now by smoothness of q x , the map is smooth, making the Lie bracket smooth as claimed.

Relationship with sheaves
In this subsection we present some results and examples which relate our pseudo-bundles of germs to more well-known objects arising in sheaf theory. The first such result, which is of crucial importance in defining the correct notion of morphism between singularly foliated bundles, is that a smooth morphism of sheaves gives rise to a morphism of the associated pseudo-bundles. Proposition 3.9. Let M be a manifold and let S 1 and S 2 be sheaves of sections of fibre bundles π B 1 and π B 2 over M respectively. Suppose thatF : S 1 → S 2 is a morphism of sheaves for which the induced morphism S 1 loc → S 2 loc is smooth (see Proposition 3.1). Then the formula defines a morphism F : Γ g (S 1 ) → Γ g (S 2 ) of diffeological pseudo-bundles.
Proof. It is clear that F preserves fibres, so we need only check smoothness. Since for each i = 1, 2 the quotient diffeology on Γ g (S i ) is inherited from the functional diffeology on S i loc , smoothness of the map S 1 loc → S 2 loc associated toF ensures smoothness of F .
where m x f denotes the function y → xf (y). Then F is smooth -indeed, if U is any open subset of R n andx : U → R andf : U → C ∞ (R, R) are any two plots, then for each x ∈ R, smoothness of (u, y) → xf (u)(y) ) is smooth. Now suppose thatF : C ∞ R → C ∞ R is a morphism of sheaves. Then for F to be induced byF , we must in particular have [F (id)] g 0 = [0 id] g 0 = 0, so that there must exist ǫ > 0 for whichF (id) vanishes identically on (−ǫ, ǫ). However, for x ∈ (−ǫ, ǫ) \ 0, we have Thus F cannot arise from any morphism of sheaves.
Morphisms of pseudo-bundles of germs which arise from morphisms of sheaves in the sense of Proposition 3.9 will play an important role in the correct notion of morphism between singularly foliated bundles. We thus record the following definition.  The next example shows that these topologies typically do not coincide -that is, the Dtopology is often strictly coarser than theétale topology.
Example 3.13. Consider again M = R and B = R × R the trivial bundle with π B : B → M the projection onto the first factor. Fix x 0 ∈ R, and consider the plot ρ : where f t is the map x → tx. Considering Γ g (π B ) with itsétale topology, the set is open in Γ g (π B ). However, is not open. Thereforeétale-open sets in Γ g (π B ) need not be open in the D-topology.

The leafwise path category
In [46], we introduced a diffeological version of the Moore path category for any regular foliation (M, F ). The objects of this category are simply points in M , while the morphisms are smooth, leafwise paths which have sitting instants in that they are constant in small neighbourhoods of their endpoints. Composition of morphisms in this category is simply concatenation of paths. In [56], the authors introduce an analogous diffeological space for singular foliations, however concatenation of paths in this space no longer defines a category. In this section, we introduce a hybrid of these two approaches -a diffeological space of integral curves of vector fields defining a singular foliation, for which concatenation of paths defines an associative and smooth multiplication.
We begin by recalling the definition of the path category of a diffeological space from [46] Definition 3.14. Let X be a diffeological space. The path category of X is the diffeological subspace P(X ) of the diffeological product C ∞ (R ≥0 , X )× R ≥0 consisting of pairs Given any diffeological space X , range and source maps r and s mapping P(X ) → X are defined respectively by (γ, d) → γ(d) and (γ, d) → γ(0), and whenever r(γ 2 , d 2 ) = s(γ 1 , d 1 ), we define the product (γ 1 γ 2 , d 1 + d 2 ) of (γ 1 , d 1 ) and (γ 2 , d 2 ) by the formula This product, together with the range and source maps, are smooth, so that P(X ) is a diffeological category [46,Proposition 3.22]. Moreover [46,Proposition 3.23] there is a smooth involution ι : P(X ) ∋ (γ, d) → (γ −1 , d) → P(X ) defined by the formula Under favourable circumstances, which will be explicated in this section, the involution ι descends to a genuine inversion on certain diffeological quotients of P(X ), giving such quotients the structures of diffeological groupoids.
We will implicitly use the notation of Equation (7) in what follows.  such that T can be written as the composite Note that smoothness of T follows from smoothness of L. If in particular P (Y) = P(F) is the leafwise path category of some singularly foliated manifold (M, F ), we refer to T as a leafwise transport functor.
An important consequence of the existence of a transport functor is the existence of an associated groupoid called the holonomy groupoid. This can be seen by the arguments of [46,Proposition 3.27].
Definition 3.17. Let X be a diffeological space, π B : B → X and π Y : Y → X diffeological pseudo-bundles, and P (Y) a diffeological subcategory of P(Y). If T : P (Y) → Aut(π B ) is a transport functor, then the quotient of P (Y) by the equivalence relation generated by the fibres of T is a diffeological groupoid called the holonomy groupoid associated to T .

Singularly foliated bundles and their holonomy groupoids 4.1 Singularly foliated bundles
Singular foliations are generalisations of Lie groupoids and of regular foliations. In each of these special cases, one has a notion of fibre bundle which is compatible with the additional structure -in the case of a Lie groupoid action, the correct notion is that of an equivariant bundle, while for a regular foliation the correct notion is that of a foliated bundle in the sense of Kamber and Tondeur [41]. We give in this section what appears to be the first definition of a fibre bundle compatible with a singular foliation, which simultaneously generalises equivariant and foliated bundles.
First, notice that projectable vector fields on a fibre bundle π B : B → M over a manifold M do not generally form a sheaf of C ∞ B -modules over B. Indeed, if X is any projectable vector field and f ∈ C ∞ (B) is any function which is non-constant along the fibres of π B , then Equation , which we call the sheaf of projectable functions. We denote by X proj,B the sheaf of C ∞ proj,B -modules which we call the sheaf of projectable vector fields.
The pushforward of projectable vector fields can now be characterised in the following sheaftheoretic fashion.
Proposition 4.2. Let π B : B → M be a fibre bundle. The pushforward of projectable vector fields induces a morphism (π B ) * : (π B ) ! X proj,B → X M of sheaves of C ∞ M -modules that preserves the Lie bracket.
Proof. Notice first that we have a canonical isomorphism C ∞  1. The morphism ℓ is a partial right-inverse to (π B ) * in the sense that on the sheaf F. In particular this implies that ℓ is injective. 3. The morphism ℓ is smooth in the sense that the induced morphism F loc → (π B ) ! X proj,B loc of diffeological spaces is smooth with respect to the diffeology of Proposition 3.1.
We refer to such a morphism ℓ as a singular partial connection.
We will usually denote a singularly foliated bundle (π B , F, ℓ) by simply π B , with F and ℓ assumed unless otherwise stated. Before discussing some examples, let us mention that completeness of ℓ does not automatically follow from ℓ being a partial right inverse to (π B ) * . Indeed, it is easy to verify that for any open set O ⊂ M and for any X ∈ F (O), we have the relationship between the flows of Fl X (x) and Fl ℓ(X) (b) wherever they are defined. In particular, that ℓ is a partial right-inverse to (π B ) * implies that for any b ∈ B x , the domain of Fl ℓ(X) (b) is contained in the domain of Fl X (x). The converse, however, does not follow without completeness of ℓ, as is easily seen by considering the standard example of B = R 2 , M = R, and with ℓ : Conversely, suppose that F is a regular foliation of a manifold M , with leaf dimension p, and that π B : B → M is a fibre bundle with a singular partial connection ℓ : F → (π B ) ! X proj,B .

In a foliated chart
then defines a singular partial connection. Completeness and smoothness are consequences of the fact that ℓ is defined in terms of a smooth action of the groupoid G.
Conversely, suppose that ℓ : F → (π B ) ! X proj,B is a singular partial connection. Let g ∈ G have source x ∈ M . Choose a smooth path g : [0, 1] → G in G x with g(0) = x and g(1) = g, and let X ∈ F(O) be any vector field extending the tangent field to the curve t → g(t) · x ∈ M . By completeness of ℓ, the formula defines an action of g on B x , which is independent of the choice of path t → g(t) by involutivity of ℓ and by simple connectivity of G x . Thus ℓ defines an action of G on B, under which Definition 4.7. Let π B : B → M be a fibre bundle, let X ∈ X proj (B), and let k denote any of the symbols 1, . . . , ∞, g. The vector field vp k (X) on Γ k (π B ) defined by
For a projectable vector field X on a fibre bundle π B : B → M and k ≥ 1, the vertical kprolongation vp k (X) is the image of p k (X) under the canonical (π k−1,k B ) * (V π k−1 B )-valued contact form θ (k) [53, Chapter 6.3] on Γ k (π B ). In local coordinates (x i , f α , f α i , . . . ), the components of θ (k) are given by Thus it is easily checked (cf. Equation (6)) that in coordinates the vertical prolongation of X = a i ∂ ∂x i + b α ∂ ∂f α is given by This can also be checked directly using the chain rule. The invariant psuedo-bundles of a singularly foliated bundle are now defined as follows.
Definition 4.8. Let π B : B → M be a singularly foliated bundle, and let k denote any of the symbols 0, . . . , ∞, g. The diffeological subspace Γ k (π B ) F of Γ k (π B ) consisting of those points Recall now [46, Definition 2.10] that if π B : B → M is a regularly foliated bundle, a locally-defined section σ of π B is said to be distinguished if, about any point in its image, there exist foliated coordinates (x α , y α , f α ), with x α and y α denoting the leafwise and transverse coordinates respectively in the base, and with f α denoting coordinates in the fibre, with respect to which σ = σ(y α ) is independent of the leafwise coordinates. We denote by D g (π B ) the diffeological bundle of germs of the sheaf of distinguished sections, and by D k (π B ) the bundle of jets of distinguished sections. The next proposition says that the F-invariant psuedo-bundles of Definition 4.8 generalise the bundles of distinguished sections appearing in the regular case, and that therefore our constructions recover those of [46] in the regular case. Proof. In foliated coordinates (x α , y α , f α ) for B, corresponding to leafwise, transverse and fibre coordinates respectively, any element X ∈ F is given by some C ∞ M -linear combination while ℓ(X) (see Example 4.5) is given by Thus, in our coordinates (x α , y α , f α ) ∈ R p × R q × R k , we have simply for small t. It follows immediately that for any smooth function σ : is constant in t for all X ∈ F if and only if σ is constant in the x coordinate. Thus Γ k (π B ) F = D k (π B ) as claimed.
Defining lifting maps and leafwise transport functors for a singularly foliated bundle is now a simple matter of putting our definitions together.
is the unique solution to the initial value problem −t (O), with π B (b) = x ′ , the right hand side of Equation (11) is the lift by ℓ of the tangent to the curve r → (Fl X −t−r ) * Y (Fl X t+r (x ′ )) at r = 0, while the left hand side of Equation (11) is the tangent to the curve r → (Fl t+r (b)) at r = 0. By uniqueness of flows therefore, we have (Fl For notational simplicity denote ϕ := Fl X t and ℓ(ϕ) := Fl ℓ(X) t . Then we use Equation (12) to by the F -invariance of σ and since ϕ −1 * (Y ) ∈ F. Therefore Equation (8) does indeed define a curve in Γ k (π B ) F . That Equation 8 defines a solution to the initial value problem of Equation is smooth. The mapγ is already a plot of P(M ) by definition. Recalling that (Γ π B ) loc denotes the space of all locally defined sections of π B equipped with the diffeology of Proposition 3.1, it suffices now to show that the map is smooth. That is, fixing (u 0 , v 0 , t 0 ) ∈ W and x 0 ∈ dom(κ(u 0 , v 0 , t 0 )), we must find an open neighbourhood W of (u 0 , v 0 , t 0 ) in W and an open neighbourhood O of x 0 in M such that O ⊂ dom(κ(u, v, t)) for all (u, v, t) ∈ W and for which the map is smooth in the usual sense. We now have the following.
1. Using Lemma 4.11 together with the smoothness of ℓ as a map F loc → (π B ) * (X B ) loc , we can find open neighbourhoods U 1 ∋ u 0 in U , ) for all u ∈ U 1 and t ∈ I 1 , and such that

By definition of the diffeology on (Γ π B ) loc , we can find open neighbourhoods
3. Again by Lemma 4.11, we can find open neighbourhoods U 2 ∋ u 0 , for all u ∈ U 2 and t ∈ I 2 , and such that Finally, therefore, setting U := U 1 ∩ U 2 , I := I 1 ∩ I 2 and is a transport functor called the k-transport functor for π B . The associated holonomy groupoid (see Definition 3.17) is called the k-holonomy groupoid of π B and denoted H(π k,F B ).
Finally, we have the following analogue of [46,Theorem 5.15] which relates all of the holonomy groupoids of a singularly foliated bundle. . Consequently we have a commuting diagram of diffeological groupoids, which we refer to as the hierarchy of holonomy groupoids for the singularly foliated bundle π B .
Proof. The proof is similar to that of [46,Theorem 5.15].

Agreement with the Garmendia-Villatoro construction
We show in this subsection that for certain trivial singularly foliated bundles, the germinal holonomy groupoid of Definition 4.12 coincides with the holonomy groupoid constructed by , hence with that of Androulidakis-Skandalis [2]. A key feature of the Garmendia-Villatoro construction is the use of slices.
In these coordinates, every F-invariant section of M × R n → M takes the form σ(x, y) = (x, y, f (y)), where f : S x → R n is an F-invariant function.
holds. Clearly in either case γ 1 and γ 2 must have the same source and range, which we denote by x and y respectively, and each X i defines a diffeomorphism x onto an open neighbourhood O y of y. We may assume that O x and O y are of the form given in Equation (13) for some slices S x and S y about x and y respectively, where S x is mapped by ϕ 1 onto S y .
Suppose first that Hol(γ 1 , [X 1 ] g , d 1 ) = Hol(γ 2 , [X 2 ] g , d 2 ). Then there exists an element Z of as maps into S y . By the arguments of [6, Lemma A.8], we can always assume that Z may be on some open neighbourhood of y. Then for any F -invariant section σ defined in an open neighbourhood of x, we have and it follows that the (γ i , [X i ] g , d i ) define the same element in H(π g,F M ×R n ). Suppose on the other hand that the (γ i , for all (a, b) ∈ R k ×S y ∼ = O y . In particular, taking f to be the inclusion of S x ∼ = R l into the first l coordinates of R n , we have for all b ∈ S y , hence Hol(γ 1 , [X 1 ] g , d 1 ) = Hol(γ 2 , [X 2 ] g , d 2 ).

Functoriality
In this final section, we show that all of our constructions are functorial for morphisms of singularly foliated bundles. Since our constructions have all been built from germs of sections of such bundles, the correct notion of morphism is one constituted by morphisms of pseudobundles. As we will see, our definition generalises that considered by Garmendia-Villatoro [56, Definition 4.16. Let π B 1 : B 1 → M 1 and π B 2 : B 2 → M 2 be singularly foliated bundles. A morphism of π B 1 to π B 2 consists of a triple (F, G, f ), where f : M 1 :→ M 2 is a smooth map, F : Γ g (F 1 ) → Γ g (F 2 ) is induced by a morphism of sheaves F 1 → f ! F 2 and G : Γ g (π B 1 ) F 1 → Γ g (π B 2 ) F 2 is a morphism of diffeological pseudo-bundles such that: 1. F preserves the Lie algebra structure in each fibre, 2. G is surjective, and 3. letting L i g denote the lifting map for π B i (see Theorem 4.10), the diagram commutes. Here P(F ) and P(G) are as in Definition 3.14.
Let us remark that in Definition 4.16, our requirement that F be induced by a morphism of sheaves F 1 → f ! F 2 appears to be necessary, since it is only in this case that the induced map P(F ) on paths can always be guaranteed to send the subspace P(F 1 ) of P(Γ g (F 1 )) to the subspace P(F 2 ) of P(Γ g (F 2 )). Indeed, by Definition 3.15, we require that F map the section of Γ g (F 2 ) over a path γ determined by a single field X ∈ (F 1 ) loc that extendsγ to a section of the same sort in Γ g (F 2 ), and Example 3.10 shows that this cannot be expected of a general Recall [56, Section 6] that Garmendia-Villatoro define a morphism of foliated manifolds commutes. Here X i is the sheaf of vector fields on M i , and ι i the inclusion. By Proposition 3.9, provided thatF is smooth with respect to the functional diffeologies on (F 1 ) loc and (f ! F 2 ) loc respectively, such anF induces a morphism F :  Proof. Denote by f −1 F 2 the submodule of X 1 generated by vector fields whose image under By Proposition 3.9, the sheaf morphismsF 1 andF 2 induce morphisms F 1 : Γ g (F 1 ) → Γ g (f −1 F 2 ) and F 2 : Γ g (f −1 F 2 ) → Γ g (F 2 ) of pseudo-bundles covering id M and f respectively. Thus to obtain a morphism of the foliated bundles π i , it suffices to construct morphisms G 1 : Let us first construct G 1 . As remarked in the proof of [56, Theorem 6.21], since F 1 ⊂ f −1 (F 2 ), any slice τ transverse to F 1 through a point x ∈ M 1 contains a slice τ ′ transverse to f −1 (F 2 ) through x. Therefore, recalling that F 1 -invariant sections σ of π 1 defined in an open neighbourhood of x are in bijective correspondence with F 1 -invariant functionsσ : τ → Q, the formula defines the required morphism G 1 : Γ g (π 1 ) F 1 → Γ g (π 1 ) f −1 (F 2 ) . Letting m denote the codimension of the leaf of F 1 through x as a submanifold of the leaf of f −1 F 2 through x, we can assume that τ ∼ = τ ′ × R m , so that f −1 F 2 -invariant functions on τ ′ can be extended trivially to F 1 -invariant functions on τ . It follows that G 1 is fibrewise surjective. Moreover, as remarked in Example 4.4, the singular partial connections for f −1 F 2 and F 1 in the trivial bundle M 1 × Q are just the restrictions of the canonical lift ℓ : X M 1 → (π 1 ) ! X M 1 ×Q to each of f −1 F 2 and F 1 respectively. SinceF 1 is just the inclusion of a submodule, a routine calculation shows that commutativity of the diagram (14) is satisfied for the triple (F 1 , G 1 , id M 1 ). Let us now come to G 2 . Again, as remarked in [56,Theorem 6.21], if τ is a slice transverse to f −1 (F 2 ), then f (τ ) is a slice transverse to F 2 , and f | τ : τ → f (τ ) is a foliation-preserving diffeomorphism. Therefore the formula defines the required morphism G 2 : Γ g (π 1 ) f −1 (F 2 ) → Γ g (π 2 ) F 2 , which is fibrewise surjective since f | τ is a foliation-preserving diffeomorphism. To show that the diagram (14) commutes for the triple (F 2 , G 2 , f ), suppose that (γ, [X] g , d) ∈ P(f −1 F 2 ) and fix a point t ∈ [0, d]. Then for each slice τ through γ(t), f | τ is a diffeomorphism onto the slice f (τ ) through f (γ(t)), and by [56, Example 6.14], F 2 (γ(t), [X] g γ(t) , d) is given over f (τ ) by (f (γ(t)), [f * X] g f (γ(t)) , d), where f * X := df • X • f −1 . The equation ) for any f −1 F 2 -invariant functionσ : τ → Q. It follows that the diagram (14) commutes for the triple (F 2 , G 2 , f ).
Due to the categorical nature of our definitions, functoriality of the holonomy groupoids of singularly foliated bundles with respect to morphisms of singularly foliated bundles is now a simple exercise.
commutes. Here the unlabelled arrows are as in Theorem 4.13. That is, the hierarchy of holonomy groupoids is functorial.
We have thus proved that each of the diagrams . It follows then that for l ≤ k, the diagram commutes, and then the result follows by Theorem 4.13.

Outlook
In the author's estimation, there are two primary questions arising from this work that have yet to be answered.
Firstly, an assumption that we have had to impose in Definitions 4.3 and 4.16 is that morphisms of sheaves of smooth sections be smooth with respect to the diffeology of Proposition 3.1. It is far from clear that this assumption is really necessary. That is, it appears possible that any morphism of sheaves of smooth sections is automatically smooth with respect to this diffeology. Indeed, the domain considerations present in Proposition 3.1 are automatically satisfied by maps arising from morphisms of sheaves, and attempts thus far to construct a morphism of sheaves which is not smooth with respect to this diffeology have proved unsuccessful. A proof that any morphism of sheaves of smooth sections is itself diffeologically smooth would allow us to remove these seemingly extraneous assumptions.
Secondly, it is clear from Examples 4.4, 4.5 and 4.6 that in many situations, a singular partial connection on a fibre bundle induces a singular foliation of its total space by projectable vector fields. This foliation cannot, however, be expected to meet the requirements of Definition 2.1 (nor the equivalent definitions using compactly supported vector fields, for instance the one used in [2]). Indeed, as pointed out in the paragraph prior to Definition 4.1, projectable vector fields are not closed under multiplication by arbitrary smooth functions on the total space. This suggests that Definition 2.1 ought to be relaxed to allow for closure of vector fields under certain subsheaves of the usual sheaf of smooth functions. Such a modification will have no effect on the integration theorem (see [55,Theorem 4.2(e)]). Having relaxed Definition 2.1, a proof that singularly foliated bundles admit foliations of their total spaces will require an understanding of how presheaves of Lie-Rinehart algebras behave under sheafification. This question does not yet appear to have been studied in the literature.