Abstract
In this article we apply ideas from homotopy theory to the study of singular foliations. We verify that a technical lemma remains valid for left semi-model categories. When applied to the category of L∞-algebroids thanks to [11] , this lemma enables to recover results very similar to those of [10] about existence and (up to homotopy) uniqueness of universal L∞algebroids associated to a singular foliation. We conclude with some open questions.
Introduction
Homotopy theory, originally developed in the realm of topology, is now used in most domains of mathematics. One can cite algebraic topology, ∞-algebras, operads, higher category theory, mathematical physics or derived geometry as examples. This tendency is in part due to the success of the concept of closed model category introduced by Quillen in [12, Chapter 1] . Indeed, a model structure on a category gives access to topological intuition and a collection of important results.
Introduction
The aim of this article is to illustrate how homotopy theory can be used to get a better understanding of singular foliations. Our motivation comes from recent results of Lavau, Laurent-Gengoux and Strobl [10] on singular foliations which are reminiscent of classical results in homotopy theory. We show that these results can actually be deduced form a left semi-model structure on the category L ∞ /A of L ∞ -algebroids.
Introduction
We first recall the results of [10] about resolutions of singular foliations and state analogous statements in the language of model categories. We then outline the structure of the paper.
1.1
Foliations are classical objects of study in mathematics. Examples arise in PDEs, Poisson geometry, Lie groups or differential geometry to cite a few. Roughly speaking, a foliation F consists in a partition of a smooth manifold M into a collection of sub-manifolds called the leaves of F . The collection D(F ) of tangent spaces to the leaves of F is called the distribution associated to the foliation F . The foliation is called regular when D(F ) forms a vector bundle. As the leaves are submanifolds, the module Γ(D(F )) of sections of this bundle is involutive under the bracket of vector fields. However, in the general case, i.e. when the tangent spaces to the leaves do not form a vector bundle, one can still consider the modules of vectors fields tangent to the leaves. Singular foliations are therefore defined in [1] as locally finitely generated submodules of the module of vector fields, closed under the bracket.
1.1
Contrary to regular foliations, which are by now well understood, singular foliations are still mysterious objects. A promising approach has been proposed in [10] to understand a given singular foliation F . The idea is to associate to F a Lie ∞-algebroid QF which is equivalent to F [10, Theorem 1.6 ]. This QF is called the universal Lie ∞-algebroid over F . The use of the adjective universal is justified by [10, Theorem 1.8] which states that two such universal Lie ∞algebroids over F are quasi-isomorphic, and that such a quasi-isomorphism is essentially unique (up to homotopy).
1.1
1.2 These results should immediately ring a bell to a reader familiar with model categories. For convenience of the reader who is not, let us briefly explain why. Model categories were introduced in order to write down a minimal set of axioms enabling to perform constructions familiar in homological algebra (projective resolutions, derived functors) and in homotopy theory (homotopy relation for maps, fibrations, cofibrations, homotopy category).
1.1
Among these constructions, the cofibrant replacement, which subsumes the notion of projective resolution, is central. The idea is, given F , to find another object QF which is better behaved (i.e. is cofibrant) and remains equivalent to F . The proper way to formulate these properties is to introduce three classes of maps, W, Cof and F ib, respectivelly called weak equivalences, cofibrations and fibrations. QF is weak equivalent to F means that there exists a map in W between QF and F . QF is cofibrant means that the map 0 ֒→ QF from the initial object is a cofibration.
1.1
Cofibrant replacements are important to derive functors. The left derived functor D(F ) of a functor F is defined by D(F )(F ) := F (QF ). A non trivial part of the theory is to ensure that this derived functor is well defined, i.e. does not depend on the choice of the cofibrant replacement. This is relevant for our purpose since key lemmas involved in proving this fact are the exact analogues of the results of [10] we are interested in (see proposition 2.1).
1.3
The natural strategy would therefore be to search for a model structure on the category of Lie ∞-algebroids. However, with the definition of [10] (in terms of complexes of vector bundles), not every singular foliation admits a cofibrant replacement (section 1.1 of [10] ). We therefore need to allow ourselves to consider what we call L ∞ -algebroids in definition 3.4, i.e. a complex L of A-modules equipped with an L ∞ -structure compatible with the A-module structure. The nuance between Lie ∞-algebroids and L ∞ -algebroids is that for a Lie ∞-algebroid, the module L consists in sections of a differential graded vector bundle.
1.3
However, as remarked by Nuiten in [11, Example 3.1.12], even in the more general setting of L ∞ -algebroids, an axiom of the definition of model category fails to be satisfied. Therefore, Nuiten has instead equipped the category L ∞ /A of L ∞ algebroids with a semi-model category, a relaxed version of the concept of model category (see definition 3.1).
1.3
We can still apply this machinery to singular foliations, but in order to do so, we need the analogues of [2, Lemma 5 .1] and [2, Lemma 4.9 ] for semi-model categories. This why we start in section 2 by stating the proposition 2.1 whose proof is essentially the same as for model categories. The only "work" consists in checking that the axioms involved in the classical case are still valid for a semi-model category. In order to facilitate this task for the reader, we recall the outline of the classical proof in the appendix.
1.3
We then give in section 3 details about Nuiten's construction. After recalling in subsection 3.1 the precise statement and the relevant definitions, we recall the lemma 3.1 which enables to transport a semi-model structure via an adjunction. The next step is to describe in the subsection 3.2 the semi model structure on M od/A that can be transported to L ∞ /A via the Free/Forget adjunction. We conclude this section by describing thoroughly in subsections 3.3 and 3.4 the free functor involved in this adjunction.
1.3
With this at hand we are able to compare in section 4.2 the results of proposition 2 applied to L ∞ /A with the results of [10] that we wish to recover. We conclude with in subsection 4.3 a list of open questions.
1.4
We would like to thank François Métayer for a discussion about filtered co-limits and Chris Rogers for mentioning the reference [11] .
Main technical tool
The results of [10] concerning universal Lie ∞-algebroids are very similar to the following, proven in the appendix. 1. Every object in C has a cofibrant replacement.
If
A is cofibrant, then l ∼ is an equivalence relation on Hom C (A, X). We denote the set of equivalence relations by π l (A, X).
If
A is cofibrant and p : Y → X is an acyclic fibration, then composition with p induces a bijection:
If
4. Any two cofibrant replacements of A are weak equivalent and any two such weak equivalences between them are left homotopic.
If
To be more precise, [ In order to flesh out in section 4 this analogy, we want to apply this proposition to C = L ∞ /O M , the category of L ∞ -algebroids over a manifold M (See Definition 3.4).
Remark 2.1.
In what follows we stop writing "left" and simply write semi-model category for a left semi-model category.
A semi-model structure on L ∞ /A
We remind the reader that throughout this section we consider categories of differential graded modules over some base field of characteristic 0 and degree preserving morphisms unless otherwise stated.
A semi-model structure on L ∞ /A
In order to be able to apply Proposition 2.1 to the category L ∞ /A, we need to equip it with a cofibrantly generated semi-model structure. In 3.1 we state Theorem 3.1, which gives the semi-model structure on L ∞ /A from the model structure on M od/A described in 3.2. In 3.3 we define the category of L ∞algebroids, which is the main focus of this work, and in 3.4 we describe the free functor LR : M od/A → L ∞ /A that allows to apply Lemma 3.1.
Statement of the result
In this subsection we present Theorem 3.1, which allows us to use the machinery available in model categories for L ∞ /A. Theorem 3.1 guarantees the existence of a semi-model structure on L ∞ /A. The definition of semi-model structure is presented in Definition 3.1.
Statement of the result
Finally we present Lemma 3.1, which is the main tool that allows to transport the cofibrantly generated semi-model structures between categories.
Statement of the result
Theorem 3.1. [11, Lemma 2.14, Theorem 3.1] Let A be a graded algebra. Then the category L ∞ /A of L ∞ -algebroids over A admits a semi-model structure, in which a map is a weak equivalence (resp. a fibration) if and only if it is a quasi-isomorphism (surjection).
Statement of the result
In fact, this structure is cofibrantly generated [5, Definition 11.1.2], [4, Chapter 11] and [7, Section 2.1], which means that there is a set of maps generating the class of trivial fibrations through the right lifting property. SM3) Every map can be factored functorially into a cofibration followed by a trivial fibration. Every map with cofibrant domain can be factored as a trivial cofibration followed by a fibration.
Statement of the result
SM4) The fibrations and trivial fibrations are stable under transfinite composition, products, and base change.
Statement of the result
Remark 3.1. The definition of a semi-model category is a weakening of the notion of closed model category which needs in addition the following modifications to the axioms:
Statement of the result
The axioms SM2 and SM3 hold without assumption of cofibrancy of the domain.
Statement of the result
One does not require for a closed model category the axiom SM4 which follows from the other axioms.
Statement of the result
In particular, any model category is a semi-model category. Theorem 3.1 is proven through the following version of the transfer lemma: Then the above classes of maps determine a tractable semi-model structure on N whose generating (trivial) cofibrations are given by F (I) and F (J).
Statement of the result
The transfer takes a model structure on the category M od/A of differential graded anchored A-modules and gives a semi-model structure on L ∞ /A through an adjunction
Statement of the result
The existence of such adjunction is stated in Proposition 3.3 below.
The model structure on M od/A
The objective of this subsection is to prove Proposition 3.1, where we give a model structure on the category of anchored differential graded modules (Definition 3.3 below) over an algebra. We give the definition of M od/A as a slice category of M od. From Theorem 3.3 the latter has a cofibrantly generated model structure and we use this with Theorem 3.2 to give a model structure to the former.
The model structure on M od/A
Proposition 3.1. The category M od/A has a cofibrantly generated semi-model structure.
The model structure on M od/A
In Section 3.1, we recall the definition of a semi-model category and in Remark 3.1 we highlight the fact that a model structure is a particular example of a semi-model structure. So, we actually exhibit a stronger, i.e. a cofibrantly generated model structure on M od/A. This becomes useful for applying Proposition 3.1.
The model structure on M od/A
We define the slice category M od/A in Definition 3.3. We then state a general result, Theorem 3.2. It enables to obtain the model structure on M od/A from the model structure on M od given in theorem 3.3.
The model structure on M od/A
Recall the classical notion of slice category : In what follows, we denote by Der(A) the A-module of derivations of A. 
The model structure on M od/A
in which the map X → Y is an element of I and 2. J Z is the set of maps in M/Z of the form (1) in which the map X → Y is an element of J, then the standard model category structure on M/Z (in which a map (1) is a cofibration, fibration, or weak equivalence in M/Z if and only if the map X → Y is, respectively, a cofibration, fibration, or weak equivalence in M) is cofibrantly generated, with generating cofibrations I Z and generating trivial cofibrations J Z .
The model structure on M od/A
We introduce two classes of morphisms in M od, the category of chain complexes of A − modules.
The model structure on M od/A
Let D A (n) denote the "disc" A[n]
The model structure on M od/A
IdA −−→ A[n − 1], and S A (n) denote the "sphere" A[n] with trivial differential (both seen as complexes of A-modules). Let J be the set of morphisms of complexes of the form 0 → D A (n) and let I be the set of inclusions S A (n − 1) → D A (n). Proof of Proposition 3.1. We apply Theorem 3.2 to the slice category M od/A and the cofibrantly generated model structure on M od given by theorem 3.3.
The category L ∞ /A
In this subsection we define the category of L ∞ -algebroids associated to a graded algebra A. In broad terms, such an algebroid is given by an L ∞ -algebra with a compatible A module structure in the same spirit as a Lie algebroid in [8] .
The category L ∞ /A
Definition 3.4. An L ∞ -algebra is a module L with an L ∞ -structure [15, Definition 2] given by degree 2 − k anti-symmetric multibrackets
The category L ∞ /A
We remind the reader that k L is the quotient of the graded vector space k L under the equivalence relation generated by
The category L ∞ /A
As well, a permutation σ in S i,j is an (i, j)-unshuffle. This is to say, a permutation of 1, . . . , k = i + j such that σ(1) < · · · < σ(i) and σ(i + 1) < · · · < σ(k).
The category L ∞ /A
An L ∞ -algebra L with an A-module structure (an A-L ∞ -algebra) is an L ∞ -algebroid if it has a compatible action on A. The action is specified by an A-module morphism (which we call the anchor)
The category L ∞ /A
of degree 0. The compatibility conditions for the anchor and brackets are given by:
The category L ∞ /A
for a in A and v i ∈ L.
The category L ∞ /A
We denote by L ∞ /A the category of L ∞ -algebroids with morphisms given by the A-module morphisms that preserve the multibrackets. An L ∞ -algebroid is a version up to homotopy of the Lie-Rinehart algebra defined in [8, 1.1]. The higher brackets and the jacobiators give the structure up to homotopy.
The free functor LR
In Proposition 3.2 we describe a right adjoint for the forgetful functor F : L ∞ /A → M od/A, which takes an L ∞ -algebroid and gives back its underlying module. The existence of this functor is implicit throughout [11, Section 3], but here we give an explicit construction.
The free functor LR
We start by presenting the L ∞ -words on a fixed M in M od/A: An L A ∞ -word of arity k is recursively defined as an element of
The free functor LR
where M ′ is an A-module of some previously defined L A ∞ -words. We represent one such word by a symbol
The free functor LR
where a is an element in A, while the v i are all previously constructed L A ∞ -words.
The free functor LR
More precisely, the recursive construction goes as follows: We start setting all the elements of L 0 = M as L A ∞ -words. We can then form the symbols that involve elements of L 0 . The set of all such symbols forms an A-module which can be expressed as
The free functor LR
Here we take 1 L 0 [1] to be L 0 [1] . Including the elements of L 0 themselves we obtain L 1 := L 0 ⊕ L ′ 1 . Next, we proceed to define the L A ∞ -words that take symbols in L 1 . The set of all such words forms the A-module
The free functor LR
which contains all the L A ∞ -words which use at least one bracket and have at most two nested brackets. We then define the A-module of all L A ∞ -words with at most two nested brackets as
The free functor LR
In general, once defined L n , the A-module with at most n nested brackets, we define
The free functor LR
and then L n+1 := L 0 ⊕ L ′ n+1 . It is clear from the construction that L n ⊂ L n+1 for all n. Given an element v ∈ L n , we say that
The free functor LR
which is the quotient of the set of L A ∞ -words of M under ∼, which is the Amodule generated by all the elements over the jacobiators
The free functor LR
The L ∞ -brackets on L A ∞ (M ) are defined in the obvious way: given k + 1 elements of L A ∞ (M ) represented by L A ∞ words v 0 , . . . , v k , their bracket is given by the class represented by
The free functor LR
These brackets are well defined on L A ∞ (M ) and their jacobiators vanish from construction.
The free functor LR
In the construction of L A ∞ (M ) we did not make use of the internal differential structure nor the anchors of the module M . These are considered below in the construction of the free L ∞ -algebroid associated to M . (2) This is clearly a morphisms of A-modules that preserves jacobiators. Therefore, it descends to the required morphism L A
The free functor LR
Since it preserves the L ∞ -brackets by construction, it is a strict L ∞ -morphism.
The free functor LR
The adjunction property can be verified using that M is an A-submodule of L A ∞ (M ). Therefore, given any A-module, bracket preserving morphism (A-L ∞morphism) F : L A ∞ (M ) → L, the restriction to M is an A-module morphism. As well, given an A-L ∞ -algebra L, Equation 2 above defines the unique strict L ∞ -extension to L A ∞ (M ) of a given A-module morphism f : M → L. The verification that these two assignations are inverse to each other is left to the reader.
The free functor LR
In fact, we obtain an A-L ∞ -morphismα : L A ∞ (M ) → T A from α through the described adjunction
The free functor LR
Here we consider the dg-Lie algebra T A as an A-module with the L ∞ -structure coming from the Lie bracket. 
The free functor LR
, and a is in A. Finally, this construction is functorial and gives a right adjoint for the forgetful functor F : L ∞ /A → M od/A. 
Applications to singular foliations
In this section we propose results parallel to those of [10, Section 1] for Lie ∞-algebroids in terms of the (semi-)model category structure on L ∞ /A given by Theorem 3.1. We first show how singular foliations and Lie ∞-algebroids [10] are examples of L ∞ algebroids. With this connection in mind, we then compare each statement of [10] with their semi-model category analogue that we indroduce and prove as corollaries of our main technical lemma 2.1. We conclude with open questions raised by the discrepancies between the two sets of results.
Universal Lie ∞-algebroids
In this subsection, in order to make the connection with the semi-model structure on L ∞ /O M , we show that both Lie ∞-algebroids and singular foliations can be seen as objects in L ∞ /O M . We fix a smooth manifold M and its ring of smooth functions O M . Recall We can also express the notion of singular foliation in terms of L ∞ -algebroid. First recall Since T OM (with the L ∞ /O M -structure given by the commutator of derivations and anchor the identity) is just X(M ), one can restate the definition in terms of the L ∞ -structure. We remind the reader of the definition of cofibrant replacements in a (semi-)model structure. The following remark is the main reason for us to work with L ∞ -algebroids instead of Lie-∞ algebroids: • The morphisms in proposition 4.3 are L ∞ -morphisms, while the morphisms in 4.4 are strict morphisms of L ∞ -algebras.
Universal Lie ∞-algebroids
• The notion of homotopy equivalence used in 4.3 is very different from the one we are using which is based on Cylinder objects (see definition 4.6.1).
Universal Lie ∞-algebroids
The semi-model category statement analogous to [10, Proposition 1.22] is the following theorem, which is a corollary of Proposition 2.1.4. Once again, the notion of homotopy equivalence used in the two statements do not coincide. The reason is that when one works in the realm of model categories, there are standard definitions of left and right homotopies between morphisms, namely Definition 4.6.3 and 4.6.4 below.
Universal Lie ∞-algebroids
Definition 4.6. Let A and X be objects in a semi-model category C.
A cylinder object for A is an object Cyl(A) together with a factorization
where the map Cyl(A) → A is a weak equivalence.
A cylinder object for A is an object Cyl(A) together with a factorization
2. Similarly, a path object for X is an object P ath(X) together with a factorization where s : A ∐ A → A ∐ A is the morphism that switches components.
A cylinder object for A is an object Cyl(A) together with a factorization
For transitivity consider maps f , g and h such that f l ∼ g and g l ∼ h through the homotopies H ′ : Cyl ′ (A) → X and H ′′ : Cyl ′′ (A) → X respectively. We can take the maps A∐A → Cyl ′ (A) and A∐A → Cyl ′′ (A) to be cofibrations from Axiom SM3. Since cofibrations are stable under pushout from Lemma A.3, and A ∐ A is obtained through the pushout The unicity of the homotopy class off follows from Proposition 2.1.3.
A cylinder object for A is an object Cyl(A) together with a factorization
Lemma A.2. In a semi-model category the maps in Cof are exactly the maps with the left lifting property with respect to W ∩ F ib.
A cylinder object for A is an object Cyl(A) together with a factorization
As well, the maps with cofibrant domain that have the left lifting property with respect to F ib are exactly those that are already in W ∩ Cof .
A cylinder object for A is an object Cyl(A) together with a factorization
The proof of this lemma is the same as for [2, Proposition 3.13 (i) and (ii)]. Since trivial fibrations determine the cofibrations and respectively, fibrations determine trivial cofibrations with cofibrant domain, we can use the argument of the proof of [2, Proposition 3.14 (i) and (ii)]. Therefore, we have the following lemma: 

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