A new definition of rough paths on manifolds

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in"Rough Paths on Manifolds"(Cass, T., Litterer, C.&Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.


Introduction
The theory of rough paths (Lyons,[28]) and its variations (e.g. Gubinelli, [19]) generalise Young's integration theory in a way that it separates the probabilistic and deterministic parts of the strongly probabilistic Itô calculus, using only the variation of paths as a way of measuring their smoothness. A crowning achievement of the theory is understanding that, as far as ordinary differential equations are concerned, a path should not be defined by its graph but rather be identified as a choice of its signature (that is, the sequence of its iterated integrals). For example, the signature of a Brownian motion could be calculated using either Itô's or Stratonovich's calculus and it is this choice that leads one to solve a stochastic differential equation driven by a Brownian motion either in the sense of Itô's calculus or Stratonovich's. More generally, the theory of rough paths provides us with a deterministic calculus constructed in a way that it does not depend intrinsically on how a signature is defined but rather on common algebraic (that can be summerized by a Hopf algebra structure) and analytical properties that all signatures are expected to satisfy. These works have, in particular, enriched the toolbox of stochastic analysis with deterministic -path by path-results and widened its scope to rougher paths than the Brownian motion. The underlying philosophy of the theory also opened the door for solving a certain class of Stochastic Partial Differential Equations that require making sense of classically ill-defined products of distributions. This was carried incrementally through the development of alternative rough path theories: branched rough paths [20], para-controlled calculus [21] and regularity structures [22] (see [16] for a succinct exposition that goes from the theory of rough paths to that of regularity structures.) In addition to the natural applications that come with stochastic analyis, the theory of rough paths highlighted the role of signatures as highly efficient transforms of paths [5,23], which led to their exploitation in recent works in machine learning (the literature on the subject being abundant, we cite [2,11,18] as varied use-case examples).
The need for such calculus on manifolds arises naturally from both the points of view of pure and applied mathematics (see for example the introductions to [8,24]). Indeed, several physical systems and theoretical constructions arise as solutions in geometric contexts to ordinary differential equations driven by non-smooth paths. As in the Euclidean case, several applications come from stochastic analysis and extensions of rough path theory to manifolds would provide deterministic tools for the understading of approximation, continuity and regularity properties of solutions to stochastic differential equations (SDEs) on manifolds (see e.g. [15,25]). A simple illustration of such equations is the construction of the Brownian motion on the unitary group (which is a Riemannian manifold) as the solution to a -1224 -diffusion equation driven by a Brownian motion on its Lie algebra. In turn, this is related to the two-dimensional Yang-Mills theory and the free unitary Brownian motion (we refer the reader for example to [4,13,30] and the references therein for more details on this specific subject). These manifoldvalued SDEs driven by vector-space-valued noise remain the most studied types of diffusions on manifolds both in theory and in practice (for example in finance [31] or for attitude estimation [33]). However, solving them in the rough analysis framework (and the consequences thereof) requires solving a Rough Differential Equation as a fixed point problem on the product of the vector space and the manifold, which itself is a "pure" manifold.
That being said, the literature on rough paths on manifolds is however still very limited compared to its counterpart in the Euclidean setting or even to that of stochastic analysis on manifolds. The main attempts to generalise these notions to manifolds are due to the seminal work of Cass, Litterer and Lyons [8] in the framework of what is called a Lipschitz manifold and Driver and Semko in [14] (for paths controlled by rough paths on Riemannian manifolds). We also refer to Cass, Driver and Litterer in [7] (for weakly geometric rough paths on submanifolds embedded in the Euclidean space) and Bailleul [3].
The aim of the present paper is twofold: to give an alternative definition that simplifies but also complements the understanding of manifold-valued rough paths according to [8] and to give a general methodology of defining similar non-canonical lifts of paths on manifolds. Our exposition is structured in the following manner: Section 2 reviews the key ingredients in the theory of rough paths and sets up the notations and conventions used in the rest of the article while Section 3 recalls parts of the theory laid in [8] that will be of use to check the consistency of our results with the classical theory. In Section 4, we show several results emphasising the local nature of rough paths which we use subsequently in Section 5 to define our new notion of rough paths on manifolds. As in [8], we avoid putting too much structure on the manifold we work on or exactly mimicking the construction of rough paths on the Euclidean space in the non-linear framework of a manifold. Finally, as there currently exist many variants of the theory of rough paths (each serving a well-defined purpose) and in order to emphasise the simple ideas we used to translate the notion of geometric rough paths to manifolds and to explain how this can be done without any further particular considerations of the class of manifolds one is working on (for example, one does not need the manifold to be Riemannian to be able to measure the smoothness of paths), we give in Section 6 a general recipe expressed in the language of category theory to motivate the introduction of local Lipschitz manifolds in the preceding sections as a natural framework and explain how one may -1225 -more generally use the same approach to define a notion of coloured paths on manifolds.

Acknowledgments
Youness Boutaib carried most of this research while at the University of Oxford and is thankful for the support of the DFG through the research unit FOR2402 in Berlin and Potsdam where important improvements on this work have been made.

Review of key elements in the theory of rough paths
We start by setting up some notations and conventions and recalling the definitions and results that will be necessary to us in the rest of this work, one of which will be the extension of the notion of Lipschitz (Hölder) maps. As rough paths have become in the past few years a widely popular and familiar subject, we will strive to keep this section short and refer the reader to the literature on (geometric) rough paths, for instance [17,27,28,29].

The concept of the p-variation
Given p 1 and a path x : [0, T ] → E taking values in a normed vector space (E, · ), we denote by x p,[0,T ] the p-variation of x over [0, T ]. When p = 1, we say that the path x has bounded variation.
Assume (E, · ) is a Banach space. The set V p ([0, T ], E) of all continuous paths from [0, T ] to E that have a finite p-variation over [0, T ] is a Banach space when endowed with the p-variation norm ( [12,29]): x t Moreover, we have the following natural embedding: The manipulation of p-variations is often made easier by the introduction of controls. For a compact interval J, we will denote by ∆ J the simplex of all pairs (s, t) ∈ J 2 such that s t.
-1226 -Definition 2.1. -A function ω : ∆ [0,T ] → R + is said to be a control if it has the following properties: • ω is continuous.
To every path x of finite p-variation, we can associate a "natural" control ω given by ω(s, t) = x p p, [s,t] . Conversely, a control ω bounds the p th power of the increments of a continuous path x, i.e: if and only if x has a finite p-variation. In this case, one also has x p p, [s,t] ω(s, t) for all pairs (s, t) ∈ ∆ [0,T ] and we say that the p-variation of x is controlled by ω (cf. [12,27,29]).

The tensor algebra
Given a vector space E, for every integer n, E ⊗n denotes the space of homogeneous tensors of E of degree n (with the convention E ⊗0 = R). The set of formal series of tensors of E is denoted by T ((E)) (see [32] for an exhaustive exposition or [29,Chapter 2] for the part of the theory that will be in use in the rest of this work). For an integer m 0, the truncated tensor algebra of order m of E is denoted by T (m) (E) while the canonical homomorphism T ((E)) → T (m) (E) is denoted by π m . A permutation σ ∈ S m acts linearly on E ⊗m by the following: Finally, we will only consider norms · on T ((E)) that are admissible, i.e. that satisfy the two following conditions: A short discussion on certain basic properties of norms on tensor product spaces and their implications on the analysis can be found for example in [6].

The signature of a path
To fix the notations, we now proceed to the formal definition of the signature, which is a well studied subject since K.T. Chen's work in the late fifties (see e.g. [9,10] Let us note that signatures satisfy several key algebraic and analytic properties. We do not recall these here as those that are of interest to us are shared within the larger class of rough paths that we will introduce next.

Rough Paths
The theory of rough paths generalizes the concept of signatures to more irregular paths and provides the tools to solving differential equations driven by these without having to build a whole new theory of integration for each one of them (as in Itô's calculus). The concept of rough paths finds its source in the signature and the main analytic and algebraic properties that it satisfies. The space of geometric rough paths is indeed simply defined as the completion of that of signatures of paths with bounded variation under a suitably chosen metric similar to the p-variation metric for paths introduced in Subsection 2.1.1. We introduce here the basic definitions, notations and results that will be extensively used in the subsequent sections.

Multiplicative functionals
The appropriate higher order generalisation of the linearity of the integral is expressed in terms of multiplicative functionals: 28,29]). -Let E be a normed vector space and T 0. Let X be a map on ∆ [0,T ] with values in T ((E)) (respectively in T (n) (E), with n ∈ N * ). X is said to be a multiplicative functional (resp. a multiplicative functional of degree n) if the following holds: (1) X is continuous.
For example, for every path x ∈ V 1 ([0, T ], E), S(x) is a multiplicative functional and for every n ∈ N * , S n (x) is a multiplicative functional of degree n ( [9,10,29]). Remark 2.4. -When no confusion is possible, we may use the term multiplicative functional with no reference to its degree being finite or not.
Notation 2.5. -X i will denote the component of X of degree i.

p-variation metric and rough paths
We generalise now the notion of p-variation: Definition 2.6 ( [28,29]). -Let E be a normed vector space and T 0.
, with Γ being the usual extension of the factorial (the Gamma function) and: For example, the signature of a path of bounded variation has finite 1variation [26].
Remark 2.7. -One can also easily note that for 1 q p, a multiplicative functional of finite q-variation is of finite p-variation.
Lyons' extension theorem states that a multiplicative functional of finite p-variation is uniquely determined by its terms of degree less than or equal to [p]: Theorem 2.8 (Extension theorem [28,29]). -Let p 1 and n ∈ N * ∪ {∞} such that n [p]. Let E be a Banach space. Let X be a multiplicative functional of degree [p] in E that has a finite p-variation over [0, T ] controlled by a control function ω. There exists a unique multiplicative functional X of degree n that has a finite p-variation over [0, T ] and such that π [p] (X) = π [p] ( X). Furthermore, the p-variation of X is also controlled by ω.
As a consequence, the signature of a path of bounded variation is then the only multiplicative functional with finite 1-variation whose component of degree 1 corresponds to the increments of said path.
We now have all the ingredients to recall the definition of rough paths: Definition 2.9 (Rough Paths [28,29]

Geometric p-rough paths
We introduce now geometric rough paths, a central notion in our paper. For more details, see [29] or [28] (or [17] for an extensive study of the subject in the finite-dimensional case motivated with examples from stochastic analysis). We recall in particular that any geometric p-rough path is a p-rough path and is then a multiplicative functional that has finite p-variation.
-1230 -Similar to the notion of concatenation of paths with values in vector spaces, we can also define the concatenation of functionals taking their values in the truncated tensor algebra: u,t] ) with values in T (n) (E). We define the concatenation of X and Y , denoted X * Y , to be the functional over ∆ [s,t] defined as follows; for The following theorem is straight-forward: . Then: • If X and Y are multiplicative functionals, then X * Y is a multiplicative functional; • If X and Y have finite p-variation, then X * Y has finite p-variation; • If X and Y are geometric p-rough paths, then X * Y is a geometric p-rough path.
It will be necessary to us to attach a starting point to our geometric rough paths as we will be mostly dealing with integrals of rough paths and rough paths on manifolds; both of which require a starting point. In the next few sections, a geometric rough path in a Banach space E will be a pair (x, X), where X is a geometric rough path in the sense of Definition 2.11 and x ∈ E is called the starting point. Hence, we identify the space of geometric rough paths with starting points with the Cartesian product E × GΩ we define a metric d p as the product metric of d p and the norm on E, i.e.: In other situations, it will be more convenient to attach a trace to a rough path instead of a starting point (i.e. a path which increments correspond the element of degree 1 in said rough path).
-Let E be a Banach space and p 1. For an open subset U of E, a local geometric p-rough path in U is a triple (x, X, J) such that: The set of local geometric p-rough paths in U defined over compact subintervals of I will be denoted GΩ I p (U ; E), that of local geometric p-rough paths in U defined over a compact interval J will be denoted GΩ p (J; U ; E).
Remark 2.15. -The trace of a geometric p-rough path is trivially a path of finite p-variation.
Remark 2.16. -It goes without saying that rough paths with starting points easily identify with local rough paths. Indeed, if J is of the form, say, [0, T ], we will identify the local rough path (x, X) (omitting the interval J when no confusion is possible) with the rough path with starting point (x(0), X).

The integral of Lipschitz one-forms along geometric p-rough paths
When trying to make sense of integrals of one-forms along rough paths, it is very important (for example in regard to solving a differential equation) to be able to control the smoothness (in terms of variation) of the image of the path under the one-form. It appears that Lipschitz maps, first introduced by H. Whitney in [35] and later studied by E. Stein in [34], are the appropriate type of maps to use in this context (for a basic study of these maps including all the results below, see for example [6]): Definition 2.17. -Let n ∈ N and 0 < ε 1. Let E and F be two normed vector spaces and U be a subset of E. Let f 0 : U → F be a map and .
The smallest constant M for which the properties above hold is called the Lip-(n+ε)-norm of f and is denoted by f Lip-(n+ε) . The set of all Lip-(n+ε) maps defined on U with values in F will be denoted Lip(n + ε, U, F ).
Remark 2.18. -Let us stress that the above definition is purely quantitative and makes sense even on discrete sets. On any non-empty open subset of U , f 1 , . . . , f n are the successive derivatives of f 0 . However, these maps are not necessarily uniquely determined by f 0 on an arbitrary set U . Keeping There exists a unique continuous map: For a geometric p-rough path with a starting point (x 0 , X) or its corresponding local rough path (x, X), we denote:

The Lipschitz geometry
In this section, we review two of the findings of [8] that are most relevant to our work in this paper: the Lipschitz structure and geometric rough paths on manifolds.

Lipschitz structures
We first recall the general definitions of Lipschitz manifolds and Lipschitz maps and one-forms on them.
Let n ∈ N * and let M be an ntopological manifold. Let I be a countable set and, for every i ∈ I, U i be an open subset of M and φ i : M → R n be a compactly supported map such that its restriction on U i defines a homeomorphism. We say that ((φ i , U i )) i∈I is a Lipschitz-γ atlas if the following properties are satisfied: With the constants above, we will say that M is a Lipschitz-γ manifold with constants (δ, L).
Example 3.2. -As one would expect, finite-dimensional vector spaces can be endowed with a natural Lipschitz-γ manifold structure of any degree γ 1.
The smallest constant C for which this property holds is called the Lip-γ norm of f and is denoted by f Lip-γ .
(1) This is a slightly stronger result than in [8] but its proof is practically the same.
if there exists a constant C such that, for every i ∈ I: The smallest constant C for which this property holds is called the Lip-γ norm of α and is denoted by α Lip-γ . Remark 3.5. -An important property that is omitted in [8] and that we underline in the above definition is that, on a Lip-γ 0 manifold, it does not make sense geometrically to define Lip-γ maps or Lip- , then, based only on the definition of a Lip-γ 0 atlas, one cannot show that, for j ∈ I, the restriction of f • φ j −1 |Uj ∩Ui : B(0, 1) → E on any non-trivial subset of U j ∩U i is smoother than Lip-γ 0 . The same principle applies to defining Lipschitz one-forms. A more rigorous way to state the above is obtained by translating it in the language of equivalent Lipschitz structures (see [8] for a definition).

Rough paths on a manifold
In this subsection, we recall the definition of rough paths on manifolds as presented in [8]. As one does not have a natural notion of linearity and iterated integrals on a manifold, one has to consider a different approach than the one arising from studying the p-variational properties of paths and signatures. As we will hint to later, integrals of Lipschitz one-forms along a rough paths do characterize said path; this is the idea behind the elegant definition of a rough path given in [8]. Additionally, in the absence of a natural translation, a rough path on a manifold comes attached with a starting point.
for every γ ∈ R such that γ 0 γ > p and every Banach space E, the following conditions are satisfied: Contrary to the classical framework, in the context of manifolds, one does not need to make a difference between rough paths and geometric rough paths (as only the latter are defined). Consequently, we drop the word "geometric" when talking about geometric rough paths on manifolds. Moreover, in the classical sense, geometric rough paths are uniquely determined by the values of the integrals of compactly supported one-forms along them. In order to make the correspondance one-to-one between the concepts of classical geometric rough paths (on finite-dimensional spaces) and rough paths on the same space when endowed with its canonical Lipschitz structure, the following equivalence relation is introduced: Let M be a Lip-γ 0 manifold. We say that two p-rough paths X and X on M are equivalent, and we write X ∼ X, if they have the same starting point and if, for every γ ∈ R such that γ 0 γ > p and for every Lip-(γ − 1) compactly supported Banach space valued one-form α on M , we have X(α) = X(α).
Let us note at this point that there exists a one-to-one correspondance between rough paths in the classical sense and in a Lip-γ manifold, when said manifold is a finite-dimensional vector space V (see [8] for the precise statement and proof). More precisely, to each "classical" geometric p-rough path (with a starting point) (x, X) on V over is associated a unique p-rough path X on V in the manifold sense in the following way: for every Banach space-valued Lip-(γ − 1) compactly supported one-form α on V , one has: When working on a manifold, one has to ensure that certain properties are invariant by the change of charts (or, in other words, by local reparametrisation). For this reason, one defines the pushforward of rough paths by conveniently chosen maps: Let M and N be Lip-γ 0 manifolds and f : M → N be a map such that there exists a constant C f such that, for all γ ∈ (p, γ 0 ] and every Lip-(γ − 1) Banach space valued one-form α on N , we have: -1236 -Then f induces a pushforward map f * from p-rough paths on M to p-rough paths on N defined as follows: for every p-rough path X over [0, T ] on M starting at x, f * X starts at f (x) and for every Lip The next proposition shows that there exists a particular class of Lipschitz maps that induce pushforwards of rough paths. In particular, one can ascertain that the pushforwards of rough paths by coordinate maps or transition maps are also rough paths: ). Then f * α is Lip-(γ − 1) and there exists a constant C γ depending only on γ such that: Like in the classical case, there exists a notion of concatenation of manifold-valued rough paths. In the context of manifolds, this is important on its own since one usually works locally on coordinate domains to solve ordinary or rough differential equations before attempting to make sense of a global solution via a concatenation procedure: . We define the concatenation of X and Y, denoted by X * Y, to be the functional over [s, t] mapping every Banach space-valued Lip-(γ − 1) compactly supported one-form α (for every γ 0 γ > p) to the classical rough path X(α) * Y(α).
Unlike the classical case, the concatenation of two rough paths on a manifold is not necessarily a rough path. This is due to the fact that rough paths on a manifold come attached with a starting point and that we have no natural notion of translation. Therefore, for this concatenation to be a rough path, we have to make sure that the two rough paths in question have starting and "ending" points that agree in the following sense:  (2) Following [6], we correct the exponent in the inequality compared to the result that appeared in [8] and drop furthermore the dependence on the dimension of the manifold. Said inequality can also be made sharper using improved estimates in [6]. every Banach space-valued compactly supported Lip-γ map f on M (for every γ 0 γ > p), we have: In this case, one can check the consistency condition for the concatenation of two rough paths and prove that it is also a rough path: Like in the classical case, the support of a rough path on a manifold can be shown to be compact: The claim of the previous theorem can be shown by writing a rough path on a manifold as the concatenation of rough paths that have their support contained in the domain of only one chart at a time as described in the following theorem. This paper will generalise this decomposition of rough paths on manifolds into a collection of classical "local" rough paths that are compatible in a suitable sense.  B(0, 1), we have: (4) X = X 1 * · · · * X N .

Intervals
We briefly present here some elementary statements on covers of intervals which will be of use when studying rough paths locally. (1) (K i ) i∈I is said to be locally finite if for each x ∈ M there exists a neighbourhood U x of x such that at most finitely many The proof of the following lemma is straightforward: As J is compact, there exists a finite subset J 0 of J such that J 0 = {U x ; x ∈ J 0 } covers J. As every K i intersects J, and hence an element in the finite set J 0 , and since every element of J 0 intersects only finitely many of the K i 's, I must be finite. Proof. -Let (J n ) n∈N be a non-decreasing sequence of compact intervals (in the sense of inclusion) such that J = n J n . Let n ∈ N and define: Then (K i ) i∈In is a locally finite collection of compact sets that cover J n such that all the K i 's intersect J n . By Lemma 4.3, I n is finite. Note now that (I n ) n∈N is a non-decreasing sequence of finite subsets of I. Moreover, I = n I n . Hence, I is countable.
The case when J is compact is covered by Lemma 4.3.
Lemma 4.5. -If (K i ) i∈I is a compact cover of an interval J, and if for each i ∈ I, (K j ) j∈Ii is a compact cover of K i then {K j | j ∈ I i , i ∈ I} is also a compact cover of J.
Proof. -First note that: The K j 's (j ∈ I I i ) are all compact subsets of J. Let x ∈ J. Let V x,J be a neighbourhood of x in J that intersects finitely many K i 's (i ∈ I). As I i is finite for every i ∈ I (Corollary 4.4), then V x,J intersects finitely many K j 's (j ∈ I I i ).
Definition 4.6. -Let (U i ) i∈I and (V j ) j∈J be two collections of sets. We say that (V j ) j∈J is a refinement of (U i ) i∈I if, for every j ∈ J, there exists i ∈ I such that V j ⊆ U i .

a cover of an interval J by open sets and K = (K h ) h∈H be a compact cover for J. Then there exists a compact cover for J that is a refinement of both O and K.
Proof. -Let h ∈ H and x ∈ K h . As x ∈ J, then there exists i x ∈ I and for every x ∈ P . Then, for every x ∈ P , I x is a compact subset of K h that is contained in O ix . Furthermore, (I x ) x∈P is a compact cover of K h . We conclude using Lemma 4.5.  For every x ∈ [0, 1), we claim that: Indeed, assume the converse is true. Then there exists x ∈ [0, 1) such that: Then necessarily I 0 I. Indeed, if I 0 = I, we let i 0 be such that 1 ∈ K i0 and then, by convexity of K i0 , we have [x, 1] ⊂ K i0 , which contradicts our assumption. Let i * ∈ I 0 . For all n ∈ N * , let x n be an element of [x, x + 1 n ] − K i * . Then, for all n ∈ N * and i ∈ I 0 , x n = x (since x ∈ K i * ) and x n / ∈ K i (by the same convexity argument used above). Hence, (x n ) is a sequence in the compact set I−I0 K i converging to x, which leads to a contradiction. Therefore, the claim is true. For every i ∈ I, we write K i = [s i , t i ]. Let i 0 ∈ I such that K i0 contains a neighbourhood of 0 with non-empty interior and for every i ∈ I such that t i = 1, let r i ∈ I be such that there exists ε i > 0 such that [t i , t i + ε i ] ⊂ K ri . We define a 0 = 0 and a 1 = t i0 . Then a 0 < a 1 and [a 0 , a 1 ] ⊂ K i0 . We define the rest of the subdivision in a recursive way: for q 1, given a q = t q * , if a q = 1, then we are done, otherwise we set a q+1 = t r q * (we have then a q < a q+1 and [a q , a q+1 ] ⊂ K r q * ). It is clear that this procedure converges in a finite number of steps and produces the desired subdivision.  Remark 4.12. -One of the main reasons for introducing locally Lipschitz maps here instead of working with Lipschitz maps that are classical in the setting of geometric rough paths is to be able to use linear maps (such as the identity maps which are pivotal in the definition of categories) which are not Lipschitz in general (3) . Another possible solution to this issue is the use of almost Lipschitz maps introduced in [6]. The results below generalise automatically to this class of maps.

Locally Lipschitz maps
We recall here two important results on Lipschitz maps. They can be found for example in [6] and [8]:   4) ).
-Let E, F and G be three normed vector spaces. Let U be a subset of E and V be a subset of F . Let γ 1. We assume that (E ⊗k ) k 1 and (F ⊗k ) k 1 are endowed with norms satisfying the projective property. Let f : U → F and g : V → G be two Lip-γ maps such that f (U ) ⊆ V . Then g • f is Lip-γ and there exists a constant C γ (depending only on γ) such that: Lemma 4.14. -Let n ∈ N, 0 < ε 1 and C 0. Let E and F be two normed vector spaces and U be an open subset of E. Let f : U → F be a map and for every k ∈ [ [1, n]], let f k : U → L(E ⊗k , F ) be a map with values in the space of the symmetric k-linear mappings from E to F . We consider the two following assertions: Using for example Lemma 4.14, one can easily show the following result: Proof. -Notice that, on every ball B(x, α) ⊆ U on which df is Lipschitz-(γ − 1), one can use for example the fundamental theorem of calculus to bound f and we deduce that the restriction of f on this set is Lipschitz-γ using Lemma 4.14. The converse is obvious.  Before we carry on, we need the following embedding theorem for which the complete statement and proof can be found for example in [6].  Let ω be a control of the p-variation of x over J. Let s, u ∈ J and let q, r ∈ [[0, n]] such that a q s · · · u a r . Since f is Lip-1 on each of the B ti 's, for i ∈ [[0, n − 1]], with a norm less than M : ω(a k , a k+1 ) 1/p + ω(a r−1 , u) 1/p which, using the super-additivity of ω and Jensen's inequality, gives the control: Therefore, f •x is of finite p-variation (M and n do not depend on s or u).

A study of some properties of rough paths
Lemma 4.20.
-Let E be a vector space and J be a compact interval. Let X and Y be two E-valued multiplicative functionals on J. Let (K i ) i∈I be a compact cover for J by compact intervals such that, for all i ∈ I, X |Ki and Y |Ki are equal. Then X and Y are equal on J.
Proof. -Let (a, b) , is a compact cover for [a, b] by compact intervals. Let (a j ) 0 j n be a subdivision of [a, b] such that for all j ∈ [[0, n − 1]], there exists i ∈ S such that [a j , a j+1 ] ⊆ K i (Proposition 4.9). Then, by assumption: ∀ j ∈ [[0, n − 1]] X aj ,aj+1 = Y aj ,aj+1 . Therefore: X a0,a1 ⊗ · · · ⊗ X an−1,an = Y a0,a1 ⊗ · · · ⊗ Y an−1,an Which, by using the multiplicativity of X and Y , gives: X a,b = Y a,b . Since this holds for all (a, b) ∈ ∆ J , then X = Y . -Let E be a vector space and J be a compact interval. Let (K i ) i∈I be a compact cover for J by compact intervals. Let (X i ) i∈I be a collection of E-valued multiplicative functionals such that, for all i ∈ I, X i is defined over K i and for i, j ∈ I such that K i ∩ K j = ∅, we have X i|K i∩Kj = X j |Ki∩Kj . Then there exists a unique multiplicative functional X defined over J such that, for all i ∈ I, X |Ki = X i .
Proof. -Let (a j ) 0 j n be a subdivision of J and (i j ) 0 j n be a finite sequence of elements of I such that for all j ∈ [[0, n − 1]], we have [a j , a j+1 ] ⊆ K ij . Let (s, t) ∈ ∆ J and let q, r ∈ [[1, n − 1]] be such that: We set: X s,t = X iq−1 (s, a q ) ⊗ · · · ⊗ X ir (a r , t) It is an easy exercise to check that X defines a multiplicative functional satisfying the requirements of the statements. The uniqueness of such functional is a consequence of Lemma 4.20.
Before we make sure that the integral of a geometric rough path against a sufficiently smooth locally Lipschitz one-form is indeed well defined, we need the following lemma quantifying the distance over an interval in the p-variation topology between two rough paths given their distances on a subdivision of that interval.
where, for a finite sequence x = (x 1 , x 2 , . . . , x n ) and j > 0, we define: , where r i and n i are such that (if they exist) sr i−1 < t i sr i and sr i+ni t i+1 < sr i+ni+1 . Then we have: Using the inequality a α +b α (a+b) α , for a, b 0 and α 1 after summing over all i's, we get: which gives the result.
We can now show that the integral of a locally Lipschitz one-form along geometric rough paths is, as expected, well defined and continuous when varying the path.
We conclude this subsection by showing that integration along locally Lipschitz one-forms is consistent with Stieltjes' classical integration theory. This will be explicitly needed in the following section.
Proof. -First note that S [p] (f (x)) is well-defined since f (x) has bounded variation by Theorem 4.19. Let s, t ∈ ∆ J : (1) S [p] (f (x)) has finite 1-variation and by definition S [p] (2) As df is continuous and x is of bounded variation, then has finite 1-variation and is simply equal to (the signature of) the Stieltjes integral df (x)dx. Therefore, for all (u, v) ∈ ∆ [s,t] : Two multiplicative functionals that have finite 1-variation and which terms of the 1 st degree agree are equal by Theorem 2.8. Therefore, the sought identity stands.

A new definition of rough paths on manifolds
Based on our findings so far, we are now able to give a minimal approach for defining rough paths on a manifold.

Example 5.2. -
• A Lip-γ manifold is a locally Lip-γ manifold. In particular, finitedimensional vector spaces are locally Lip-γ manifolds. • Let n ∈ N * . A C n manifold is a locally Lip-n manifold.  γ, U, V ), we denote by f * the following map: ) i∈I of p-rough paths on R n satisfying the following conditions: Remark 5.5. -For this definition to be geometrically sound; for example stable under the change of atlas or charts by equivalent ones, one needs to show that the lift of transition maps (φ k • φ −1 i ) → (φ k • φ −1 i ) * is functorial. As our main aim is to study rough paths on manifolds rather than locally Lipschitz manifolds, we choose to carry on with our constructions in the current section and delay presenting and proving such geometric consideration until Section 6.
We identify similar local rough paths in the following way: This, of course, defines an equivalence relation. We will henceforth only consider the equivalence classes associated to this relation.
We can now easily define the lift of a manifold-valued path into a (local) rough path: i∈I on a compact interval J is said to be a p-rough path extension for the path x : J → M if the following holds: If such a rough path exists, we say then that x admits a p-rough path extension.
Following Definition 5.6, we will consider that two rough path extensions of a given path to be the same if their union is also a local rough path extension of that path.

Consistency with previous definitions
We make sure that our notion of rough path is compatible with the classical one (Definition 3.6): . Let X = Z 1 * · · · Z n−1 . One can check that the latter concatenation is indeed well-defined as we have, for every j ∈ [[0, n − 1]] and f Banach space-valued compactly supported Lip-γ map on M : By construction X satisfies the required conditions.
, we have, first by the identification between (y i , Y i ) and (X i , x i ), then the consistency of the endpoint of X i with the starting point of X i+1 : This proves that Y is indeed a local p-rough path on M over [0, 1]. Lemma 5.8 shows that the mapping X → Y constructed above is onto and one-to-one.

Coloured paths on manifolds
The constructions and the procedure followed above are a mere illustration of a general recipe that can be used to define any notion of colouring already existing on the Euclidean space to a manifold, assuming that we can find a suitable functorial rule. The lift of a path into a rough path can be indeed seen as a colouring: an extra bit of information that cannot necessarily be learned by looking at the base path only (the Brownian motion as an example). In this concluding section, we aim to showcase a general methodology expressed in the language of category theory that first enables one to identify a suitable notion of manifold on which one can build such coloured paths and gives then the definition of such paths.

Elements from category theory
The goal of this subsection is to recall the notion of functors which summarises in a single word many of the properties satisfied by the integrationof (local) rough paths along (locally) Lipschitz maps. We refer for example to [1] for the definitions below which may also be gathered in [24].
satisfying the three following axioms: In this case, for all i, j, k ∈ I, U i is called an object of C; an element in hom(U i , U j ) (also denoted by hom C (U i , U j )) is called either an arrow, a morphism or a homomorphism in C. The collections (U i ) i∈I and (hom C (U i , U j )) i,j∈I are respectively denoted ob(C) and hom(C). The binary operation ψ i,j,k is called a composition of morphisms and is simply denoted by •. i.e. for i, j, k ∈ I, f ∈ hom(U i , U j ) and g ∈ hom(U j , U k ), is a category, and with the notations of the previous definition, the axioms of associativity and the existence of identities can be rewritten in the following way: • By taking a family of sets considered as objects and all maps between these sets considered as arrows and the composition of maps as binary operations we obtain a category, usually called the category of sets.
• Let (G i ) i∈I be a family of groups. For every i, j ∈ I, let hom(G i , G j ) be the set of all group homomorphisms from G i to G j . Then C = ((G i ) i∈I , (hom(G i , G j )) i,j∈I , •) is a category (called category of groups). • Let (M i ) i∈I be a family of topological spaces (respectively smooth manifolds). For every i, j ∈ I, let hom(M i , M j ) be the set of all continuous maps (resp. smooth maps) from M i to M j . Then C = ((M i ) i∈I , (hom(M i , M j )) i,j∈I , •) is a category.
Definition 6.4. -Let C 1 and C 2 be two categories. A functor (or a functorial rule) from C 1 to C 2 is a pair of mappings ob(C 1 ) → ob(C 2 ) and hom(C 1 ) → hom(C 2 ), which we will denote by the same letter F , satisfying: (2) For every two objects U and V in C 1 and an arrow f ∈ hom C1 (U, V ), For all objects U, V and W in C 1 , and arrows f ∈ hom C1 (U, V ) and Example 6.5. - • Given any category, there exists a trivial functor from this category to itself preserving all objects and arrows called the identity functor. • Given a category of groups C 1 , let C 2 be a category of sets whose objects are the underlying sets to the objects of C 1 and whose arrows are the group homomorphisms in C 1 taken as maps between the underlying sets. The natural map associating to each object and arrow in C 1 their set-counterparts in C 2 is a functor. Functors constructed in a similar manner are called forgetful functors. • Let (G i ) i∈I be a family of Lie groups. For every i, j ∈ I, let: hom(G i , G j ) denote the set of all Lie group homomorphisms from G i to G j , -Lie(G i ) denote the Lie algebra of G i , -hom(Lie(G i ), Lie(G j )) denote the set of all Lie algebra homomorphisms from Lie(G i ) to Lie(G i ). Then C 1 := ((G i ) i∈I , (hom(G i , G j )) i,j∈I , •) and are categories. The mapping associating to each object in C 1 its Lie algebra and to each arrow in C 1 its pushforward at the identity defines a functor from C 1 to C 2 :

A functorial rule for rough paths
Let E be a Banach space and 1 p < γ. Let (U i ) i∈I be a family of open subsets of E. It is obvious that is a category. For i, j ∈ I, we denote by hom(GΩ p (U i ), GΩ p (U j )) the set of maps that assign in a continuous way (in the p-rough path topology) to each rough path in U i a rough path in U j defined over the same compact interval. It is also straightforward that: Theorem 6.6. -The rule that assigns to every object U in C 1 the object GΩ p (U ) and to every morphism f ∈ Lip loc (γ, U, V ), where U and V are objects in C 1 , the map f * , is functorial. Proof.
-For every open subsets U , V and W of E that are objects in C 1 , we need to prove the following: The first assertion regarding the identity map is straightforward. Let U , V and W be open subsets of E that are objects in C 1 . Let f ∈ Lip loc (γ, U, V ) and g ∈ Lip loc (γ, V, W ). Let x be a U -valued path over a segment J with bounded variation. By Lemma 4.24 As f (x) has bounded variation (Lemma 4.19), we similarly have: and as g • f is locally Lip-γ: Therefore ((g • f ) * ) |GΩ1(U ) = (g * • f * ) |GΩ1(U ) . As both (g • f ) * and g * • f * are continuous in the p-variation metric (Theorem 4.23) and as GΩ 1 (U ) is dense in GΩ p (U ) for this metric we deduce that (g • f ) * = g * • f * . Remark 6.7. -As remarked earlier, we have chosen in the previous sections to focus on the definition of rough paths on suitably chosen manifolds; and that for these definitions to be geometrically sound, the lifts of Lipschitz maps need to be functorial in the sense of Theorem 6.6. This now ensures for example (among several other things) that our definition of a rough path is stable across equivalent atlases (the transformation of the Euclidean space under a diffeomorphism being the simplest illustration). These properties being of relatively common knowledge in the context of Lipschitz maps and rough paths, we highlighted them here in view of the generalised procedure we present next.

A general recipe
We can define a notion of coloured charts and atlas over any n-topological manifold provided that the transition maps are arrows in an appropriate category. We will call such a manifold a coloured manifold. Definition 6.8. -Let n ∈ N * . Let C be a category whose objects are all open subsets of R n and for which inclusion and restriction maps are also arrows. Let M be a topological manifold and A an atlas on M . We say that (M, A) is a coloured manifold with respect to C if for any two charts (U, φ) and (V, ψ) in A such that U ∩ V = ∅, we have ψ • φ −1 ∈ hom C (φ(U ∩ V ), ψ(U ∩ V )).
We easily retrieve some familiar constructions of manifolds using this language: Example 6.9. -Let n ∈ N * . Let C 1 be the category whose objects are all open subsets of R n and whose arrows are continuous maps between these sets. Topological n-dimensional manifolds are exactly the coloured manifolds with respect to C 1 . Example 6.10. -Let n ∈ N * . Let S 1 be the category whose objects are all open subsets of R n and whose arrows are smooth maps between these sets. Smooth n-dimensional manifolds are the coloured manifolds with respect to S 1 . Example 6.11. -Let n ∈ N * and γ 1. Let L 1 be the category whose objects are all open subsets of R n and whose arrows are locally Lipschitzγ maps between these sets. Locally Lip-γ n-dimensional manifolds are the coloured manifolds with respect to L 1 .
Suppose now that we have a notion of coloured paths on R n that have underlying base paths which we will call traces (rough paths are an example).
Denote by T (U ) the sets of coloured paths whose traces lie in an open subset U of R n . Denote by C a category as in Definition 6.8 and let C be a category whose objects are the sets of coloured paths with traces in open subsets of R n . We assume there exists a functor from C to C: U → T (U ) ; f → f * such that for each arrow f between objects U and V of C, the arrow f * between T (U ) and T (V ) associates to each coloured path X on U a coloured path on V whose trace is the image by f of the trace of X. We can now define coloured paths on a coloured manifold in the same manner as in Definition 5.4 and the existence of coloured path extensions for manifold-based paths as in Definition 5.7.
Definition 6.12. -Let n ∈ N * . Let C be a category whose objects are all open subsets of R n and for which inclusion and restriction maps are also arrows.Let M be a coloured n-manifold w.r.t. C. A coloured path on M over a compact interval J is a collection (X i , J i , (φ i , U i )) i∈I of coloured paths on R n satisfying the following conditions: • (J i ) i∈I is a compact cover for J by segments; • For every i ∈ I, (φ i , U i ) is in the atlas of the coloured manifold M . • ∀ i ∈ I : X i ∈ T (φ i (U i )) and X i is defined over J i ; • (Consistency condition) If i, k ∈ I such that J i ∩ J k = ∅, then we have: A coloured path (X i , J i , (φ i , U i )) i∈I on an interval J is said to be a coloured path extension for the path x : J → M if the following holds: If such a coloured path exists, we say then that x admits a coloured path extension.
Remark 6.13. -As the study of the classical examples below will show, we emphasize that the rule linking C and C need be functorial in order to have sound geometric definitions and for these definitions to make sense on their own and be consistent with the definitions of coloured paths on the Euclidean space seen now as a coloured manifold. Example 6.14 (Example 6.9 continued). -Take C 2 to be the category whose objects are the sets of continuous paths with values in open subsets of R n . The colouring map associated to an arrow f in C 1 is a map that assigns to every continuous path x the path f • x. Then our new definition of a continuous path on M (as a coloured path) can be identified with the classical one which relies only on the topology on M : every continuous path on M in the classical sense can be seen as the concatenation of pushforwards of continuous paths on the Euclidean space that are consistent among themselves.
Example 6.15 (Example 6.10 continued). -Take S 2 to be the category whose objects are the sets of smooth paths valued in open subsets of R n . The associated functorial rule from S 1 to S 2 is the same as above by replacing continuity with smoothness. Finally, one can see the definitions of smooth maps in the classical sense and when described as coloured paths are equivalent. Example 6.16 (Example 6.11 continued). -Let 1 p < γ. Let L 2 be the category whose objects are sets of local geometric p-rough paths supported in the open subsets of R n and whose arrows are the set of mappings between the rough path in these objects defined over the same interval in a continuous way (in the p-rough path topology). We showed in the previous sections that rough paths can be seen as coloured paths in this setting.