Robert Viseur
email: robert.viseur@cetic.be

Gregorio Robles

First Results about Motivation and Impact of License Changes in Open Source Projects

Keywords: open source, intellectual property, license, contributor agreement, business model

Free and open source software is characterized by the freedoms and criteria that are warranted by specific licenses. These licenses describe the rights and duties of the licensors and licensees. However, a licensing change may be necessary in the life of an open source project to meet legal developments or to allow the implementation of new business models. This paper examines the motivations and impacts of license changes in open source projects. After a state of the art on the subject, a set of case studies where projects changed their license is presented. Then a set of motivations to change licenses, the ways to legally make this change, the problems caused by this change and a set of benefits of the license change are discussed.

Introduction

Licenses are thought to be selected at the beginning of the project with no posterior change [START_REF] Fogel | Producing open source software: How to run a successful free software project[END_REF]. They give the rules of the collaboration which everybody agrees to if participating in the project. To some extent, they provide a sort of "constitution" or legal agreement of how the project is developed and distributed. A change in the license is something very complex, not only because the aforementioned rules are changed, but also because it requires all contributors to agree on the license change.

One of the first, and most notorious license changes was performed by Mozilla in the late 90s (www-archive.mozilla.org). Due to incompatibility issues with the GPL noticed by the FSF, Mozilla decided to change from its dual NPL/MPL license to MPL/GPL/LGPL. This meant that all developers that had contributed to Mozilla had to explicitly give their consent to the change. Mozilla's website listed a number of contributors that could not be contacted or were missing. The license change finished when only around a dozen of missing confirmations from developers were still required; their code was identified and rewritten or removed. Another famous relicensing project is KDE whose purpose was to solve some compatibility issues with GPLv3. The projects leaders were obliged to identify and get agreement from several hundreds of contributors to that major free and open source project.

Despite the importance of the issue for the governance of open source projects, there are few scientific papers dedicated to the motivations and the impact of licenses changes in open source projects. Our paper proposes to contribute to thinking on that issue. It is organized in four sections. The first one includes a brief state of the art about motivation and impact of the license changes. The second one is dedicated to the presentation of the methodology and the analysis of several case studies. The third one consists in a discussion of our first findings. The last one proposes some conclusions and perspectives.

Background

Free and open source projects (also said FOSS or FLOSS) are covered by specific licenses that warrant the free and open source nature of software as defined by the Free Software Foundation (fsf.org) and the Open Source Initiative (opensource.org). Those licenses can be divided in two main families (de Laat, 2005; St. Laurent, 2004; Viseur, 2013b). The first one includes the academic or permissive licenses; the second one, the copyleft, reciprocal or restrictive licenses. The permissive licenses allow the use of the source code in proprietary software. The Apache, BSD and MIT licenses are famous examples. Contrariwise, copyleft licenses impose limitations on licensees of any derivative work, such as the conservation of the license or the availability of the source code. The AGPL, CDDL, CPL/EPL, GPL, LGPL, MPL and OSL licenses are famous examples of copyleft licenses.

Ten years ago the copyleft licenses represented more or less 80% of all the open source projects. However the success of permissive licenses has grown over time. For example, [START_REF] Hofmann | A dual model of open source license growth[END_REF] noted that "the combined effect of increased commercial investment with the need for competitively differentiated products built on top of that shared investment has lead to an increase of permissively licensed projects".

We identified several authors working on the license changes and their impact on the open source projects. The studies are not numerous and are often recent.

The main study was processed by Di Penta et al. (2012). The authors identified several motivations for license changes such as the "changes in the legal landscape, commercial code licensed as FOSS, or code reused from other FOSS systems" and the "evolution of a license per se" (i.e. GPL v2 → GPL v3). They draw their conclusions from a set of examples and empirical study analyzing, including ArgoUML, Eclipse-JDT, FreeBSD and OpenBSD, Mozilla and Samba. They found a wide variety of licenses, even in source codes that are supposed to be homogeneous, due to a period of transition after the decision to change from a license to another or a version to another one.

Savola and Anttila (2012) published a seminar report dedicated to open source software license changes. They based their seminar report on examples from scientific literature including Di Penta et al. (2012). For each example they identified the motivations and the result of the license change. Their selection of license changes includes Netbeans IDE, MySQL, Mono and Java. The report identifies reasons for the license changes (including better defendability, more clear license text, gain more users, better suitability for commercial usage, more paying customers, prevent commercial usage without paying license fee, reduce viral effect, partner's requirement and wider distribution) together with the problems (including license incompatibilities and push-back from the community) and the benefits (including wider distribution, user concerns about copyright and higher money income) after the license change.

Santos et al. (2011) worked on projects that had changed their type of license over the years. Their final sample is of 756 free and open source software for a total of 1012 intellectual property policy (IPP) interventions (i.e. changes of license type). They studied the impact of the IPP intervention on FLOSS attractiveness. The authors explored the transitions. They calculated their frequency and their positive or negative impact on the project attractiveness. The results revealed a bias in favor of the confidential Academic Free License. Note that the impact of the license type of open source software success is always a debated topic (Viseur, 2013a).

Viseur (2013b) studied the evolution of business models of open source editors facing cloud computing. The study was focused on e-business software including Enterprise Resource Planning (ERP), Customer Relationship Management (CRM) and e-commerce software. The author showed evolutions in license choices. A first pattern is related to companies switching to dual licensing model, often with technical differentiation between community and commercial release of the software (e.g., Compiere). A second pattern is related to companies facing to the rise of business software in SaaS mode and switching to copyleft licenses with network effect (e.g., OpenERP). Broadly speaking, the copyleft licenses with network effect impose the option to download the source code if users communicate with the software interface. Famous examples are the GNU Affero General Public License (AGPL) and the Open Software License (OSL).

Methodology and Cases Studies

We list a set of project that encountered at least one license change (see Table 1). The list was manually fed by the authors. Some information collected in the context of a study about forks was also used [START_REF] Viseur | Fork impacts and motivations in free open source projects[END_REF]. The list was completed by new examples that the authors obtained after Twitter requests. We analyze each case by focusing on a set of criteria: the application domain, the original and the new license, the projects leaders' motivation to change, the way the license change was conducted and the impact on the open source project. Our work is mainly based on scientific or professional sources such as Bert-Erboul (2013), de Laat (2005), Di Penta and German (2010), [START_REF] Hamerly | Freeing the Source: The Story of Mozilla, In Open Sources: Voices from the Open Source Revolution[END_REF], [START_REF] Välimäki | Dual licensing in open source software industry[END_REF], [START_REF] Viseur | Fork impacts and motivations in free open source projects[END_REF], Viseur (2013a), Viseur (2013b) and [START_REF] Willis | Relicensing VLC from GPL to LGPL[END_REF]. The issue of license change has usually not been the main subject of the scientific papers; however, professional sources document some aspects for specific projects. We describe below four representative cases: Mozilla, OpenERP, Trolltech and VLC. Mozilla Domain -The Mozilla Foundation develops several tools for the World Wide Web. The most famous are the discontinued Mozilla suite and the lighted Firefox Web browser. License -The project was published in 1998 under NPL license. A conflict with the community led to the publication under the first version of MPL license. The LGPL and the GPL later appeared in the development (triple license scheme), in order to address problems of incompatibilities with third-party projects. The project also published an evolution of the MPL license that solved incompatibility issues with the GPL license. Scope -The license changes impacted the whole project. How -There is no copyright assignment. The license changes needed contributor agreement. The Mozilla Foundation asked for the agreement of committers by choosing the acceptable licenses. Impact -Simplifying reuse in some third-party products is assumed. How -The process of the license change is not clearly documented. Impact -The OpenERP company evolved its business model and was able to respond to new market trends. However the second license change was associated with the emergence of tensions within the community.

Trolltech (Qt) Domain -The Qt software is a cross-platform application framework. License -The software was published under QPL license. Then a dual licensing scheme (QPL + GPL) was applied under the pressure of the KDE community that used Qt intensively. Finally, after Nokia bought Trolltech, the decision to publish the software under the LGPL license in order to spread the technology was taken. The project is currently managed by another company, Digia, and "some modules are available under LGPLv2. [START_REF] Bert-Erboul | VideoLan: un cas d'école dans l'industrie informatique[END_REF]

, LGPLv3 or GPL v3 and some other modules under

LGPLv3 only" (qt.io). Scope -The whole library was impacted by the successive license changes. However the actual release licensing scheme is heterogeneous. How -The company was copyright owner. Impact -The publication under GPL satisfied the community and the Harmony fork was abandoned. The impact of the publication under LGPL has not been quantified. The current license schema is complex. VLC Domain -The VLC project offered a cross-platform media player and a streaming media server. The project was initiated by students from Ecole Centrale de Paris. License -The project has been covered by GPL v2 (or higher) since 2001. The license was changed for LGPL v2.1 (or higher) in 2011. Scope -The license change was implemented for modules impacting third party vendors. Motivations -There were several motivations, including "making VLC compatible with various gadget-vendor application stores (e.g., Apple's)" [START_REF] Willis | Relicensing VLC from GPL to LGPL[END_REF]. How -Two hundred and thirty contributors had to be contacted. It was difficult to contact them and get response. With 150 responses, the project owned an agreement for 99.99% of the code for libvlc and libVLCcore. It was more difficult for other parts, for which it was necessary to delete, reimplement, refactor into separate files or drop some code. The code was kept under GPL if there was no impact on third party vendors.

Impact -Some modules had to be reimplemented (less than 5). The impact on Apple is unknown (opacity).

Discussion

Beyond the four cases that we have presented so far, we identified a set of motivations for license changes (see Table 2):

1. The license can be changed for compatibility. The open source project can thus be distributed in new packages (e.g., Java), be integrated in wider source code (e.g., WebM) or reduce the license impact on third party add-ons development (e.g., Odoo, VLC or SharpDevelop2). 2. The license can be changed by the editor with the hope that the revenues will increase. This is the case when an editor organizes a transition to a dual licensing scheme (e.g., MySQL). 3. The license can be changed from the GPL license to a more permissive one in order to avoid how "derived work" is defined (e.g., Mono). 4. The license change can be motivated by the simplification of a multiple licensing scheme (e.g., JQuery). 5. The license change can be motivated by the desire to improve the relationships to community and limit the risk of fork (e.g., Trolltech). 6. The license change can be the consequence of an ego-conflict. That reason is often alleged but not proofed (e.g., IP Filter). 7. The license change can be an adaptation to evolutions in the competitive environment. For example, a lot of open source projects are preparing themselves for the cloud computing rise (e.g., OpenERP, now Odoo, or SugarCRM). 8. The license change can be require by a commercial partner (e.g., Mono). 9. The license change can be motivated by legal arguments. It can be to clarify the original license (e.g., IP Filter) or to adopt a more suitable license (e.g., OpenStreetMap).

Table 2. Motivations (the cumulated percentage must be higher than 100% as a license change can be associated to several motivations). The GPL license is often in the center of the license changes; either the project changes to the GPL for financial or compatibility reason, or the project moves away from GPL to a more permissive one in order to avoid the side effects of the GPL. Note that a license change can also be the result of several motivations at once.

Expected benefit

The license change can be hindered by the shared property (among all contributors) of the source code. The issues can be managed in two ways:

1. The editor ensures that he owns the copyright on the source code (by contributors' agreement or by rewriting of contributions) and can change the license. This is the case of MySQL or Qt. 2. The responsible must get agreement from all the contributors (if necessary, by contacting them one by one). This has been the case for Mozilla or VLC.

We identified three problems that the license change can cause:

1. The project license can impact strongly or become incompatible with other projects (e.g., MySQL with PHP). The incompatibilities are often caused by copyleft licenses such as the GPL. An exception clause (e.g., classpath exception, FLOSS or FOSS exception, etc.) allows to reduce the virality effect of the license. 2. The new license, if it is permissive, can allow the appropriation of the source code by the editor, and may be undesired for community members (e.g., Mono). 3. The license change can irritate the community and lead to a fork (e.g., ath5k).

We identified three benefits the license change can bring:

1. The updated project can gain a wider distribution due to the license change (e.g., Squeak or Java). However, the updated project can contrariwise be affected by a smaller distribution due to the license change (e.g., IP Filter).

2. The editor who updated the license can gain more revenue (e.g., MySQL). However there is insufficient data to objectify the effect of license change on the company profitability (e.g., Mono). 3. The license change can lead to the abandonment of a fork (e.g., Trolltech Qt and Harmony) if it is part of community claims.

The case studies showed that the issue of license change was not specific to software. Indeed OpenStreetMap (openstreetmap.org) shows a similar problem in the field of databases. The ODBL license (opendatacommons.org/licenses/odbl/) was adopted because it was considered more suitable compared to CC-BY-SA license (creativecommons.org).

Conclusion and perspectives

As preliminary results, our paper allowed to identify nine motivations, three negative impacts and three positive impacts for license changes.

We intend to continue this study with some future research. First, we plan to add new cases of license changes. For example: KDE (relicensing to be compatible with GPL v3), Samba (adoption of GPL v3) and Wikipedia (move to CC-BY-SA). The discovery of new cases might be facilitated by the use of free and open source projects data. For example, the FLOSSmole datasets (flossmole.org) contain license information (see database schema) that would be useful to automate the search of new interesting cases. Second, we want to study in detail the identified cases. In particular we found there was often a lack of documentation about the license change procedure (even for well known projects such as Qt), and that the impact was rarely objectified. We would like to enrich the cases with facts and figures allowing to better quantify the impact of license changes. So we wish to obtain a deeper understanding of the phenomena. Our final goal is to propose guidelines to open source project managers and entrepreneurs since there is already for license selection.

OpenERP

  Domain -OpenERP (now Odoo) is an open source Enterprise Resource Planning (ERP) software including Customer Relationship Management (CRM) features. License -The software moved from GPL to AGPL in 2009 in order to avoid the use in SaaS mode without publishing the modifications. The software editor proposes its own SaaS offer based on OpenERP. In 2011 the company introduced a new license, an AGPL with Private Use License (dual licensing) allowing to keep the modifica-tions private in case of SaaS use. A new license change is planned (from AGPL to LGPL). Scope -The license change affects all the software except the OpenERP Web Client under MPL license.

Table 1 .

 1 Documentation of license changes.

	Project	License (from)	License (to)
	Alfresco	GPL with FLOSS exception	LGPL
	ath5k	BSD	BSD + BSD/GPL (fork)
	Compiere	MPL	GPL + proprietary
	Dimdim	MPL	GPL
	ExtJS	LGPL-style	GPL + proprietary
	IP Filter	IPFilter (BSD-Like)	IPFilter (modified)
	Java	SCSL (hybrid)	GPL with classpath exception
	Joomla (framework) GPL	LGPL
	JQuery	GPL/MIT	MIT
	Mongoose	MIT	GPL + proprietary
	Mono	GPL	MIT/X11
	Mozilla	NPL	MPL
		MPL	MPL/LGPL/GPL
	MySQL	LGPL	GPL
	OpenERP	GPL	AGPL
		AGPL	LGPL
	OpenStreetMap	CC-BY-SA	ODBl
	SharpDevelop2	GPL	LGPL
	Paint.Net	MIT	Proprietary
	Squeak	Squeak	Apple Public License
		Apple Public License	Apache License
		Apache License	MIT/Apache License
	SugarCRM	SPL (MPL-like)	GPL + proprietary
		GPL	AGPL
	Talend	GPL	LGPL
	Trolltech	QPL	GPL + proprietary
		GPL	LGPL


Wolfgang Bertram
email: wolfgang.bertram@univ-lorraine.fr

A PRECISE AND GENERAL NOTION OF MANIFOLD

Keywords: Mathematics Subject Classification. 18B10, 18F15, 58A05, 58H05, ordered groupoid, manifold, atlas

come

A precise and general notion of manifold

Introduction

When talking about manifolds, physicists usually are consequent, and mathematicians are not: physicists usually do not try to give a formal definition of what a manifold is, whereas mathematicians often start in a formal way, but, following the example of Elie Cartan, from a certain point on end phrases with "and so on", "whenever defined", or similarly. As far as I know, a fully formalized definition of a sufficiently general notion of "manifold" has never been written up. 3 Today, I will try to be consequent (to allow myself to be inconsequent tomorrow). The desire for consequence and full formalisation arose from a discussion with Anders Kock on Appendix D of my paper [START_REF] Bertram | Conceptual Differential Calculus. I : First order local linear algebra[END_REF]: he remained sceptical about the general notion of "primitive manifold (with atlas)" defined there. Indeed, when working over completely general base rings, where analytic constructions in charts may be replaced by "formal" constructions, our intuition coming from real, finite-dimensional manifolds may fail, and one needs to care about complete formalization. In the present note I shall fill in, rather pedantically, the missing details of Appendix D loc. cit. However, when doing so, I realized that this is not quite a simple exercise of rewriting: some changes have to be made, in particular with regard to notation -namely, in loc. cit., I followed the classical notation φ ij : V ji → V ij (see e.g. [START_REF] Husemoller | Fibre Bundles[END_REF], or the wikipedia-article) for transition maps between charts (φ i , U i ) and (φ j , U j ). But it turns out that this notation is cumbersome since it mixes up two different aspects of transition maps:

(1) the aspect of restricting things ("wherever defined", i.e., on

U ij = U i ∩ U j ),
(2) the aspect of transition between local coordinates (φ ij is a bijection). I think that, for better understanding the formal structure of the notion of manifold, it is advisable to seperate (1) and (2) notationally: aspect (1) corresponds to a partial order L on the index set I (one chart may be included in another one, we write i ′ ≤ i or (i ′ , i) ∈ L), and aspect (2) corresponds to an equivalence relation E on the index set I (two charts are equivalent if their chart domains coincide, we write (i, j) ∈ E). There is a natural compatibility condition turning the index set I into what we call an "e-pos". This is an interesting mathematical structure in its own right (in fact, it is a special case of an ordered groupooid, cf. Appendix A): Definition 1.1. An equivalence-partially ordered set (e-pos) is a set I together with an equivalence relation E and a partial order L (or ≤) on I satisfying (Epos) if i ′ ≤ i and (i, j) ∈ E, then there exists a unique j ′ ∈ I such that: (i ′ , j ′ ) ∈ E and j ′ ≤ j.

Adopting the convention that horizontal dashes mean "in relation E" and nonhorizontal dashes "in relation L (with i ′ placed lower than i)", property (Epos) is represented by figures like these:

i j i ′ j ′ i Y Y Y Y Y Y Y Y j i ′ j ′ i Y Y Y Y Y Y Y Y j i ′ j ′
Every manifold M with atlas A has an underlying e-pos. If the atlas is finite, we get a finite e-pos. For instance, the projective plane over a field K with its 3 "canonical" charts φ ν with domains [x 0 , x 1 , x 2 ] where x ν = 0 for ν = 0, 1, 2, leads to an e-pos with |I| = 12: each of the three canonical charts can be restricted to the intersection with the remaining 2 charts domains (giving rise to 6 charts on the next lower level), and finally all three canonical charts can be restricted to

U 0 ∩ U 1 ∩ U 2 .
This e-pos is visualized as follows:

2 1 0
On the other hand, maximal atlases give rise to very big, often uncountable, eposes (Section 4). In the general case, the transition functions φ ij for every pair (i, j) ∈ E, together with restrictions for every pair (i ′ , i) ∈ L, define a morphism of the underlying e-pos into the pseudogroup of the model space V of the manifold (Section 2). Conversely, every such morphism can be seen as "gluing data" of an abstract manifold M, that can be constructed by taking some kind of quotient with respect to E, taking account of all restrictions by L (Section 3, Theorem 3.2). For instance, the finite gluing data for the projective plane shown above satisfy our conditions when K is an alternative division algebra, and hence the theorem yields a very natural construction of the octonion projective plane OP 2 (see Example 4.3). This equivalence also allows to perform constructions, like the one of tangent bundles, in complete generality: whenever we have a functorial construction associating to the pseudogroup of V a pseudogroup on another space W , then the gluing data yield gluing data modelled on W , and thus give rise to a manifold modelled on W (Theorem 3.4). On a next level, one wishes to describe morphisms of manifolds (i.e., smooth maps, if the manifold is smooth) via the gluing data. A formal analysis of the usual construction makes it clear that we won't get a plain "morphism of e-poses", since a map f : M → N won't induce a map assigning to a transition function on M another one on N (unless f is a bijection). We rather get a natural relation between e-poses. Indeed, this is a special instance of the more general notion of natural relation between groupoids or small categories (Appendix B): natural relations are very closely related to natural transformations from general category; just like these, they give rise to double categories and 2-categories ([M98]), and thus may be of general interest for category theorists (cf. remark B.2). We give some comments and mention some topics for further work in the final section 7.

Acknowledgment. Besides Anders Kock for his critical and constructive remarks, I would like to thank the students of the lecture series "Concepts géométriques" for their patience -some of the material of this paper was presented there.

A → B is Γ f = {(f (x), x) | x ∈ A}.
This notation is best compatible with writing the function symbol f on the left of its argument x.

Throughout, V is a non-empty set, called the model space, and I is a set, called the index set, which will serve as "chart index" for atlases. In practice, V will often be equipped with a topology, whereas the set I should be considered as "discrete".

From atlases to atlas data

Definition 2.1. A chart of a set M, modelled on V is a bijection φ : U → W of a subset U ⊂ M onto a subset W ⊂ V . An atlas on M is a collection of charts A = (φ i : U i → V i ) i∈I , or (U i , φ i , V i ) i∈I , such that: (1) [φ i = φ j ] ⇒ [i = j] (atlas without repetion), (2) M = ∪ i∈I U i (the atlas covers M), (3)
(intersection and restriction of charts) whenever (U i ∩ U j ) = ∅, then there exists a unique k ∈ I (which we denote by k = [ij]) such that

U k = (U i ∩ U j ), ∀y ∈ U k : φ k (y) = φ i (y) .
Sometimes, one may prefer to work with the following slightly weaker version of (3):

(3') for all x ∈ (U i ∩ U j ), there exists k ∈ I such that

x ∈ U k ⊂ (U i ∩ U j ), ∀y ∈ U k : φ k (y) = φ i (y) .
If V and M are equipped with topologies, we generally assume that the sets U i and V i are open, and that the φ i are homeomorphisms. In this case, we will say that the atlas is saturated if every homeomorphism from a (non-empty) open subset of M onto an open subset of V is of the form φ k with some k ∈ I.

One may note that (3) implies (1), but (3') doesn't. See Section 4 for the definition of atlases with certain properties (such as smoothness), and of maximal such atlases.

Definition 2.2. For (M, A) as above, we define (a) a relation E ⊂ (I × I) by: (i, j) ∈ E iff U i = U j (i.e., both charts have same domain, but need not coincide as charts), (b) a relation L ⊂ (I × I) by:

(i ′ , i) ∈ L iff [U i ′ ⊂ U i , and ∀y ∈ U i ′ : φ i ′ (y) = φ i (y)] (i.e., one chart is included in the other; this implies V i ′ ⊂ V i ).
Lemma 2.3. Assume (1), (2), (3') hold. Then: (i) The relation E is an equivalence relation on I, and L is a partial order on I. (We shall henceforth write i ′ ≤ i instead of (i ′ , i) ∈ L.) Assume moreover that (3) holds. Then (ii) the triple (I, E, L) forms an e-pos in the sense of Definition 1.1, (iii) if i ≤ m and j ≤ m and (V i ∩ V j ) = ∅, then there exists a unique k ∈ I with k ≤ i and k ≤ j and

V k = V i ∩ V j .
We represent (ii) by a diagram, as in the Introduction, and (iii) as follows: They can be used to relate our approach to the one of V.V. Vagner who desrcribes manifolds via inverse semigroups, cf. [START_REF] Schein | On the theory of inverse semigroups and generalized grouds[END_REF][START_REF] Lawson | Ordered groupoids and étendues[END_REF].

m c c c c c c c c i a a a
Example 2.1. The sphere S n with charts s, resp. n, given by stereographic projection from the south pole and from the north pole, along with the resrictions s n and n s to the intersection domain, gives rise to the epos I = {n, s, n s , s n } with

s n s n n s
Thus E has three equivalence classes. Condition (Epos) is meaningless in this case.

Example 2.2. The e-pos of the projective plane over a (skew)field K has been described in the Introduction (Section 1): there are the 3 "canonical" charts φ ν , ν = 0, 1, 2, and 9 other charts given by restricting them to all possible intersections of chart domains. Every quadrangle appearing in this graph stands for a configuration given by three indices i, j, k, namely: if there are two horizontal edges, the trapezoid stands for

φ j | U k , φ k | U j (upper edge), φ j | U k ∩U i , φ k | U j ∩U i (lower edge),
and if no edge is horizontal, the quadrangle stands for

φ k (top vertex), φ k | U i , φ k | U j , φ k | U i ∩U j (bottom vertex;
for esthetical reasons, we try to represent such quadrangles by parallelograms, if possible).

Example 2.3. For n ≥ 2, the e-pos of KP n consists of n + 1 different n-hypercubes of restrictions of the n + 1 "canonical" charts to all possible intersections of chart domains (so |I| = (n + 1)2 n ); the vertices of different hypercubes are linked among each other if they correspond to the same intersection of domains. An elegant description of this e-pos is by identifying its index set I with the set of edges of an n + 1-cube (cf. [START_REF] Bertram | Conceptual Differential Calculus. II : Cubic higher order calculus[END_REF] for notation): the edge (β, α) with α = β ∪ {i} is identified with the restriction of the canonical chart φ i to the intersection

U β = U β 0 ∩ . . . ∩ U β ℓ where i / ∈ β. Two edges (β, α), (β ′ , α ′ ) are equivalent iff α = α ′ . They are in relation L, (β ′ , α ′ ) ≤ (β, α), iff [β ′ ⊂ β and i = i ′ (same "direction")]. Definition 2.4. If (i, j) ∈ E, we define the transition map φ ij := φ i •φ -1 j : V j → V i .
Lemma 2.5. The transition maps satisfy, whenever (i, j), (j, k) ∈ E,

(1)

φ ii = id V i , (2) φ ij • φ jk = φ ik , (3) φ -1 ij = φ ji .
Proof. This follows directly from the definitions.

The preceding definitions and lemmas can be summarized by saying that

(I, E; L) → (P(V ), B loc (V ); ⊂), (k, (i, j); (i ′ , i)) → (V k , φ ij , (V i ′ ⊂ V i ))
is an (injective) morphism of ordered groupoids. This is what we are going to call "atlas data" in the following section.

Definition 2.6. Assume given a pseudogroup G = (G 0 , G 1 ) ⊂ (P(V ), B loc (V )) of transformations of the model space V (cf. Def. A.2), we say that atlas data, and the atlas A, are of type

G if V k ∈ G 0 and φ ij ∈ G 1 whenever k ∈ I, (i, j) ∈ E. In case (G 0 , G 1 )
is the pseudogroup of locally defined diffeomorphisms of a topological K-module V , we say that A is a smooth atlas.

3. From atlas data to atlases Definition 3.1. We call atlas data (with model space V and chart index I) the following: the index set I is an e-pos (I, E, L), and

(1) to each i ∈ I, is associated a set

V i ⊂ V ("chart range"), (2) to each pair (i, j) ∈ E is associated a bijection φ ij : V j → V i such that φ ii = id V i , φ ij • φ jk = φ ik , φ -1 ij = φ ji , (3) if i ′ ≤ i, then V i ′ ⊂ V i , and if (i, j) ∈ E, (i ′ , j ′ ) ∈ E, i ′ ≤ i, j ′ ≤ j, then ∀x ∈ V j ′ : φ i ′ j ′ (x) = φ ij (x) . (4) if i ≤ m and j ≤ m and (V i ∩ V j ) = ∅, then there exists a unique k ∈ I with k ≤ i and k ≤ j and V k = V i ∩ V j .
We write [i, j] := k. We say that atlas data are topological if V carries a topology, all V i are open and all φ ij are homeomorphisms.

The idea how to reconstruct the manifold M from these data is simple: a point x ∈ V i shall be identified with a point y ∈ V j iff, possibly after restricting chart ranges to smaller sets V i ′ , resp. V j ′ , there is a transition function such that y = φ j ′ i ′ (x). That is, we shall define M as a quotient of the set

S := {(x, i) | x ∈ V i } ⊂ V × I
under a suitable equivalence relation which arises from combining E and L: Theorem 3.2. The following defines an equivalence relation on S:

• (x, i) ∼ (y, j) iff:

• ∃i ′ < i, ∃j ′ < j: (i ′ , j ′ ) ∈ E and (x, i ′ ) ∈ S, (y, j ′ ) ∈ S, φ i ′ j ′ (y) = x: symbolically, (x, i) (y, j) (x, i ′ ) (y, j ′ )
The quotient set M := S/ ∼ carries an atlas A (satifying the condition (3') of Def. 2.1), with index set I, and charts defined by

U i := {[x, i] | x ∈ V i }, φ i ([x, i]) := x.
Proof. Symmetry and reflexivity of ∼ are clear. Let us prove transitivity: assume (x, i) ∼ (y, j) and (y, j) ∼ (z, k).

There exist i ′ ≤ i, j ′ ≤ j, j ′′ ≤ j, k ′ ≤ k such that (i ′ , j ′ ), (j ′′ , k ′′ ) ∈ E and φ i ′ j ′ (y) = x, φ j ′′ k ′ (z) = y. Using (4), we let m := [j ′ j ′′ ].
By property (Epos) there exist i ′′ and k ′′ such that i

′′ ≤ i ′ , (i ′′ , m) ∈ E, k ′′ ≤ k ′ , (k ′′ , m) ∈ E.
By (3) and transitivity of L and E, it follows that (x, i) ∼ (z, k). The whole argument is summarized by the following diagram:

(x, i) v v v v v v v v v (y, j) v v v v v v v v v s s s s s s s s s (z, k) s s s s s s s s s (x, i ′ ) r r r r r r r r r (y, j ′ ) r r r r r r r r r (y, j ′′ ) u u u u u u u u u (z, k ′ ) u u u u u u u u u (x, i ′′ ) (y, m) (z, k ′′ )
Now let M = S/ ∼ and define the U i as in the claim. Then each [x, i] ∈ M belongs to some U i , so the U i form a covering of M. Let us show that φ :

U i → V i is well-defined. When U i is defined as in the claim, the map V i → U i , x → [x, i] is surjective. Let's show that it is injective: assume [x, i] = [y, i], so there exists i ′ ≤ i, j ′ ≤ i with φ i ′ j ′ (y) = x.
But according to Remark 2.1, this implies i ′ = j ′ , and so x = φ i ′ i ′ (y) = y. It follows that V i → U i is bijective, and hence its inverse

φ i : U i → V i , [x, i] →
x is well-defined. Moreover, it follows that (using (3) for the last equality)

φ j (φ -1 i (x)) = φ j [x, i] = φ j [φ j ′ i ′ (x), j] = φ j ′ i ′ (x) = φ ji (x)
, so the φ ij indeed describe the transition functions between the charts. This implies that, if (i, j) ∈ E, then

U i = U j . Let's show that (3') from Def. 2.1 holds: assume [x, i] = [y, j] ∈ (U i ∩ U j ). There exist i ′ ≤ i, j ′ ≤ j such that (i ′ , j ′ ) ∈ E (whence U i ′ = U j ′ ) and φ i ′ j ′ (y) = x. Thus (3') holds by taking m = i ′ .
Remark 3.1. We get "almost" an equivalence between atlasses and atlas data, the only difference being that, starting with (1), (2), (3) from Def. 2.1, we only recover (1), (2), (3'). The reason for this is that, in the procedure of reconstrucing, we loose control over size of chart intersections (we juste require that they are non-empty). It would be a bit technical to impose conditions allowing to keep such control, and we refrain from this.

Theorem 3.3. Assume that atlas data are topological and equip M with the final topology with respect to all maps φ -1 i :

V i → M (so U ⊂ M is open iff, for all i ∈ I, the set φ i (U ∩ U i ) is open in V ). Then all charts φ i : U i → V i are homeomorphisms.
Proof. The maps φ -1 i are continuous by definition of the topology on M. To see that φ j : U j → V j is continuous, let W ⊂ V j be open. We have to show that, for all i ∈ I, the set

Z := φ i (U i ∩ φ -1 j (W )) is open in V j . When (i, j) ∈ E, then we have Z = φ ij (W )
, which is open since φ ij is a homeomorphism. Else, using property (3') of Definition 2.1, restrict to smaller charts i ′ , j ′ with (i ′ , j ′ ) ∈ E, and the same argument implies that each point of Z is an inner point, and so Z is open.

See [START_REF] Bertram | Projective completions of Jordan pairs. Part II: Manifold structures and symmetric spaces[END_REF], Theorem 5.3, for examples of smooth manifolds obtained by the construction described in the theorem. Note that, already for usual, real manifolds, M need not be Hausdorff, even if V is Hausdorff. A simple counter-example is given by atlas data

V 1 = V 2 = R, V 12 = R × , φ 12 (x) = x, so M is "R with origin doubled".
Theorem 3.4. Assume M has an atlas A of type G = (G 0 , G 1 ), and assume that T is a functor from G to a pseudogroup H = (H 0 , H 1 ) acting on a space W = T V . Then we may define a manifold T M with atlas T A given by all (T V k , T φ ij ) k∈I,(i,j)∈E , which is modelled on W and of type H.

Proof. T A = (T V k , T φ ij ) k∈I,(i,j)∈E are again atlas data, for the same e-pos (I, E, L), and hence give rise, by the preceding theorem, to an atlas.

For instance, if all φ ij are smooth, then T may be the tangent functor, or any other Weil functor. Thus one constructs the tangent bundle, or other Weil bundles, of a manifold M (cf. [START_REF] Bertram | A general construction of Weil functors[END_REF]).

Examples; maximal atlases

From an economical viewpoint, one is interested in keeping atlases of manifolds as small as possible. On the other hand, for theoretical purposes, most mathematicians are used to work with maximal atlases.

V = V 0 = V 1 = V 2 = K 2 , and
V 01 = {(u, v) | u = 0} = V 02 , V 12 = {(u, v) | v = 0}, V 012 = {(u, v) | u = 0, v = 0}. φ 01 (u, v) = (u -1 , u -1 v), φ 02 (u, v) = (u -1 v, u -1 ), φ 12 (u, v) = (v -1 u, v -1 ).
By direct computation, the reader may check that each of these maps is of order two (so

φ νµ = (φ µν ) -1 = φ µν ) iff in K the identities (x -1 ) -1 = x,
x(x -1 y) = y are satisfied. Likewise, we have φ 01 • φ 12 = φ 02 on V 012 iff, moreover in K we have:

(xy) -1 x = y -1 .
Now, it is well-known that these identities hold in any alternative field (in particular, for K = O, the octonions). Thus our reconstruction theorem implies that for all alternative fields we may glue together copies of K 2 to get a projective plane over K. (Essentially, this way of constructing the octonion plane is the one described by Aslaksen, [START_REF] Aslaksen | Restricted homogeneous coordinates for the Cayley projective plane[END_REF].) If K is associative, similar formulas permet to describe the groupoid of transition functions. It is special for this example that this groupoid embeds into a finite subgroup of the projective group PGL(n + 1; K).

Example 4.4. Assume M = V . The biggest possible atlas is given by all possible local bijections of V , that is, I = B loc (V ), where an index g is identified with the chart g : dom(g) → im(g) it describes. Then L is given by inclusion: f ≤ g if f is a restriction and corestriction of g, and

E = {(f, g) ∈ B loc (V ) 2 | dom(f ) = dom(g)}
The transition functions are φ f g (x) = f g -1 (x). It follows that the morphism of the e-pos (I, E, F ) to (P(V ), B loc (V )) is given by

I → P(V ), f → im(f ), E → B loc (V ), (f, g) → f • g -1 .
This atlas is maximal in the sense to be described next. If V carries a topology, then of course one will take only open sets and homeomorphisms.

Example 4.5. If M = V , we may also take I = Bij(V ), the group of all bijections (or some subgroup, like the group Gl(V ) if V is a linear space). Then E = I × I, and L is trivial. Transition functions are as in the preceding example.

Theorem 4.1. Assume A is a G-atlas on M, modelled on V , where G is a pseudogroup of transformations on V . Then there exists a maximal G-atlas à containing A, which is defined by:

à = (U, φ) φ : U → W bijection, W ∈ G 0 , U ⊂ M, ∀i ∈ I : [φ i • φ -1 : φ(U ∩ U i ) → φ i (U ∩ U i )] ∈ G 1
Proof. This statement is standard in many differential geometry textbooks (e.g., [START_REF] Kobayashi | Foundations of Differential Geometry[END_REF], p.2). The main point is to check that φ • ψ -1 ∈ G 1 , for any two charts φ, ψ having same domain (or having non-empty intersection). This is locally true, by intersecting with suitable charts from A, and the local-to-global property (PsG) of a pseudogroup (Def. A.2) permits to conclude that φ • ψ -1 ∈ G 1 . We leave it to the reader to formalize these arguments (much like the proof of Theorem 3.2). Definition 4.2. A (V, G)-manifold is a set M with a maximal G-atlas. A smooth manifold (over K) is a (V, G)-manifold, where V is a topological K-module over a topological ring K, and G the pseudogroup of locally defined diffeomorphisms of V .

From morphisms to morphism data

In this section, assume (M, A, V ) and (M ′ , A ′ , V ′ ) are manifolds with atlas, and f : M → M ′ a map. We describe f with respect to the atlases. Definition 5.1. Let (I, E, L), resp. (I ′ , E ′ , L ′ ) be the e-pos of M, resp. of M ′ . Then f induces binary relations F ⊂ (I × I ′ ) and R ⊂ (E × E ′ ) by

F := (i ′ , i) ∈ I × I | f (U i ) ⊂ U i ′ , R := (i, k), (i ′ , k ′ ) ∈ E × E ′ | (i ′ , i), (k ′ , k) ∈ F . Elements (i, k, i ′ , k ′ ) ∈ R may be represented by a parallelogram: i E F `k F b b b b b b b i ′ E ′ k ′
Lemma 5.2. The relation F respects the partial order L in the sense that:

(1) if

(i ′ , i) ∈ F and k ≤ i, then (i ′ , k) ∈ F , (2) if (i ′ , i) ∈ F and i ′ ≤ m ′ , then (m ′ , i) ∈ F . Proof. (1) f (U i ) ⊂ U i ′ and U k ⊂ U i implies f (U k ) ⊂ U i ′ .
(2): similar.

Definition 5.3. Whenever (i ′ , i) ∈ F , we define the (i ′ , i)-component of f by

f i ′ i := φ i ′ • f • φ -1 i : V i → V i ′ . Lemma 5.4. Whenever (i, k), (i ′ , k ′ ) ∈ R, then φ k ′ i ′ • f i ′ i = f k ′ k • φ ki : V i f i ′ i / / φ ik V ′ i φ ′ k ′ i ′ V k f k ′ k / / V ′ k Proof.
Immediate from the definition of the φ ij and f j ′ j .

If we want to reconstruct f from the data (f i ′ i ) (i ′ ,i)∈F , then every x must admit a neighborhood on which f is described by some component. This is not automatic -it is a condition, very much like "ordinary continuity": Definition 5.5. We say that f is atlas-continuous if, for all x ∈ M and all j ∈ I ′ with f (x) ∈ V j , there exists i ∈ I with x ∈ V i and f

(V i ) ⊂ V j (that is, (j, i) ∈ F ).
Remark 5.1. If the atlas of M is saturated, then this condition amounts to saying that f is continuous in the usual sense.

From morphism data to morphisms

We shall reconstruct f from the "morphism data" f ij . More precisely, we shall see that morphism data are certain natural relations of certain ordered groupoids (cf. Def. B.1). Written out, this means: Definition 6.1. Assume (I, E, L, (φ ij ) (i,j)∈E ) and (I ′ , E ′ , L ′ , φ i ′ j ′ ) (i ′ ,j ′ )∈E ′ are atlas data belonging to manifolds with atlases (M, A), (M ′ , A ′ ). Morphism data between them are given by: a special natural relation (F, R) between the e-pos (I, E, L) and

(I ′ , E ′ , L ′ ), that is, a pair of binary relations (F, R) ⊂ (P(I × I ′ ) × P(E × E ′ )) such that: whenever ((i, k), (i ′ , k ′ )) ∈ R, then (i ′ , i) ∈ F and (k ′ , k) ∈ F , and, for each (i ′ , i) ∈ F , a map f i ′ i : V i → V i ′ such that, whenever (i, k), (i ′ , k ′ ) ∈ R, then V i f i ′ i / / φ ik V ′ i φ ′ k ′ i ′ V k f k ′ k / / V ′ k ( * )
Moreover, the following restriction and co-restriction properties shall be satisfied:

(1) if (i ′ , i) ∈ F and k ≤ i, then (i ′ , k) ∈ F and f i ′ k = f i ′ i | V k , (2) if (i ′ , i) ∈ F and i ′ ≤ m ′ , then (m ′ , i) ∈ F and ∀x ∈ V i : f i ′ i (x) = f m ′ i (x). Morphism data are called full if ∀k ∈ I, ∀x ∈ U k : ∃i ≤ k, ∃i ′ ∈ I ′ : x ∈ U i , (i ′ , i) ∈ F .
Theorem 6.2. Let (M, A), (M ′ , A ′ ) be manifolds with atlas. Then, given full morphism data, there is a unique map f : M → M ′ such that, whenever x ∈ M,

(i ′ , i) ∈ F and x ∈ U i , f (x) = φ -1 i ′ (f i ′ i (φ i (x))) . Proof.
Uniqueness is clear from the last formula, since (by the fullness condition) for every x ∈ M there exist (i ′ , i) ∈ F with x ∈ U i . To prove existence, we define f (x) by that formula, with respect to some choice of (i ′ , i) ∈ F with x ∈ U i , and we have to prove that with respect to another such choice, (j ′ , j),

φ -1 i ′ (f i ′ i (φ i (x))) = φ -1 j ′ (f j ′ j (φ j (x))). If (i, j) ∈ E and (i ′ , j ′ ) ∈ E ′ , so (i, j), (i ′ , j ′ ) ∈ R, then this follows from φ j ′ ,i ′ •f i ′ i = f j ′ j • φ ji . If (i, j) or (i ′ , j ′ )
are not in E, resp. E ′ , then using fullness and restriction and corestriction properties, we may find smaller charts with the corresponding property, and we get the same result.

Example 6.1. Assume that M = M ′ . Then f ij := φ ij defines morphism data. The preceding theorem shows that these morphism data belong to the identity map id M . Thus one may say that manifold data are "morphism data of a would-be-identity". Definition 6.3. Morphism data, and morphisms, are said to have some property, such as smoothness, if all components f ij have this property. Theorem 6.4. If g and f are (smooth) composable morphisms (having some property, such as smoothness) of manifolds with atlas, then so is g • f .

Proof. One has to check that (g • f ) ℓi = g ℓk • f ji , together with the relation G • F from I to I ′′ , define (smooth) morphism data. No new ideas are involved here, and we may leave it to the reader.

One may say that the proof of Theorem 6.4 consists of putting diagrams of the type (*) from Def. 6.1 next to the other (horizontally), and thus define new diagrams. But one may also put them one over the other (vertically), and this again defines new diagrams. Indeed, this can be done for general natural relations (appendix B), and then defines a double category. However, in the present case the second category structure is not relevant (since atlas data are a "would-beidentity" morphism, as said above, and hence the second composition law somehow only reflects the composition of identity maps). Nevertheless, these remarks show that morphisms of manifolds sit inside very natural and bigger double categories. There is also a link with 2-categories: Example 6.2. Consider the case V = M, V ′ = M ′ with their atlases described in example 4.4. Let f : V → V ′ be a map (continuous if data are topological). Then φ gh (x) = gh -1 (x) and f kℓ (x) = kf ℓ -1 (x). Following the standard terminology from matrix theory, let us say that all components f kℓ are equivalent to each other. In our example, f can be recovered directly from its equivalent pictures, via f = f id V ′ ,id V . For a general manifold, there is no such formula, and one has to use the definition given in Theorem 6.2. Example 6.3. Assume now that, with notation as in the preceding example, V = V ′ . Then we may recover f already from the restricted data f kk = kf k -1 . Following usual terminology from the linear case, we say that these components are similar to each other. Definition 6.5. Wih notation as in Definition 6.1 and Theorem 6.2, assume that

M = M ′ , V = V ′ , A = A ′ . Then the collection (f k ′ k ) with (k ′ , k) ∈ F such that k ≤ k ′ are called restricted morphism data, or similarity data.
Morally, if f is "sufficiently close to the identity map of M", then f can be recovered from its restricted morphism data, as in Theorem 6.2, just as in linear algebra, where we describe endomorphisms by using the same base in the domain and in the range space.

Comments

Some short remarks and comments on related topics and open problems:

(1) (Finite atlases.) Like projective spaces (example 4.3), many algebraic varieties come together with, more or less "canonical", finite atlases. In particular, this is true for Grassmann and Lagrangian varieties, and more generally, symmetric R-spaces (the "Jordan geometries" from [START_REF] Bertram | Projective completions of Jordan pairs. Part II: Manifold structures and symmetric spaces[END_REF][START_REF] Bertram | Jordan Geometries -an Approach via Inversions[END_REF]) where such atlases have a direct relation with Jordan theory. It should be interesting to describe and study the interaction between the abstract theory and the combinatorial and algebraic structure of such atlases. (2) (Conceptual calculus.) As said in the introduction, this is the starting point for the present work [START_REF] Bertram | Conceptual Differential Calculus. I : First order local linear algebra[END_REF]). The first order difference groupoid M {1} of a manifold M (cf. loc. cit.) is a natural example for the following item: (3) (3-categories.) As said above, 2-categories are naturally related to the present approach. What about 3-categories? When the space M carries itself the structure of some kind of groupoid or pregroupoid (e.g., principal bundles, Lie groupoids), then such structure together with those described in the present work should be compatible, and thus give rise to (strict) higher order categories. (4) (Categorical aspects.) For a discussion of purely categorial aspects of the notion of manifolds and their morphisms, see the n-lab, in particular https://ncatlab.org/nlab Our definition of manifolds via e-poses is also related to Lawson's construction of ordered groupoids via combinatorial groupoids, [START_REF] Lawson | Constructing ordered groupoids[END_REF].

Appendix A. Ordered groupoids Notation. I use notation as in [START_REF] Bertram | Conceptual Differential Calculus. I : First order local linear algebra[END_REF], Appendix B: a groupoid is an algebraic structure G = (G 0 , G 1 , π 1 , π 0 , δ, * ), where G 0 is the set of objects, G 1 is the set of morphisms, π 1 : G 1 → G 0 the target, and π 0 : G 1 → G 0 the source projection, δ : G 0 → G 1 the unit section, and * : G 1 × G 0 G 1 → G 1 the composition map. Our convention is that g * h is defined iff π 1 (h) = π 0 (g). The inverse of g is denoted by g -1 . A small category (small cat) is defined like a groupoid, without assuming existence of inverses. The definition of ordered groupoid goes back to Charles Ehresmann, see references in [START_REF] Lawson | Constructing ordered groupoids[END_REF]. The following form of the axioms is taken from [AGM14]: Definition A.1. An ordered groupoid is a groupoid (G 0 , G 1 , π 0 , π 1 , δ, * ) together with partial order relations L (or ≤) on the sets G 0 and on G 1 , such that:

(OG0) ∀x, y ∈ G 0 : x ≤ y iff δ(x) ≤ δ(y), (OG1) g ≤ h iff g -1 ≤ h -1 , (OG2) if g ≤ h, g ′ ≤ h ′ and if g * g ′ and h * h ′ are defined, then g * g ′ ≤ h * h ′ , ( OG3
) for all morphisms g : x → y and all objects x ′ with x ′ ≤ x, there exists a unique morphism g ′ : x ′ → y ′ with g ′ ≤ g, y ′ ≤ y:

x L g / / y L x ′ g ′ / / y ′
One may think of g ′ as a kind of restriction of g to x ′ .

Remark A.1. Every partially ordered set gives rise to a small category (subcat of the pair groupoid), and hence the set [x] = {x ′ | x ′ ≤ x} is the object set of a small cat. Every g ∈ G 1 then defines a functor from [π 0 (g)] to [π 1 (g)].

Example A.1. (e-poses) Recall that an equivalence relation E on a set I is the morphism set of a groupoid with object set I. Thus an e-pos (I, E, L) (Definition 1.1) is an ordered groupoid. Here the partial order on E is completely determined by the partial order on I: necessarily, (i ′ , j ′ ) ≤ (i, j) iff [i ′ ≤ i and j ′ ≤ j].

Example A.2. (local bijections) Recall that endorelations on a set V form a small category (C 0 , C 1 ) = (P(V ), P(V × V )). Morphisms are relations R ⊂ (V × V ) with composition being relational composition, source and target

π 0 (R) = dom(R) = {x ∈ V | ∃y ∈ V : (y, x) ∈ R} π 1 (R) = im(R) = {y ∈ V | ∃x ∈ V : (y, x) ∈ R}.
Objects and morphisms carry a natural partial order L given by inclusion of sets.

The full pseudogroup of V , or groupoid of local bisections of the pair groupoid of V , is the subcat (P(V ), B loc (V )) of (C 0 , C 1 ) whose morphisms are precisely the local bijections, that is, the injective and locally functional relations R:

(y, x), (y ′ , x), (y, x ′ ) ∈ R ⇒ y = y ′ , x = x ′ .
These are the graphs Γ f of bijections f : V ′ → V ′′ with V ′ , V ′′ ⊂ V , and f ≤ g means that f is a restriction of g. Properties (OG0) -(OG3) are easily checked.

Example A.3. (pseudogroups of smooth maps) Assume now that V carries additional structure, e.g., V is a topological vector space over a topological field K.

Then we may define the subgroupoid of B loc (V ) of all locally defined diffeomorphisms of class C k : its objects are open subsets of V , and its morphisms graphs

of C k -diffeomorphisms f : V ′ → V ′′ with V ′ , V ′′ open in V .
It is again an ordered groupoid, since restriction (together with co-restriction to the image) of f is again a C k -diffeomorphism. This defines an ordered groupoid B k loc (V ), called the pseudogroup of local C k -diffeomorphisms. Similarly, any kind of structure on V with structure preserving maps that can be restricted (and co-restricted) to suitable subsets, defines a certain pseudogroup, which is an odered subgroupoid of B loc (V ). If the property is defined by "local" properties (such as smoothness), the following pseudogroup property is ensured:

Definition A.2. A subgroupoid G = (G 0 , G 1 ) of the full pseudogroup (P(V ), B loc (V )) is called a pseudogroup of transformations of V if: (PsG) assume that U = ∪ α∈J U α with all U α ∈ G 0 and that f : U → U ′ ⊂ V is a bijection such that ∀α ∈ J: f | Uα ∈ G 1 . Then f ∈ G 1 . Appendix B. Natural relations between groupoids Assume G = (G 0 , G 1 ) and G ′ = (G ′ 0 , G ′ 1 ) are groupoids. A morphism between G and G ′ , or functor, is a pair of maps h 0 : G 0 → G ′ 0 , h 1 : G 1 → G ′
1 preserving all structures. However, there are other ways to turn groupoids into a category, such as the following: Definition B.1. A natural relation between G and G ′ is given by a pair (F, K) of binary relations on objects, and a binary relation R on morphisms

(F, K) ∈ P(G ′ 0 × G 0 ) 2 , R ∈ P(G ′ 1 × G 1 ) such that: (NR) if (g, h) ∈ R, then letting x := π 0 (g), x ′ := π 0 (g ′ ), y := π 1 (g), y ′ := π 1 (h),
we have (x, x ′ ) ∈ F and (y, y ′ ) ∈ K:

π 0 • R = F • π 0 , π 1 • R = K • π 1 .
Example B.1. In an ordered groupoid, L is a special natural endorelation of G. If we wanted to draw diagrams of natural relations between ordered groupoids, we would need a third dimension for representing them! It should be clear that composition of natural relations gives again a natural relation: if (F, K; R), (F ′ , K ′ ; R ′ ) are natural between G and G ′ , resp. between G ′ and G ′′ , then (F ′ • F, K ′ • K; R ′ • R) is natural between G and G ′′ . For the purposes of the present paper, this remark suffices. However, it is certainly useful to note that just as for natural transformations in general category theory, there are in fact two compositions, and that we get a double category (see, e.g., [START_REF] Bertram | Conceptual Differential Calculus. I : First order local linear algebra[END_REF][START_REF] Bertram | Conceptual Differential Calculus. II : Cubic higher order calculus[END_REF] for definitions): where the two compositions on 2-morphisms are given by:

(1) relational composition: if (F, K; R) and (F ′ , K ′ ; R ′ ) are composable pairs of natural relations, then (F ′ • F, K ′ • K; R ′ • R) is again a natural relation, (2) pointwise * -composition: if (F, K; R) and (F ′ , K ′ ; R ′ ) are natural relations on the same groupoid G and

K = F ′ , let R ′ * R := (g ′′ , h ′′ ) ∈ G 1 × G 1 ∃(g ′ , h ′ ) ∈ R ′ , (g, h) ∈ R, (g ′ , g), (h ′ , h) ∈ G 1 × G 0 G 1 , g ′′ = g ′ * g, h ′′ = h ′ * h .
Then (F, K ′ ; R ′ * R) is again a natural relation.

Proof. The shortest proof is probably by remarking first that the pair groupoid functor PG applied to a small category C = (C 0 , C 1 ), yields a small double cat PG(C):

C 1 × C 1 / / / / C 1 C 0 × C 0 / / / / C 0 .
(Indeed, PG is a cat rule in the sense of Appendix B in [START_REF] Bertram | Conceptual Differential Calculus. II : Cubic higher order calculus[END_REF].) Applying the power set functor P to this, we get again a small double cat P(PG(C)). Now observe that the structure defined in the theorem, when G = C, is nothing but a sub-double cat of this small cat. Indeed, composition of 2-morphisms corresponds to putting parallelograms as depicted in Def. B.1, side by side, resp. one over the other.

It is obvious that the special natural endorelations of G form a sub-doublecat, where the upper horizontal double arrows are replaced by a single arrow. On the other hand, to every natural endorelation (F, K; R) we may associate a relation Ř between objects and morphisms of C by restricing to units: Remark B.2. It seems that the concept of natural relation has so far not yet been considered in category theory (the reason might be that people generally work in the context of "large" categories, where one is not used to consider binary relations that are not necesarily functional). See, however, a short remark in the n-lab, https://ncatlab.org/nlab/show/natural+transformation ("An alternative but ultimately equivalent way..."), pointing into the direction pursued here.

Ř := {(x, g) ∈ G 0 × G 1 | (δ(x), g) ∈ R},

  Notation. By P(X) we denote the power set of a set X. Relational composition of binary relations R ⊂ (C × B), S ⊂ (B × A) is defined by R • S = {(c, a) | ∃b ∈ B : (c, b) ∈ R, (b, a) ∈ S}, and the graph of a map f :

  Example 4.1. An atlas is trivial, I = {e} and φ ee = id, if and only if M = U e = V e ⊂ V , with just one chart. Example 4.2. An atlas has e-pos of the form given in Example 2.1 (sphere) iff M arises from "gluing together" two subsets of V along certain proper subsets which are identified via a bijection φ. Example 4.3. Consider the e-pos with 12 elements belonging to the projective plane (Introduction and Example 2.2), and let

/

  / y ′ If, moreover, F = K, we speak of a special natural relation. Remark B.1. Condition (NR) implies that, under relational composition,

Theorem B. 2 .

 2 The set N (G) of natural relations on a groupoid G forms (the set of 2-morphisms of ) a small double category N (G)

  and define Ň (G) ⊂ P(G 0 × G 1 ) = { Ř | R ∈ N (G)}.The relation Ř is the precise analog of what one might call a natural transformation from the functor F : C 0 → C ′ 0 to the functor K : C 0 → C ′

  Remark 2.3. In terms of relational composition, (Epos) implies that E • L ⊂ L • E. This implies that both H 1 := L • E and H 2 := E • L are transitive relations. In case of Definition 1.2, (i, k) ∈ H ν means, for ν = 1, 2, "U i is included in U k ", respectively, "U i is included in U k , and φ i extends to chart onto U k ". Thus H 1 and H 2 are different relations, in general. We will not work with them in this paper.

	a a a a a		j
		k
	Proof. It is clear that E is an equivalence relation and L a partial order. Uniqueness
	in (ii) is clear since necessarily U j ′ = U i ′ and φ i ′ (x) = φ j ′ (x), and existence follows
	from (3): j ′ := [j, i ′ ] satisfies (Epos). Similarly, (iii) is proved: uniqueness follows
	from the given conditions, and existence by taking k := [ij].
	Remark 2.1. By uniqueness, in every e-pos, [ i ≤ j and (i, j) ∈ E ] implies i = j.
	Likewise, [ i ≤ m, j ≤ m, (i, j) ∈ E] implies i = j. Thus in an e-pos, triangles such
	as the following are always degenerate in the sense that i = j:
	m i	c c c c c c c c	j
	Remark 2.2. En e-pos is a special instance of an ordered groupoid, see Appendix A.