Lie Groupoids as generalized atlases

LIE GR OUPOIDS AS GENERALIZED A TLASES JEAN PRADINES Abstract. Sta rting with some motiv ating examples (classsical atlases for a manifold, space o f lea v es of a foliation, gr oup orbits), we prop ose to view a Lie groupoid as a generalized atlas f or the “vir tual structure” of its orbit space, the equiv a lence b et we en atlases being here the smo oth Morita equiv alence. This “structure” keeps memory of the isotropy groups and of the smo othness as we ll. T o tak e the s m oothness into accoun t, we claim that w e can go very far by retaining just a few f ormal properties of embeddings and surm er sions, yielding a very p olymorphous unifying theory . W e suggest further developmen ts. Contents 1. Int ro duction. 2 2. Some motiv ating exa mples. 3 2.1. Regular equiv ale nces. 3 2.2. Classical atlases. 5 2.3. Group and group oid actions. 6 2.4. F oliations on B . 7 3. Dipt ychs. 9 3.1. Deﬁnition of dipt ychs. 9 3.2. Some v ar iants for data and axioms. 10 3.3. A few examples of basic large dipt ychs. 11 3.4. (Pre)dipt ych str uctures on simplicial (and related) categor ies. 13 4. Commutativ e s quares  in a dipt yc h D = ( D ; D i , D s ). 14 4.1. Three basic t ype s of squares . 14 4.2. Basic dipt ych structures on  D . 15 5. Diagrams of type T in a dipt ych. 16 5.1. Deﬁnitions : ob jects and morphisms. 17 5.2. Dipt yc h structures on the catego r y of diag rams. 17 5.3. The expo nential law for diagrams. 18 6. Group oids as diagra ms in E = Set . 19 6.1. Some remarks ab out N + c and N + o . 19 6.2. Characteriza tion o f the nerv e of a group oid. 19 7. D -group oids . 21 7.1. Deﬁnition of D -group oids. 21 7.2. Principal and Gode ment D -group oids. 21 7.3. Regular group oids. 23 8. The category Gp d ( D ). 24 Date : 04/11/03. 1991 Mathematics Subje c t Classiﬁc ation. 58H05. Key wor ds and phr ases. Lie groupoids, s paces of l ea ve s, orbit spaces. 1 2 JEAN PRADINES 8.1. D -functors. 24 8.2. Actors. 24 8.3. D -equiv alence s. 25 8.4. Dipt yc h structures on Gp d ( D ). 25 8.5. The category Gp d ( N + ∗ c ). 26 9. Double functoriality o f the deﬁnition of D -group oids. 28 9.1. Biv aria nce of D -group oids and D -functors. 28 9.2. Examples for the left functoriality . 29 9.3. Examples for the right functoria lit y . 30 9.4. D − natural transforma tions, holo morphisms. 30 10. The butterﬂy diagra m. 31 10.1. Generalized “structure” of the orbit space. 31 10.2. Inv e rting equiv alences. 31 10.3. Example. 33 11. Epilogue. 33 References 34 1. Introduction. The aim of the present lecture is, rather than to present new r esu lts , to sketc h some unifying c onc epts and gener al metho ds wished to b e in Char les Ehresmann’s spirit. As usual I a m exp ecting t hat ge ometers will think t hese sorts o f concepts ar e to o gener al and too abstract for being useful, while c ate goricists will es timate they are too sp ecial a nd too concr ete for being interesting. How ev er let us g o. In the following, I shall b e c o ncerned with a cer tain structure B (basically thought as a manifold) endo w ed with a certain equiv alence relation denoted by ∼ or R , and I would like to describe what kind of s mo othness or structure is inher- ited from B b y the quotient set Q = B /R . The canonic a l pro jection will b e denoted by B q → Q . The r elation R will b e ident iﬁed with its graph, deﬁned by the following pullback squar e, in which β = pr 1 , α = pr 2 . W e also denote by R τ R → B × B t he R β > B pb B α ∨ q > Q q ∨ canonical injection, with τ R = ( β , α ). In case when the structure B is just a top olog y , the well known answer is given by the so-ca lled quotient or identiﬁcation top olog y on Q , which owns the go o d exp ected universal pr o p erty in the ca tegory T op . How ev er we notice that, when given other similar data B ′ q ′ → Q ′ , o ne ha s not in g eneral, in spite of a famous er ror (in B ourbaki’s ﬁrst edition), a homeomo rphism b etw een the pro duct Q × Q ′ and the quotient space of B × B ′ by the pro duct o f the tw o equiv ale nce r elations, thoug h this is true in tw o impo rtant cases , when q and q ′ are b oth op en o r prop er, since q × q ′ has the same prop erty . GENERALIZED A TLASE S 3 On the opposite, when B is a ma nifold, it is well known that there is no such satisfactory answer when staying inside the category D = Dif of (smo oth maps betw een) smo oth manifolds, i.e. there is no suitable manifold structur e for Q . Now for facing this situation there may b e tw o opp osite, or b etter complemen- tary , st yles of approaches. The ﬁrst one consists in “co mpleting” D , i.e. em b e dding D in a large r categ ory b D by a dding new ob jects in such a wa y than b D has b etter categoric al prop erties, i.e. has enough limits for a llowing to deﬁne a go o d universal quotient. F or instance one can wish b D be a top os. V ario us interesting so lutions do exist, the study of which is out o f our present scop e. W e just mention, b esides Ehresmann’s appro aches, tw o dual ways (con- sidered, under v arious asp ects, b y several lecturers at the presen t Conference) of deﬁning ge ner alized smo oth structures on Q , one (ﬁrst s tressed b y F r¨ olicher) con- sisting in deﬁning the smo oth curves, while the second metho d (emphasized b y Souriau with his diﬀeolo gies) consider s the smo o th functions on Q . Alain Connes’ “non-commutativ e” a pproach is also rela ted. W e follow here an o ppo site path, avoiding to add to o many (necessarily patho - logical) new ob jects, a nd trying to sta y within D . W e do not attempt to deﬁne a gener a lized smo oth s tr ucture (in the set-theoretical sense) on t he mo st genera l quotients, and limit ours e lf to o b jects which are s uﬃcien tly close to manifolds in the sense that they can b e descr ib e d by means of equiv alence classe s of some simple t yp es of diagra ms in D ; we do not try to introduce the limits of s uch diagrams in the categoric al sense. Inde e d we t hink that the classic al c ate goric al c onc ept of limit involves in gener al a c ertain loss of the info rmation enc apsulate d in the c onc ept of a su itable e quivalenc e class of diagr ams, but we shal l not attempt to develo p mor e formal ly su ch a gener al c onc ept her e, though we think it a very pr omising way, b eing c ont ent with il lustr ating this p oint of view by the imp ortant s p e cial c ase sketche d pr esently. 2. Some motiv a ting examples. Before g oing to abstra ct genera l deﬁnitions, I s tart by giving some elementary examples (to be made more precise later) of the kinds of ob jects I have in mind. 2.1. R e gular equi v alences. The ideal situation is of course that of the so- called regular equiv alences. This means that there ex ists on Q a (necess a rily unique) manifold structure such that B q ◮ Q is a s urmersion (= surjectiv e submersion). (Here w e start anticipating some pieces of notation for ar rows to b e systematized later within a more general setting). Go dement’s the or em gives a c haracteriza tion of those equiv alence s b y prop erties of the graph R summarized by the following notations : R α ◮ β ◮ B and R ◮ τ R > B × B where again the black triangle head for an ar row stands for “surmer sion”, while the black tria ng le tail means “ embedding” (in the s e nse of Bo ur baki), or “ prop er embedding” when dealing with Hausdorﬀ manifolds. 4 JEAN PRADINES These conditions expr e ss tha t R , regarded (in a seemingly p edantic wa y) as a subgroup oid of the (banal) gro up oid B × B , is indeed a smo oth (or Lie) gro upo id in the sense intro duced by Ehresma nn, embedded in B × B , and the manifold Q may b e viewed as the orbi t sp ac e of this gr o up o id. A Lie g r oup oid R satisfying the framed conditions will be called a princip al o r Go dement gro upo id. W e shall see in the next exa mple why it is con venien t to consider R α ◮ β ◮ B q ◮ Q as a “ gener alize d (non ´ etale) atlas ” for Q . If we have ano ther “atlas” R ′ α ◮ β ◮ B ′ q ′ ◮ Q of the sa me manifold Q , we can take the ﬁbred product of q a nd q ′ , and w e get a co mm utative diagra m expr ess- S R ◭ f ∼ E β S H α S H R ′ f ′ ∼ ◮ B β R H α R H ◭ p B ′ β R ′ H α R ′ H p ′ ◮ Q r H ◭ q ′ q ◮ ing the “ c omp atibility ” of these “atla s es”, which means that they deﬁne the same (manifold) structure on Q . More pr ecisely the graph S (which in turn may b e viewed as a group oid) can be obtained in the following wa y by means of the commutativ e cub e b elow, the b o ttom S f ∼ ◮ R R ′ g ′ ∼ ◮ f ′ ∼ ◮ H Q g ∼ ◮ h ∼ ◮ E × E τ S ∨ H p × p ◮ B × B τ R ∨ B ′ × B ′ τ R ′ ∨ H q ′ × q ′ ◮ p ′ × p ′ ◮ Q × Q τ Q ∨ H q × q ◮ r × r ◮ face of whic h is the pullback of q × q and q ′ × q ′ , and whic h is constructed step by step by pulling ba ck along the vertical arrows, s tarting with τ Q . The last one is just the diago nal o f Q , a nd may b e co ns idered as the anchor map o f the “ nul l ” gr oup oid GENERALIZED A TLASE S 5 Q (consisting of just units). The upp er face is then also a pullback, as well a s all the six fac es, and also the v ertical diagona l s quare with three dashed e dg es. The ∼ symbols emphasize (very sp ecial instances of ) “sur mersive equiv alences ” b etw een Lie group oids . (One can obser ve on this diagr am the general prop erty of “parallel transfer b y pulling back” fo r the embeddings and surmer sions). Thu s we s ee that the “compatibility” of the tw o (generalized) atlases R α ◮ β ◮ B q ◮ Q and R ′ α ◮ β ◮ B ′ q ′ ◮ Q for Q is expressed by the exis tence of a common “reﬁnement” (pictured a bove with dashed arrows) : S α ◮ β ◮ E r ◮ Q . One might prov e dire ctly a con v erse, which indeed follows from mor e genera l consideratio ns. 2.2. Cl assical atlase s. An imp ortant specia l cas e of the pre vious one (which ex- plains the terminology) will be given by the following diagra mma tic descr iption of atlases and co vers of a manifold Q . Let ( ϕ i : V i → U i ) ( i ∈ I ) be an atlas of the manifold Q , where ( V i ) ( i ∈ I ) is a cover of Q and the co domains U i ’s o f the c harts ϕ i ’s a re o p en sets in some mo del space (whic h may b e R n or a Banach space). Let V ij be V i ∩ V j and U ij be the image of V ij in U i by the restr iction of ϕ i . Set V = ` i ∈ I V i , with its c a nonical pro jection r : V → Q (whose da tum is equiv alent to the datum of the cov ering), U = ` i ∈ I U i (a tr iv ial ma nifold), R = ` ( i,j ) ∈ I × J U ij , a nd S = ` ( i,j ) ∈ I × J V ij . The charts ϕ i ’s de ﬁne a bijection ϕ : V → U as w ell as a bijection φ : S → R . Note that S , together with its canonica l pro jection on to Q , deﬁnes the intersec- tion cov ering, while, with its tw o canonica l pr o jections onto V , it can b e view ed also as the graph of the equiv alence relation ass o ciated to the surjection r . Using the bijectio ns ϕ , φ , we hav e analogo us consider ations for R a nd U , but moreov er the latter ar e (trivia l) ma nifolds, and the equiv alence is r e gular, so that we recover a (very) specia l instance of the situation in the ﬁrst exa mple. Here the pro jections α R , β R are no t only sur mer sions but moreov er ´ etale maps (of a sp ecial t yp e, which migh t b e called trivia l) ; here they will be pictured by arrows of type ⊲ . More precisely their restr ictions to the comp onents of the copro duct R deﬁne homeomor phisms onto the open se ts U ij ’s, and the datum o f the smooth group oid R with base U is precisely equiv ale nt to the datum o f the pseudo gr oup of changes of charts . The situation is summed up by the following diagr a m (with q = r ◦ ϕ − 1 ), which S . . . . . . . . . . . . φ ≈ > R V β S ∨ . . . . . . . . . α S ∨ . . . . . . . . . . . . . . . . . . . . ϕ ≈ > U β R ▽ α R ▽ Q r ∨ . . . . . . . . . = = = = = Q q ▽ 6 JEAN PRADINES describ es the gener alize d atlas R α ⊲ β ⊲ U q ⊲ Q asso ciate d to a classic al atlas (whence the terminolog y). The dotted arrows in the dia gram are to remind tha t the left column lies in Set , while the right column lies in Dif . Now if we deﬁne a reﬁnement of the pr evious atla s, denoted by U q ⊲ Q for brevity , as a n atlas W r ⊲ Q such that r admits a surjective 1 factorization W p ⊲ U , it is easy to see that the compatibilit y of tw o atlases U q ⊲ Q a nd U ′ q ′ ⊲ Q ma y be expressed b y the existence o f a common reﬁnemen t W r ⊲ Q , and w e get a special case of the notion of c o mpatibility introduced in the previo us subsection, where the general surmersions are replaced by (v ery spec ial) surjective ´ etale maps. The compatibility diagram now rea ds as below, and this explains the terminolog y Z R ⊳ f ∼ W β Z ▽ α Z ▽ R ′ f ′ ∼ ⊲ U β R ▽ α R ▽ ⊳ p U ′ β R ′ ▽ α R ′ ▽ p ′ ⊲ Q r ▽ ⊳ q ′ q ⊲ of the previous section. 2.3. Group and group oid actions. An action o f a Lie gro up G on a manifold E , deﬁned by the (smo oth) map : G × E β → E , ( g, x ) 7→ g · x = β ( g , x ), c a n b e describ ed by the graph o f the m ap β . Mo difying the order in the pro ducts, this graph deﬁnes a n embedding H = G × E ◮ ι > G × ( E × E ), ( g , x ) 7→ ( g , ( g · x, x )). Regarding G × ( E × E ) a s a (Lie) group oid with base E (pro duct of the group G by the “ banal” gro up oid E × E ), the asso ciativity prop e rty of the action law ma y be expressed by the f act t hat H is an (embedded) subgroup oid of G × ( E × E ). Comp osing ι with pr 1 yields a (smo o th) functor H f → G which o wns the prop erty that the comm utative diag r am generated b y the source pro jections is a pullback. This c o nstruction ex tends for the action o f a (smo oth) group oid G α ◮ β ◮ B acting o n a manifold ov er B : E p ◮ B , replacing the pr o duct G × ( E × E ) by the 1 This surjectivity is not im pl ied by the usual deﬁnition of a reﬁnement of a co ve ring, but one can alwa ys imp ose it by the following slight mo diﬁcation of the deﬁnition, which c hanges nothing for the common use made of it : one demands in addition that a reﬁnement of a co v ering con tains this latter co ve ring, which can alwa ys b e ac hiev ed by taking their union. GENERALIZED A TLASE S 7 ﬁbred pro duct of the anchor map τ G and p × p , and one gets a pullback squa r e : H f ◮ G pb E α H H f (0) ◮ B α G H where f (0) = p (induced by the functor f on the bases E = H (0) , B = G (0) ). The action la w ma y be recovered by β = β H , using the isomo rphism of H with the ﬁbred pro duct of G and E ov er B , which results from the pullbac k prop erty . Note that the pullback pro pe r ty remains meaningful even when E p → B is not a surmersion, since α G still is, (though the ﬁbred pro duct of τ G by p × p may then fail to exist), and β H still deﬁnes an action of G on p . F or that reason w e call a functor H f act > G owning the previous pullba ck prop- erty an actor , whic h is empha sized by the framed la be l of the a rrow. Such functor s received v arious unfortunate names in the categor ical literature, among which “dis- crete opﬁbra tions” a nd “foncteur s d’hypermor phisme” (Ehre s mann), and, b etter, “star-bijective” (Ronnie Br own), but note that the present co ncept encapsulates a smo othness informa tio n, included in the pullback prop erty , and not o nly the pur e ly set-theoretic or algebraic conditions (see below for a more general setting). There is an equiv alence of categories b etw een the categ o ry of equiv ariant maps betw een action la ws a nd the catego ry a dmitting the actor s as ob jects and commu- tative squa r es of functor s as arrows. Note than in the literature H is curr ently called the a ction gr oup oid , but it is only the whole datum of the a ctor f : H → G which fully describes the action law, whence our terminology . Here we let Q b e the (set-theoretic) quotient of the manifold E by the action of the Lie gr oup(oid) G . By the pr evious construction it app ears to o as the orbit space of the Lie gro upo id H , so we ha ve aga in for Q = E /G = E /H a generalized atlas H α ◮ β ◮ E . . . . . . . . ◮ Q , the dotted a rrow meaning her e that we hav e now just a set-theoretic surjection (w e ha ve here to g o out of Di f , since Q is no more a manifold). 2.4. F o liations on B . Her e we need a more restrictive no tion for o ur surmer sions, called r etr o c onne cte d (it is in a certain s e nse prec is ely the opp osite of being ´ etale, which migh t b e called as well “ re tr o discr et e ”), and, in the present s ubsection, unlike the previous one, a nota tion suc h as E p ⊲ B will indica te that the sur mersion p is retro c o nnected. This means that the inv erse image of a ny x ∈ B is co nnected, or, equiv alently , that the inv erse image of a ny connected subset of B is connected. In the following, when we hav e to use simultaneously ´ etale and retro co nnected surmersions , we shall distinguish them b y means of circled labe ls : A ´ e  ⊲ B or A c  ⊲ B . There is an obvious notion of folia tion indu c e d by pulling bac k a foliatio n along such a retro co nnected surmersio n, and the induced foliation keeps the same s et- theoretical space of leav es. Then the no tion of F -equiv a lence (in the sense introduced by P . Molino in the 70’s) b etw ee n tw o folia tions ( B , F ), ( B , F ′ ) can be expressed by the existence of 8 JEAN PRADINES a comm utativ e diagram a s below, whic h means that F a nd F ′ induce the same ( E , G ) ( B , F ) ⊳ p ( B , F ′ ) p ′ ⊲ Q r ▽ . . . . . . . . . . . . . . . . . . . . . . . ⊳ . . . . . . . . . . . . . . . . q ′ . . . . . . . . . . . . . . . q ⊲ foliation G on the manifold E . Now we note t hat aga in Q = B / F may be view ed a s the orbit spac e of a Lie group oid (here with connected source ﬁbres ), to kno w the Ehresmann holo nomy gr oup oid H α ⊲ β ⊲ B ( sometimes renamed muc h later as the g r aph of the foliatio n), and Molino equiv alence may alter na tively be describ ed b y the commutativ e diag ram below (with all maps retr o connected), where f (0) = p , f ′ (0) = p ′ , and the symbols K H ⊳ f ∼ E β K ▽ α K ▽ H ′ f ′ ∼ ⊲ B β H ▽ α H ▽ ⊳ f (0) B ′ β H ′ ▽ α H ′ ▽ f ′ (0) ⊲ Q r ▽ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⊳ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q ⊲ ∼ mean (sur mersive retro connected) equiv a lences of Lie gr oup oids, in a sense to b e made more precise later. As we shall see also la ter, suc h a diagram deﬁnes a (smo oth in a v ery prec ise sense) Morita equiv ale nce betw een H and H ′ . Remark. An y surmersion A f ◮ Q admits of an essent ially unique 2 factoriza- tion A q ′ c  ⊲ Q ′ e ´ e  ⊲ Q with q ′ retro connected and e ´ etale. Though this might be prov ed directly , it is better t o apply th e gener al theo ry of Lie group oids, in the sp ecial case of principal (or Go dement) group oids (see 2.1 ab ov e): 2 i.e. up to i somorphisms. GENERALIZED A TLASE S 9 if R is the gr aph o f the regular eq uiv alence on A deﬁned by q , its neutral (or α -connected) component R c is an op en (pos sibly non clos ed 3 , but automatically inv ar iant) subgro up o id of R , hence it is still a Go dement g roup oid, with now con- nected α -ﬁbr es, a nd it deﬁnes the r egular foliation admitting Q ′ 4 for its spa ce of leav es. Then Q ′ > Q is a s urmersion w ith discrete ﬁbres, i.e. ´ e tale, and the graph of the equiv a lence on Q ′ th us deﬁned is is omorphic to the tw o-sided quotient group oid R//R c (cf. [P2]). 3. Diptychs. I am now enoug h mo tiv ated for intro ducing more dog matically some genera l ab- stract deﬁnitions mo delling a nd unifying the prev ious situations, as well as myriads of others. 3.1. Deﬁni tion of di pt yc hs. The notions presented in this s ection have a m uch wider range than it w ould be strictly ne c essary for the sequel, if one wants to stay in Di f , but g ive to it a muc h wider sco pe, even when aiming o nly at applications in Dif , as illustrated b y some of the previous examples. W e introduced them a long time ago, in [P 1], and think they deser ve b eing better known and used. In the pres ent ation of the examples of the previous section, we emphasized the role play ed by embeddings/surmer sions (these are special mono/e pimo rphisms of Dif , but no t the most gener al ones, which would indeed b e pathologica l), with po ssibly some more restrictive conditio ns added. Our claim is that an incredible amount of v a r ious constructions can b e p er formed without using the speciﬁcity of these conditions, but just a f ew v ery s imple and apparently mild stability pro pe r ties (of categor ical nature) fulﬁlled in a surprising ly wide ra ng e of situatio ns encountered by the “working ma thema ticians”. The p ow er of these prop erties comes from their conjunction. Then the le ading ide a (il lustr ate d b efor ehand i n the p r evio us se ction) wil l b e to describ e the s et -the or etic al c onstructions by me ans of diagr ams, emphasizing inje c- tions/surje ctions, and then r er e ading these diag r ams in the c ate gory involve d, u sing the distinguishe d given mono/epi’s. 3.1.1. Diptychs data. A “ diptych ” D = ( D ; D i , D s ) (which ma y be sometimes de- noted loo sely b y D a lone) is deﬁned by the following data : • D is a category whic h comes equipped with ﬁ nite non void 5 pr o ducts . The subgroup oid of inv ertible arrows (called isomorph isms ) is denoted by D ∗ . • D i / D s is a sub ca tegory of D , the a rrows of which are mono /epi-morphisms (b y axiom (iii) b elow), called go o d mono/epi ’s and denoted generally by arrows with a triangular tail/head such as ⊲ > / ⊲ , or ◮ > / ◮ , and so on (here / is written loo sely for resp.). The arrows b elong ing to D r = D i D s (i.e. comp osed of a go o d mono and a go o d epi) are called r e gular . In gener al D r wil l not b e a su b c ate gory . Wh en D r = D , the dipt ych may be called r e gular . 3 As in [B] , we ha ve to deal with p ossibly non Hausdorﬀ manifolds. 4 Po ssibly non Hausdorﬀ. 5 See below. 10 JEAN PRADINES 3.1.2. Diptychs axioms. These data have to satisfy the following axioms (whic h lo ok nearly self dual, but not fully , and that has to b e noticed 6 ) : (i) D i ∩ D s = D ∗ ; (ii) D i and D s are stable by pr o ducts ; (iii) (a)/(b) the arrows of D i / D s are monos/strict 7 epis ; (iv) ( “str ong/we ak sour c e/r ange -stability” of D i / D s ) : (a) 8 ( h = g f ∈ D i ) ⇒ ( f ∈ D i ) ; (b) (( h = g f ∈ D s ) and ( f ∈ D s )) ⇒ ( g ∈ D s ) ; (v) (“ tr ansversality ”, denoted by D s ⋔ D i ) : (a) (“ p ar al lel tr ansfer ”) : given A s ◮ B a nd B ′ ◮ i > B (whic h mea ns : s ∈ D s , i ∈ D i ), there exists a pullback with moreover s ′ ∈ D s , i ′ ∈ D i (the que s tion marks ¿ A ′ ? ◮ ¿ i ′ ? > A ¿ pb ? B ′ ¿ s ′ ? H ◮ i > B s H frame the ob jects or pr op erties, as well as the das hed ar rows, which app ear in the conclusions, as consequences of the data). (b) (conv ersely : “ desc ent ” , or “ r everse tr ansfer ”) : given a pullback s quare as below (with i ′ ∈ D i , s,s ′ ∈ D s , i ∈ D ), A ′ ◮ i ′ > A pb B ′ ¿ s ′ H ◮ ? i > B s H one has i ∈ D i (conclusion pictured by the q ue s tion mark s a round the dashed triangular tail). 3.1.3. F u l l sub dip tychs. Remark. Let be given a sub class C of the class of ob jects of D , which is st able by pr o ducts and let D ′ , D ′ i , D ′ s be the full subca teg ories of D , D i , D s th us genera ted. In order they deﬁne a new dyptich D ′ (full sub diptyc h), one o f the following conditions is suﬃcien t : (1) g iven a go o d epi A ◮ B , the condition B ∈ C implies A ∈ C ; (2) g iven a go o d mono A ◮ > B , the conditio n B ∈ C implies A ∈ C ; 3.2. Som e v ariant s for data and axioms. 6 It i s said that the same thing happened at the very instant of the big bang, w i th analogous consequenc es. 7 This means that they ar e co equalizers [McL]. See b elow for an alternative for mulat ion. 8 W e just mention that it may be sometimes useful to work with only the “weak source-stability ” condition, dua l of (b). It is then possible to deﬁne a suitable ful l sub c ate g ory of D i n which the strong axiom is satisﬁed. The ob jects of this subcategory own (in particular) the prop erty that their diagonal maps ar e in D i , which is formally a Hausdorﬀ (or separation) ty p e prop ert y . These ob jects may be called i -sc atter e d . GENERALIZED A TLASE S 11 3.2.1. Pr e diptychs. When dealing with dia grams, we sha ll need a weak er notion : Deﬁnition. A pr e diptych is a triple T = ( T ; T i , T s ) where one just demands T i , T s to be sub ca teg ories of T , such that : T i ∩ T s = T ∗ (isomorphisms). Most of the predipt ychs we shall consider are r e gular , i.e. T = T r = T i T s . 3.2.2. Al ternatives for axioms. F or any arrow B f > B ′ , we hav e the “ gr aph factorization ” : B ◮ i > B × B ′ pr 2 > B ′ , with i = (1 B , f ) ∈ D i by a xioms (iv) (a) and (i) (as a se ction o f pr 1 ), whence it is r eadily deduced that one has D s ⋔ D ; this means : • (v)(a) r emains valid when omitting D i b oth in assumptions and c onclusions . W e ha ve a lso D s ⋔ D s . In particula r the pullbac k squa re generated b y tw o go o d epis p , q , has its four edges in D s . Such pullback squares will be ca lled p erfe ct squar es . The case p = q : B ◮ Q covers the genera l situation em bracing v arious examples ab ov e. It can b e s hown, using compo s ition of pullback squares, that the axiom (iii) (b), in pre s ence of the other ones, may be rephra sed in the tw o following equiv alent w ays (using the notations of (2.1)) : • (iii) (b’) any go o d epi q : B ◮ Q is the c o e qualizer of the p air R α ◮ β ◮ B , wher e R is the ﬁbr e d pr o duct of q by q ; • (iii) (b”) any p erfe ct squar e is a push out to o (this last proper ty is very impo rtant and r emark a ble). 3.2.3. T erminal obje ct. The existence o f the void pro duct (i.e. of a terminal obje ct ), is not alwa ys required, in view of impo r tant examples ; when it do es exis t, it will be denoted b y a plain dot • (though its support ha s not to be a singleto n). Though, in ma ny exa mples, not o nly there e xists a terminal ob ject, but moreover the canonical ar r ows A → • are in D s , it ma y b e useful how ev er not to require this pro pe r ty in general. Then those ob jects owning this proper t y will be called s -c ondense d (see examples below). If A ◮ B is in D s , then A is s -condensed iﬀ B is. Observe that, if A is s -co ndensed, then, for a ny ob ject Z , the ca nonical pro jection pr 2 : A × Z ◮ Z will b e in D s (whic h is alwa ys true whenever all ob jects are required to be s -condensed). If D ′ , D ′ i , D ′ s are the ful l s ub c ate gorie s gener ate d by the s - c ondense d obje cts it can be check ed that they deﬁne a (ful l) sub diptych D ′ = ( D ′ ; D ′ i , D ′ s ). 3.3. A few examples of basic l arge diptyc hs. In f act most o f the categ ories used by the “ working ma thematicians” own one or several natural diptyc h struc- tures, and chec king the axioms may sometimes b e a more or less substantial (not alwa ys so well known) and often non trivial part of their theor y , which is in this wa y encapsula ted in the (pow erful) statemen t that one gets a dipt ych s tructure. 9 9 It might b e also advisable to lo ok for a wa y of adapting the axioms wi thout weak ening their pow er in order to include some notew orthy exceptions such as m easurable space s or Riemannian or Poisson manifolds. 12 JEAN PRADINES This is all the more remark able since a g e ne r al categ ory (with ﬁnite pro ducts) bea rs no cano nical non trivial (i.e. with D s 6 = D ∗ ) diptych structure, the cr u- cial po in t being that in general the pro duct of tw o epimorphisms fails to b e an epimorphism. 3.3.1. Sets. The catego ry E = Set of (applications b etw een) sets o wns a canonica l dipt ych structure E = Set , which is regular, by taking for E i / E s the s ubca tegories of injections/surjections (here these are exactly all the mono/e pi -mo rphisms). The same is true for the dual catego ry (exchanging injections and surjectio ns), but the dual dipt yc h E ∗ is not isomorphic to E . 10 3.3.2. Two gener al examples. There a re tw o r e mark a ble and imp ortant cas es when one g ets a (ca no nical) diptych structure, whic h is moreov er r e gular , b y tak ing a s go o d monos /epis al l the mono/epi -morphisms, to k now : the ab elian c ate gories and the top oses 11 . Of course these tw o very g eneral ex amples embrace in turn a hu ge lot of sp ecial cases in Algebra and T op olog y . As a conse q uence al l the c onstructions we c arry out by using diptychs ar e working for gener al top oses , but the conv erse is false, since the most interesting and useful dipt ychs ar e n ot top oses . 3.3.3. T op olo gic al sp ac es. In T op (res p. Haus 12 ), we can take as go o d mono s the (resp. pr op er 13 ) top olog ical embedding s, and as g o o d epis the surjective op en maps. All ob jects are then s -co ndensed. This dipt ych is n ot r e gular . These canonica l diptyc hs will be denoted b y T o p and Haus . W e may alternatively take as go o d epis in T op the ´ etale/retro co nnected (or , in Haus , pr op er) s urjective maps. Then the s -c ondense d obje cts are the dis- crete/connected (compact) spaces. This is illustrated by ex amples ab ov e. 3.3.4. Banach sp ac es. In Ba n , the category of (contin uous line a r maps b etw een) Banach spaces, one can take as go o d monos/ epis the left/rig h t invertible arrows. All ob jects are s -co ndensed. This dipt ych is not r egular (sav e for the full s ub diptyc h of ﬁnite dimensional spaces). It will be denoted b y Ban . 3.3.5. Manifol ds. In Dif , the category of (smo o th maps b etw een) possibly Bana ch and p ossibly no n Hausdorﬀ manifolds (in the sense of Bo urbaki [B]), the basic dip- t ych structure, denoted by Dif , is deﬁned by ta king as g o o d monos the em b eddings and as go o d epis the surmersions. But ones g ets a very lar g e num ber of very us eful v a riants and of full sub diptyc hs, as in T op , when suitably adding a nd combinin g extra conditions for ob jects or arrows such as being Haus dorﬀ, prop er, ´ etale, retro connected, and also v arious c oun t ability c onditions , either on the manifolds (e.g. existence o f a countable dense subset) or on the maps (for ins tance ﬁniteness or coun tability of the ﬁbres : retro - ﬁniteness, retro-countabilit y). 10 Owing to the lac k of symmetry for the axioms, one cannot in general deﬁne the dual of a dipt yc h. 11 See for instance, f or a goo d part, but not t he whole, of the p rop erties in v olv ed i n this statemen t, the textb ooks by M ac Lane and Pe ter Johnstone. 12 The ful l sub category of Hausdorﬀ spaces. 13 If we had tak en these ones for T op , w e would hav e b een in the s ituation alluded to in f ootnote 8 and the Hausdorﬀ spaces might be constructed as the i - scattered ob jects. GENERALIZED A TLASE S 13 This basic diptyc h is not r e gular 14 . 3.3.6. V e ctor bund les. Let V e cB denote the categ ory of (mo rphisms b etw ee n) vec- tor bundles (for instance in Dif ) ; the ar r ows are comm utative squa res (see (4.1): E ′ f > E B ′ p ′ H f 0 > B p H There are several useful dipt yc h str uctures. The basic one, denoted by VecB , takes for goo d monos the squares with f , f 0 ∈ D i , and for go o d epis thos e with f , f 0 ∈ D s , and wh ich mor e over ar e s -ful l in the sense to b e deﬁned b elow (4.1). (One can also use pullbac k squares, which means E is induc e d by B along f 0 ). 3.4. (Pre)diptyc h structures on simpl i cial (and related) categories . Small dipt ychs (and es pec ially prediptyc hs) may be a lso of in terest : 3.4.1. Finite c ar dinals. The category N c of all maps b etw een ﬁnite c ar dinals (or int egers) deﬁnes a r egular diptyc h N c by taking for ( N c ) i / ( N c ) s all the injec- tions/surjections . The set of ob jects is N . The s -co ndens ed ob jects ar e those which are 6 = 0. The dual o r opp os ite ca tegory , denoted by N ∗ c , deﬁnes a lso a r egular diptyc h N ∗ c with ( N ∗ c ) i = (( N c ) s ) ∗ , ( N ∗ c ) s = (( N c ) i ) ∗ . The pro duct in ( N c ) ∗ is the sum in N c . The terminal ob ject is now 0, and all the ob jects ar e no w s -condensed. When dropping 0 one gets still diptyc hs denoted by N + c and N + ∗ c 15 , the o b jects of whic h will b e denoted b y · n = n + 1 ( n ∈ N ) 16 . The diptych N + ∗ c wil l b e b asic for our diagr ammatic description of gr oup oids 17 . 3.4.2. Finite or dinals. Considering now the integers as ﬁn ite or dinals , we get the sub c ategories of monotone maps which will be denoted here b y N o , N + o , (denoted by ∆, ∆ + in [McL]) ( simplicial c ate gories ), as well as their duals, but these have no (car tesian) pr o ducts and therefore c annot deﬁne diptychs . They deﬁne only pr e diptychs deno ted here b y : N o , N + o , N ∗ o , N + ∗ o . Remark. Though N o is by no means isomor phic to its dual N ∗ o , how ever there are t wo cano nica l isomorphisms Φ, Ψ = (Φ ∗ ) − 1 ( deﬁne d only on t he privile ge d sub c ate gories ! ), which can b e deﬁned using the canonical ge ne r ators a s denoted in [McL]: Φ : ( N o ) i → ( N ∗ o ) i = (( N o ) s ) ∗ , n 7→ · n , δ n j 7→ ( σ · n j ) ∗ , Ψ : ( N o ) s → ( N ∗ o ) s = (( N o ) i ) ∗ , · n 7→ n , σ · n j 7→ ( δ n j ) ∗ , where a star b ea ring o n an ar r ow means that this very arr ow is regarded as belo nging to the dual category (with source and target exchanged). 14 This is i ndeed an imp ortan t source of diﬃculty , but also of richness for the theory . 15 The latter wi th a termi nal ob ject, the former without such. 16 The notation suggests that the element s of such an ob ject have to b e numb er e d fr om 0 to n , the dot symbolizing the added 0. 17 W e stress again that it is not isomor phic to N c . 14 JEAN PRADINES 3.4.3. Canonic al pr e diptychs. T o each (small or no t) category T , we can asso ciate thr e e c anonic al pr e diptychs (the last t w o being regular): T ( ∗ ) = ( T ; T ∗ , T ∗ ) , T ( ι ) = ( T ; T , T ∗ ) and T ( σ ) = ( T ; T ∗ , T ) . 3.4.4. Sil ly pr e diptychs. Let I (o r so metimes, more picto rially , ↓ or → ) deno te the (seemingly silly) catego ry wit h t w o o b jects 0, 1, and one non unit a rrow 0 ε → 1 (it owns pro ducts and sums) 18 . It is canonically isomorphic with its dual I ∗ , b y exchanging the tw o ob jects. All the arrows are b oth mono- a nd epi-morpis ms , but ε is not strict, and cannot be accepted a s a go o d epi. As we sha ll s ee, it w ill b e co nv enien t to endow I with one of its canonical pr e diptych str uctures (3.4.3): I ( ∗ ) , I ( ι ) , I ( σ ) , accor ding to what is needed. 4. Commut a tive s quares  in a diptych D = ( D ; D i , D s ) . 4.1. Three basic types of s quares. Let  D 19 denote the categ ory of co m- m utative squares in D with the horizontal comp os ition, which can b e regarde d (peda n tically) as the category of natural trans formations betw een functors fro m I to D , with the (unfortunately so-calle d !) vertical comp osition. Its arr ows might alternatively b e descr ibe d as functors from I × I to D . It turns out that the following prop er ties of a commutativ e square ( K ) play a A ′ f ′ > B ′ ( K ) A u ∨ f > B v ∨ basic role (the terminology will b e explained by the applica tio n to functors). Deﬁnition : the co mm utative squa re ( K ) is said to be : . a) i -faithful if A ′ ◮ ( u,f ′ ) > A × B ′ is a goo d mono ; notatio n : A ′ · ◮ > f ′ > B ′ ( K ) A u ∨ f > B v ∨ or A ′ f ′ > B ′ ﬁd A u ∨ f > B v ∨ . b) a go o d pul lb ack 20 if it is a pullbac k squar e (in the usual catego rical sense) and moreov er i - fa ithful 21 ; one writes f ⋔ v ( f and v are “ we akly tr ansversal ” ) to mean that the pair ( f , v ) can be co mpleted in suc h a square ; notation : 18 This is the category denoted by 2 in [McL], since it represents the order of the ordinal 2. 19 Ehresmann’s notation. 20 In Dif , there are plen t y of useful pullbac k squares generate d by pairs ( f, v ) which are not transv ersal i n the classi cal sense of [B] (for instan ce t wo curves intersect ing ne atly in a high dimensional manifol d), but there are also (actu ally pathological) pullbac k squares existing without ( f , v ) b eing weakly transversal. W e think the wea k transversalit y is the most useful notion. 21 Observe that p erfect squares are go od pullbacks, as well as those ari sing from axiom (v), or more generally from (3.2.2), but it may happen that more general ones are needed. GENERALIZED A TLASE S 15 A ′ f ′ > B ′ ( K ) A u ∨ f > B v ∨ or A ′ f ′ > B ′ pb A u ∨ f > B v ∨ . c) s -ful l 22 if one has f ⋔ v 23 and if moreover the canonica l ar r ow A ′ ◮ A × B B ′ (whic h is then deﬁned) is a go o d epi ; notation as below. A ′ · ◮ f ′ > B ′ ( K ) A u ∨ f > B v ∨ or A ′ f ′ > B ′ ful A u ∨ f > B v ∨ 4.2. Basi c diptyc h structures o n  D . The thre e previous kinds of square s hav e rema rk able comp osition stability pr op erties resulting from the axioms, whic h we shall not state he r e (cf Pro p. A 2 o f [P3 ], with a diﬀerent terminology). The (purely diagrammatica l) pro ofs a re nev er v ery hard, but may b e lengthy . A substantial par t of these prop erties is ex pressed by the f ollowing imp o rtant (non exhaus tive) statemen ts, whic h deser ve to b e co nsidered a s theorems, as they bring together a very large n umber of v a rious pr o p e rties, which acquir e muc h power by b eing ga thered. Let b e given a diptyc h D = ( D ; D i , D s ) . 4.2.1. Canonic al structure . On  D , w e ge t a ﬁrst diptyc h str ucture , called c anon- ic al by s e tting :  D = (  D ;  ( D , D i ) ,  ( D , D s )) , which is pictured by : · > · · ∨ > · ∨ ; · ◮ > · · ∨ ◮ > · ∨ , · ◮ · · ∨ ◮ · ∨ . and also, using rema rk (3.1 .3), the full sub diptyc h (still deﬁned for the dipt yc hs of the following para graph): i  D = (  ( D i , D );  ( D i , D i ) ,  ( D i , D s )) , pictured b y : · > · · ∨ H > · ∨ H ; · ◮ > · · ∨ H ◮ > · ∨ H , · ◮ · · ∨ H ◮ · ∨ H . Deﬁning an analo gous full sub dipt ych with D s replacing D i r e quir es a mor e r e- strictive choic e for the squar es taken as go o d epis . (These full sub diptyc hs will be essential to get ﬁbred pro ducts of diagra ms, he nc e of group oids). 4.2.2. F u l l and pul lb ack epis. W e hav e the tw o basic dipt ychs:  ( i , ful ) D = (  D ;  ( D , D i ) , ful ( D , D s ))  ( i , pb ) D = (  D ;  ( D , D i ) , pb ( D , D s )) which can b e pictured by : 22 Suc h a square owns the p ar al lel tr ansfer pr op erty : ( v ∈ D s ) ⇒ ( u ∈ D s ). 23 This was not demanded in a). 16 JEAN PRADINES · > · · ∨ > · ∨ ; · ◮ > · · ∨ ◮ > · ∨ , · · ◮ ◮ · · ∨ ◮ · ∨ · > · · ∨ > · ∨ ; · ◮ > · · ∨ ◮ > · ∨ , · ◮ · · ∨ ◮ · ∨ . and in this way , thanks to a pa rallel transfer prop erty , we get the exp ected b asic ful l sub diptychs : s  ( i , ful ) D = (  ( D s , D );  ( D s , D i ) , ful ( D s , D s )) s  ( i , pb ) D = (  ( D s , D );  ( D s , D i ) , pb ( D s , D s )) pictured b y : · > · · H > · H ; · ◮ > · · H ◮ > · H , · · ◮ ◮ · · H ◮ · H · > · · H > · H ; · ◮ > · · H ◮ > · H , · ◮ · · H ◮ · H . One also deﬁnes tw o basic sub diptychs b y taking : pb D = ( pb D ; pb ( D , D i ) , pb ( D , D s )) s ful D = ( ful ( D s , D ); ful ( D s , D i ) , ful ( D s , D s )) pictured by : · > · · ∨ > · ∨ ; · ◮ > · · ∨ ◮ > · ∨ , · ◮ · · ∨ ◮ · ∨ · > · ful · H > · H ; · ◮ > · ful · H ◮ > · H , · ◮ · ful · H ◮ · H . 4.2.3. Iter ation. One immediately notices than this theorem a llows an iter ation of the c o nstruction of commutativ e squar es giv ing ris e to new diptyc h structures for commutativ e cub es, and so o n, whic h w ould be v ery diﬃcult to handle directly . This is esp ecia lly interesting when dea ling with comm utative cubes since such a diag r am gives r ise to t hr e e c ommutative squar es in  D , each edge of w hich (whic h is ac tua lly a square of D ) b elonging to two diﬀer ent squar es of  D , and this gives a powerful metho d for deducing prop erties of certain fa c e s (for instance being a pullback), or edges fro m prop erties of the o thers, using diptyc h prop erties of parallel transfer. But w e cannot develop more here . 5. Diag rams of type T in a diptych. W e a re now going to r eplace the silly ca tegory I of 3 .4.4 b y a more general no- tion gener alizing the constructio n of c o mm utative squares, and a llowing to p erfor m v ar io us set-theoretic constructions in a general dipt yc h D . GENERALIZED A TLASE S 17 Let T be a smal l pre diptych 24 (3.2.1), and let D = ( D ; D i , D s ) be a dipt yc h. 5.1. Deﬁni tions : ob jects and morphisms. W e denote b y  T D : = D T the category ha ving : – as obje cts the ele ments of I T D : = Ho m ( T , D ) , called diagr ams of D of typ e T , whic h means those functors F from T to D such that one has : (1) F ( T i ) ⊂ D i and F ( T s ) ⊂ D s (these last conditions are void when T = T ( ∗ ) (3.4.3)) 25 ; – as arr ows the morphisms (i.e. the natur al transforma tions) betw een such functors 26 . These morphisms may b e descr ibed : - either as functors Φ from T × I to D such that (denoting by (:) the set of the t wo units of I ): Φ( T i × (:)) ⊂ D i and Φ( T s × (:)) ⊂ D s - or as functors Ψ from T to  vert D 27 such that, with the notations of (4.2) : Ψ( T i ) ⊂  ( D i , D ) = i  D and Ψ( T s ) ⊂  ( D s , D ) = s  D . In w ords this means that such a morphism may be viewed : - either as a diagr am in D of typ e T × I ( ∗ ) (cf.3.4.3 and 3.4.4) - or as a diagr am of t he same t yp e T in  vert D , reg arded with its c anonic al (vertical) diptyc h structure (4.2). How ev er be careful that, in such a description, though the previous deﬁnition of the mor phisms uses the vertic al c omp o sition of squares, the c omp osition o f these morphisms (called vertical in [McL]!!) in volv es the horizontal co mpo sition of squares. 5.2. Dip tyc h struct ures on the category of di agrams. 5.2.1. Deﬁnitions. Using the latter interpretation, and taking now in to account the horizontal comp osition, it is clea r that any prediptyc h structur e on  D determines a pr edipt ych structure on  T D . F or instance, using on T its tr ivial pr ediptyc h structure, we can consider the canonical pr e diptych str ucture  T D , but this not in general a dyptich structure (one needs para llel transfer prop erties for go o d epis in order to ensure conditions (1) of (5.1)). Using the par a llel transfer prop erties of s -full and pullback squares , o ne ca n g e t three useful diptych structur es denoted by: (  ( i , ful ) ) T D , (  ( i , pb ) ) T D and pb T D . 24 It may b e sometimes useful to ext end the foll o wing deﬁnitions when T is just a graph with t wo giv en s ubgraphs. 25 F orgetting the prediptyc h structure of T and dropping conditions (1) would ov ersimplify the theory , but deprive it of all its strength. 26 Again with the “v ertical comp osition”, which we prefer here to write horizon tally , dr awing the diagr ams verti c al ly , and the morphisms horizon tally . Of course one can exchange everywhe re simultane ously “ ve rtical” and “horizontal”, since the distinction i s purel y psycholog ical and no- tational, and since  vert D and  hor D are canonically i somorphic. 27 This notation means that  D has to b e considered here with its vertical comp osition law. 18 JEAN PRADINES Of sp e cial interest, a s we sha ll see, will b e those diagr ams which preserve certain pullbacks, since gr oup oids ma y be descr ib e d b y diagrams of this t yp e . 5.2.2. The sil ly c ase. The c ase of c ommutative squar es is just the sp e cial c ase when one tak es fo r T the sil ly c ate gory I , with one of its pr e diptych structu r es (3.4.4), since one has: I I ( ∗ ) D = |D| , I I ( ι ) D = |D i | , I I ( σ ) D = |D s | , 28  I ( ∗ ) D =  D ,  I ( ι ) D = i  D ;  I ( σ ) D = s  D ; more precisely :  I ( ∗ ) D =  D ,  I ( ι ) D = i  D (  ( i , ful ) ) I ( ∗ ) D =  ( i , ful ) D , (  ( i , pb ) ) I ( ∗ ) D =  ( i , pb ) D . and also : ( s  ( i , ful ) ) I ( σ ) D = s  ( i , ful ) D , ( s  ( i , pb ) ) I ( σ ) D = s  ( i , pb ) D . 5.3. The exp onential la w for diagrams. Given another similar da tum S , w e hav e the exp onential law , which, with our no tations, reads (at lea st when S a nd T are trivial dipt ychs) :  S (  T D ) =  S × T D (with a lot o f p o ssible v ariants), a nd the ca nonical is omorphism : S × T ≈ T × S yields a canonical isomorphism :  S (  T D ) ≈  T (  S D ) . Particularly one has: (  T ) n =  T n , and, specia lizing for T = I ( ∗ ) : (  ) n =  ( I ( ∗ ) ) n . Remarks ab out notations. The r eader may ha ve observed that the symbols  and I used a b ove are in tendedly am biguous and pro tea n, since this allows to memorize and visualize a lot of v ario us pr op erties : – the symbols I or I ( • ) , where • stands for ∗ , ι , or σ : - denote the silly category , poss ibly with v arious predipt ych structures on it; - denote the bifunctor Hom (?,?) , with the ﬁrs t argument tr eated as an exponent); - often behaves formally as a 1; - ma y sometimes suggest the functor T 7→ T × I ( • ) ; – the symbol  with po ssibly lab els in v ario us pos itions ⋄  ? ⊤ ( ♯,♮ ) : - may picture or s uggest the product I × I or mo re precise ly v a rious insta nces o f I ( △ ) × I ( ▽ ) ; - may create the commutativ e squares of a catego r y or of a diptyc h with p ossible extra conditions : ∗ on the left, they b e ar on the v ertical edges ; ∗ on the right, they describ e the s ub ca tegories for a (pre)dipt ych structure ; ∗ inside, they describ e global prop erties of the squar es ; - ma y b ecome the bifunctor H om (?,?), with possible restrictions on the square s inv o lved. 28 In the second m em b ers of this l ine, the signs | | denote the for getful functor forgetting the composition laws, since they ar e not deﬁned by the ﬁrst members. GENERALIZED A TLASE S 19 6. Groupoids as diagrams in E = Set . W e sha ll b e concerned only with smal l gro up o ids, v ie wed as gener alizing b oth groups and g r aphs of eq uiv alence rela tio ns, as well a s sp ecializing (sma ll) categor ies. According to o ur pro gram, we need a dia g rammatic description of b oth group oid data a nd a xioms, whic h will b e trans ferred fr om the diptyc h E = Set to a general dipt ych D . This can be achiev ed in tw o complemen tary w ays : - either using ﬁnite diagrams (called sketches in Ehresmann’s terminology) ; - or using the simplicial description b y the nerve . Though seemingly mo r e abstract, the latter turns out to b e often the most con- venien t for theoretical purp os es, while the former is a dapted to practical handling. W e refer to [McL] and also to (3.4) for genera l prop erties and notations concern- ing simplicial categorie s and simplicial ob je cts , which are just functor s fro m N + ∗ o to E , and more precisely diagr ams of t yp e N + ∗ o in E . 6.1. Som e remarks ab out N + c and N + o . (see 3.4) 6.1.1. The arr ows as functors. F or each n ∈ N , we shall denote b y : ∗ − → · n the o rdinal · n = n + 1 viewed as a small c ate gory (deﬁning the order) 29 ; ∗ ˙ ∧ n the sub category consisting of ar r ows of this order category with source 0 30 ; ∗ ← → · n = ( · n ) 2 the banal gr oup oid ( · n ) × ( · n ) , which igno res the ordering 31 . With these structure s on the o b jects, the arrows of N + o and N + c may b e rega r ded (though this lo o ks peda nt ic) as functors ( or morphisms) b etwe en smal l c ate gories . It may b e sometimes sugg estive to write : ← → · 0 = ← → 1 = · , ← → · 1 = ← → 2 = ↔ , ˙ ∧ 2 = ∧ , ← → · 2 = ← → 3 = △ , ← → · 3 = ← → 4 = ⊠ . 6.1.2. Pul lb acks in N ∗ c . Pullbacks in N ∗ c come from pushouts in N c . O ne can chec k that al l the c ommutative squar es describing the r elations b etwe en the c anon- ic al g ener ators of N o (see [McL]) b e c ome pushouts when writt en in N c , and they gener ate, by c omp osition, al l the pushouts of N c . Mo re precis ely (with the nota- tions in [McL]) the squares inv olving the injections δ j ’s alo ne or mixing b oth δ j ’s and σ k ’s ar e pul lb acks to o , but not those with t he su rje ctions σ j ’s alone 32 . This is still v alid when dropping 0 (but o ne looses the squa res describing the sums as pushouts). 6.2. Characterization of the nerv e of a group o id. The sp ecial prop erties of a gro upo id among categ ories yield very sp ecia l and remark able prop erties a s well as alternative descr iptions for its nerv e. 6.2.1. Thr e e descriptions for the nerve. Given a (small) g r oup oid G , denoted lo os ely by G or G ⇒ B , we can asso ciate to it thr e e c anonic al ly isomorphic simplicia l ob- je cts (among which the ﬁrst o ne is the ner ve of G rega rded as a categor y). Fir st we deﬁne and denote the three images of the generic ob ject · n = n + 1 33 : (1) ↓ G ( n ) = hom( − → · n , G ) = { paths of G of length n } ; 29 − → · 1 = − → 2 is just what we called in (3.4.4) the si lly category I = ↓ = → . 30 And of course the uni ts. 31 Though the num ber ing of the base i s kept! 32 This induces a strong dissymmetry b etw een the dual dipty ch s N c and N ∗ c . 33 Though these images may b e regarded as instances of diagrams of ﬁnite type in G , the notations used b elow diﬀer sli gh tly f rom those used in the previous section for the general case. 20 JEAN PRADINES (2) ∧ G ( n ) = hom( ˙ ∧ n, G ) = { n -uples of arrows of G with the same source } ; (3) l G ( n ) = hom( ← → · n , G ) = { commutativ e diagrams o f G with · n v ertices } 34 . Identifying t hese three sets, we can write lo o sely: G ( n ) = ↓ G ( n ) = ∧ G ( n ) = l G ( n ) . W e shall also sometimes feel free to write lo os ely and suggestively : G (0) = B , G (1) = ↓ G = l G = G , G (2) = ∧ G = △ G , G (3) = ⊠ G . The imag es of the canonica l generators of N + c are then es p ecia lly ea sy to deﬁne with the int erpretatio n (3 ), since they consist in repeating or for getting o ne vertex. But, with the descriptions (1) and (3), one ca n immediately interpret the required contra v ar ia nt functors from N + o to E as be ing just s pec ial instances of the classical contra v ar ia nt hom-functors hom(? , G ), which consis t in letting the arr ows of N + o act by right morphism c omp osition (in the category o f morphisms or functors betw een small categories ) with the elements of ↓ G ( n ) or l G ( n ) . 6.2.2. Pr op erties of the nerve of a gr oup oid. . - ( Extension of the nerve ). With the in terpretation ( 3), we get a b onus, since it is now obvious that this co ntrav ariant functor ext en ds t o N + c , and so deﬁnes a diagram in G o f type N + ∗ c . (W e remind that, in this interpretation, int egers ar e int erpreted as small banal group oids, and the arr ows o f N + c as morphisms.) . - ( Exactn ess pr op erties ). Mo reov er this ex tended functor, which is now deﬁned on a diptych , is “ exact ” , there meaning that it pr eserves pul lb acks 35 . It would b e enough to c heck this for the gene r ating pullbac ks of N ∗ c (6.1.2). . - Conversely one mig ht chec k that the datum of an e x act dipt y ch morphism G = ( G ( n ) ) ( n ∈ N ) 36 from N + ∗ c to E = Set determines a gro upo id, the nerve of which is the restriction of this morphism to N + ∗ o . . - ( Sketch of a gr oup oid ). 37 Actually one might chec k that the gr oup oid data are fully determined by the r estriction of G to the (truncated) full subca tegory [ · 2 ] o f N + ∗ c generated by · 0 , · 1 , · 2, while the gr oup oid axioms ar e expresse d by its restriction to [ · 3 ] (a nd t he c o nditions tha t the imag es o f the previous gener ating squares be pullbacks) 38 . 6.2.3. Concr ete description. Actually the previo us data are somewhat redundant, and it is conv enient for our purp ose to obse r ve that G may b e fully describ ed by the data ( G, B , ω G , α G , δ G ) (satisfying axioms which w e shall not mak e explicit), where : • B (“bas e”) and G (“set of arrows”) are ob jects of E ; • ω G : B ◮ > G is an injection (“unit law”) ; • α G : G ◮ B is a surjectio n (source map) ; 34 With t wo-sided edges and num bered vertices. 35 But not pr o ducts (and not the pushouts whic h don’t arise from perfect squares). When some risk of confusion migh t ar i se, it wou ld be more correct to sa y someth ing like “diptyc h-exact” or “p.b.-exact”, since here the term “exact” has to be understo o d i n a muc h w eak er sense than the general meaning for functors b etw een categories : i t has not to preserve all (co)limits. 36 Relaxed notation, usi ng only the i mages of the ob jects. 37 This notion was introduced by Ehresmann f or general categories. 38 W e can not give more details here. GENERALIZED A TLASE S 21 • δ G : ∧ G = △ G ◮ G is a surjection (“division map” 39 ), with : ∧ G = G × B G (ﬁbred pro duct of α b y α ). One often writes α for ω α , and we set : • τ G = ( β G , α G ) : G → B × B (“anchor map” 40 , or “ tr ansitor ” ). F ro m these data, it is not diﬃcult to get the r ange map β G , the inv erse la w ι G , and the compo sition law γ G (deﬁned on the ﬁbred pro duct of α G and β G ). With notations fro m Mac Lane: ω G , α G , β G , δ G would b e the resp ective ima ges of : ( σ 1 0 ) ∗ , ( δ 1 1 ) ∗ , ( δ 1 0 ) ∗ , ( δ 2 0 ) ∗ ( in the dual N ∗ c ). 7. D -groupoids. 7.1. Deﬁni tion of D -group o ids. 7.1.1. Gr oup oids in a diptych. The diagrammatic description of a set-theoretica l group oid giv en ab ove le a ds to deﬁne a D -gro up oid as an exact diptych morhism (6.2.2) , i.e. pr eserving go o d monos, go o d epis, and go o d pul lb acks : G : N + ∗ c → D . As a bove, the e x actness pr op erties allow to characterize G by r estriction to v ar io us subca tegories, and to rec ov er in this wa y the s implicia l descr iption as w ell as the skecth ones and the ( G, B , ω G , α G , δ G ) presen tation. W e shall als o use as pre viously the relaxed nota tio ns ( G ( n ) ) n ∈ N (omitting the eﬀect o f G on the arr ows), o r br ieﬂy G (meaning G (1) ), B for G (0) , and G α ◮ β ◮ B and its v aria nt s. W e shall denote by Gpd( D ) the cla ss of D -group oids. 7.1.2. Examples. This very g eneral no tio n may b e sp ecialized using for instance the v ar io us examples giv en in (3.3). Note that the T op -gr oup oids a re those for whic h the sourc e ma p has to be op en , a condition whic h can hardly b e av oided for ge tting a useful theory . Lie gr oup oids 41 are of course the Dif -gro upo ids, with the usual diptyc h structur e on Dif , but, using some of the ab ove-men tioned v ar iants, the theo ry will include, for instance, among others, ´ etale or α -connected group oids as w ell. V ecB - gr oup oids w ere used in [P6]. 7.1.3. Nul l gr oup oids. F o r any ob ject B of D , the constant s implicial ob ject B is a D -group oid, called “ nu l l ” 42 , and denoted by ◦ B . All ar rows a re units, and the map ω G is an isomorphism. 7.2. Principal and Go deme n t D -group oid s . 39 This means the map : ( y , x ) 7→ y x − 1 . 40 Mac ke nzie’s terminology . 41 In troduced b y C. Ehresmann under the name of diﬀer entiable gr oup oids . 42 W e cannot accep t the traditional terminology “discrete”, whi ch has another topological meaning, and which moreov er do es not agree with the group terminology , unlike ours. 22 JEAN PRADINES 7.2.1. The tr ansitor τ G . F rom a pur ely set-theore tical p oint of vie w , the transitor τ G : G → B × B mea sures the (in)tr ansitivity 43 of the group oid G . It turns out that, in the diptyc h s etting, its prop er ties encapsulate a very rich “structured” informatio n. W e just mention, witho ut developing here, that, for in- stance in Dif , one can fully c haracter ize, just by v ery simple prop er ties of τ G , not only the graphs of r egular equiv ale nces, but (amo ng o thers) the gauge gr oup oids of principal bundles, the Poincar ´ e group o ids of Galois coverings, the holono m y group oids of foliations, the Barre Q-manifolds, the Satake V-manifolds (or orb- ifolds). 7.2.2. Princip al D -gr oup oids. The notion o f gr aph of r e gular e quivalenc e (2.1) may be carried ov er in any diptyc h as follows. Given a g o o d epi B q ◮ Q of D , we can construct the iterated ﬁbr e d pro duct of q , deno ted by R ( n ) = · n × Q B ( n ∈ N ) and chec k that ( n 7→ R ( n ) ) ( n ∈ N ) allows to deﬁne a D -simplicial ob ject whic h is a g roup oid. W e sha ll say that R = ( R ( n ) ) n ∈ N is the princip al gr oup oid asso ciate d t o q (with base B ). Moreov er we have an “ au gmen t ation ”, which means an extended “ e xact” (6 .2.2) dipt ych mor phism (6.2.2) from N ∗ c to D . This augmentation carries 0 44 to R ( − 1) = Q , and the added generator η ∗ = ( δ 0 0 ) ∗ 45 to B q ◮ Q . This situation gives rise to a p erfect sq ua re (3.2.2): R β R ◮ B pb po B α R H q ◮ Q q H The pusho ut prop erty of this s quare shows that the go o d epi q is uniquely de- termine d by the kno wledge of the D -group oid R . Applied to G α G ◮ B , this construction allo ws cons ide r ing ∧ G as a principal group oid (see be low (8.2.2)). 7.2.3. Banal gr oup oids. This co nstruction a pplies in particular when B is a n s - c ondense d obje ct (3.2.3). Then one ha s R ( n ) = B · n ( · n times itera ted pro duct). It is called the “ b anal gr oup oid ” 46 asso ciated to B ; it is principal. It is deno ted b y B × B or B 2 . More generally such a ba nal gr oup oid (p ossibly non principa l in the a bs ence of a terminal ob ject) is asso ciated to an ob ject B whenever the ca nonical pro jections pr i : B × B → B are go o d epis 47 . One might call “ pr op er ” those ob jects B for which t he banal group oid B 2 is deﬁned. When such is the ca se, the transitor τ G may b e viewed as a D -functor. Given an integer n , we can attac h to it, besides (for n 6 = 0) the banal group oid ← → n = n 2 (whic h uses the pro duct in N c ), a banal co g roup oid 2 n or 2 n , using the pro duct × ∗ of N ∗ c (whic h is the copr o duct + of N c ). 43 Whence the terminology adopted here, preferably to “anc hor map”. 44 Whic h might b e wri tten 0 = · ( − 1) . 45 Dual ar row, f rom Mac Lane’s notations. 46 W e can not accept the term “coarse”, often used in the l iterature for the same reason as for “discrete”. 47 W e ga ve abov e examples of diptyc hs in which s uc h i s not alwa ys the case. GENERALIZED A TLASE S 23 7.2.4. Go dement gr oup oids and Go dement diptychs. Deﬁniti o n. A D -gr oup oid G is called a Go dement gr oup oid if τ G : G ◮ > B × B is a go o d mono. Every principa l group oid is a Godement group oid. • W e say the Go dement axiom is fulﬁlled, and D is a Go dement diptych if c on- versely ev ery Go demen t D -group oid is principal. The fact that the dipt yc h Dif (3.3.5) is Godement is the con tent of the so -called Go dement the or em , proved in Serre [LALG] 48 . But it is hig hly remark able t hat nea rly all of the exa mples of diptyc hs giv en ab ov e are indeed Go dement diptyc hs, as w ell as most of the diagra m diptyc hs we constructed ab ov e, provided one star ts with a Go dement diptyc h. Such a statement includes a long list of theorems, which are not alwa ys classical. • F r om now on, we shal l assu me D is Go dement whenever this is u s eful. 7.3. R e gular groupoi ds. More gene r ally , a D - group oid G is called r e gular if τ G is regular (3.1.1). Then we hav e the follo wing factorization of τ G : G π ◮ R ◮ τ R > B × B , where R is a Go demen t D -group oid, hence principa l, so that we can construct the per fect square (7.2.2) ; then the orbit space Q exists as an ob ject of D , a nd is also the pushout of α G and β G . How ev er the ob ject Q inherits an “ extr a st ructur e ” from the arrow G π ◮ R , a go o d epi which m e asur es how much the s -ful l pushout squar e GB B Q , fails to b e a pul lb ack . • In the gener al c ase, when G is neither princip al nor even r e gular, the aim of the pr esent p ap er is to deﬁne a kind of “virtu al augmentation ” (as a substitute for the failing one), which is the D -Morita e quivalenc e class of G , and which has to b e c onsider e d intuitively as deﬁning the “ virtua l structure ” of the orbit sp ac e. 7.3.1. Plurigr oups. An imp ortant sp ecial case of r egular D -gro up o id is when R = ◦ B (7.1.3) (this is indeed equiv alent to α G = β G ). When suc h is the case, w e sha ll say G is a “ D - plurigr oup ” 49 . 7.3.2. s -tr ansitive D -gr oup oids. The o ppo site degeneracy is when τ G is a g o o d epi: we shall say G is s - transitive. When D = Set , this just mea ns that the orbit space is reduced to a singleton, but in D if , for instance , this has very str o ng implica tions, since this means ess ent ially that G may b e view ed as the gaug e g roup oid of a principa l bundle 50 . This ca n be proved in a purely dia grammatic way , which allow to ex tend these concepts to all (Godement) dipt ychs. In fact the or bit space has to be thought as “ a singleton structured by a gro up”. One o f the basic reaso n of the strength of the notion of D -gro upo id is t hat it uniﬁes and gathers in a single theory all these v ario us deg eneracies. 48 Where the f ormal asp ect of this theorem is clearly visible, and inspired our Go demen t axiom. 49 W e keep the term D -gr oup for the case when m oreo v er B is a terminal obje ct . On the other hand, the p ossible term “m ultigroup” would create confusion with the multiple categories, whi c h ha ve nothing to do with the present case. 50 The term “Lie group oid” w as ﬁrst reserve d to that s pecial case (see for instance the ﬁrst textbo ok by K. Mack enzie), till A. W einstein and P . Dazord c hanged the terminology , with m y full agreement . 24 JEAN PRADINES 8. The ca tegor y Gp d ( D ) . 8.1. D -functors. 8.1.1. D -functors as natu ra l tr ansformations. D - functors (or morphisms ) b etw een D -group oids are of cours e sp ecial cas es of diagram mor phisms (5) and, a s such, are deﬁned as natural transformations be tw een the D -group oids viewed as functors . But the pullbac k proper t y of the generating sq uares of the d iagra ms deﬁning group oids (6.2.2) has very strong implications (a rising from the pr eliminary study of co mm utative squa r es in a dipt ych) which w e cannot develop her e , refer ring to [P4] for more details. W e just mention a few basic facts. A D -functor f : H → G is fully determined by f (1) : H (1) → G (1) and hence often denoted lo osely by f : H → G ; we s ha ll write : H (0) = E , G (0) = B . It is called princip al if H is pr incipal. W e s ay f is a i / s - functor when f lies in D i / D s ; as a consequence, one can chec k this is still v alid for all the f ( n ) . W e g et in this way the catego ry Gp d ( D ) , with Gp d ( D ) a s its bas e , of ( D - functors or mor phisms b etw een) D -group oids , a nd tw o sub categor ies Gp d i ( D ) and Gp d s ( D ), but, a s anno unced ear lier, the second one will not be the right candidate for go o d epis in Gp d ( D ) (see below). An arrow o f Gp d ( D ) is s a id to be split if it is right inv ertible, in other words if it admits of a section. 8.1.2. D -functors as (  vert D ) -gr oup oids. F ollowing (5.1), a D -functor may be viewed as a (  vert D )-group oid 51 . 8.2. Actors. It turns out that the basic (algebraic) 52 prop erties of a D -functor f : H → G are encapsulated into two fundamental squar es , written b elow, which immediately acquire a “s tr uctured” (and hence mor e pre c ise) meaning when written in a dipt ych. It is enough to write these pr op erties at the low est level ( f (0) , f (1) ), since it tur ns out that the s p ecia l pullback prop erties o f the nerve, as a diagram, a llow to carry them ov er to all levels. 8.2.1. The activi ty indic ator A( f ) : in/ex/-actors. W e co nsider ﬁr st the comm uta- tive squar e g enerated by the sour c e maps : H f > G A( f ) E α H H f (0) > B α G H . W e shall say f is : an actor 53 , an inactor , an exactor 54 , dep e nding on whether the square A( f ) is a pullback, i -faithful, or s - full (4.1). 51 One has to c hec k the exactness prop ert y . 52 Most of these prop erties own v arious names i n the literature, depending on the authors, and equally unfortunate for our purp ose, since these purely algebraic prop erties received often names issued from T op ology , which cannot b e ke pt when working in T op or Dif . 53 See (2.3) for the terminology . 54 The underlying al gebraic notion is kno wn in the categorical literature under the name of “ﬁbering functors” (or “star-surj ectiv e” f unctors for Ronnie Bro wn), which cannot b e kept when wo rking in a topological setting. GENERALIZED A TLASE S 25 One can show that a ny exactor f : H → G , one can deﬁne its kernel K ◮ > H , which is null when f is an actor. A princip al actor is an actor R f → G with R principal. F or instance , taking for R the graph asso ciated to a cov ering, as describ ed in (2.2), we r ecov er the notion of G -c o cycle , including co cy cles deﬁning a principal 55 ﬁbration (when G is a Lie group) and Haeﬂiger coc y cles deﬁning a foliation (when G is a pseudogro up). 8.2.2. The c anonic al actor δ G . The map δ G : ∧ G = △ G ◮ G may b e viewed as a functor, and indeed a principal s -actor, asso ciated to the right action of G on itself. This w ill b e enligh tened by the functorial considerations to b e develop ed below. 8.3. D -e quiv alences. 8.3.1. The ful l/fa ithfulness indic ator T( f ) . The sec o nd bas ic sq ua re is built w ith the tr ansitors (anchor maps) . H f > G T( f ) E × E τ H ∨ f (0) × f (0) > B × B τ G ∨ . 8.3.2. Equivalenc es and ext ensors. W e shall say f is a n inductor/ i - faithful/ s -full, depe nding on whether the square T( f ) is a pullback/ i -faithful/ s -full 56 (4.1). When f is s -full/a D -inductor, and moreover f (0) lies in D s , then f is an s - functor, and we say f is an s -e quivalenc e /an s -extensor 57 . While the concept of D - inductor derives from the diagrammatic description of “full and faithful”, the general concept of D -equiv alence demands to add a diagram- matic description of the “essential (o r gener ic) surjectivity”, which uses the A ( f ) square and will not b e given here (w e refer to [P 4]). This genera l notion a llows to sp eak of i -equiv ale nc e s to o. Parallel to the notion of canonical actor , we hav e those of canonical equiv alences. These will b e deﬁned b elow, when we have g iven a diagra mma tic c o nstruction of  G and of the canonical morphisms: G →  G ⇒ G . 8.4. Dip tyc h structures on Gp d ( D ) . The following result , whic h we ca n just men tion here, is of basic impor tance for a uniﬁed study of structured group oids. Using the preliminar y study o f diagrams in a dipt ych, one can deﬁne several useful (Go dement ) diptyc h structures on Gpd ( D ). W e s tr ess the fact that the s -functor s are not the rig h t candidates for go o d epis 58 . Among v a rious poss ibilities, one can take: 55 Whence the terminology . 56 As announced, this explains the terminology used for the squares. 57 The terminology derives from the fol lowing fact : it turns out that such functors are the exact generalizations of Lie gr oup extensions (sav e for the f act that one has to use tw o- s ided cosets, which, in general, don’t coincide with right or l eft cosets.). The smo oth case is treated in [P2], which is written in order to b e read possi bly in any dipty ch without any c hange. 58 This problem doesn’ t arise i n the set-theoretic case. Many authors seem to b elieve that pullbac ks along s - functors alwa ys exist in the Di f case , but actually the delicate p oint, often forgotten, is to pr ov e the sur m ersion condition for the source map. 26 JEAN PRADINES • for go o d monos : – either the i -functors – or the i -actors ; • for go o d epis those s -functors which moreov er b elong to one of the following t yp es: – s -exacto rs – s -actor s – s -equiv alences. 8.5. The category Gp d ( N + ∗ c ) . D ∗ -group oids ma y be ca lled D -cogr oup oids. Some constructions for D -gro upo ids may b e b etter understo o d from a study of N + ∗ c -group oids or N + c -cogro upo ids (whic h ar e not Set -gro upo ids) (3.4.1). Some pieces of notations are needed to av oid confusions ar ising from dualit y . 8.5.1. Notations for End( N + c ) and End( N + ∗ c ) . In con trast to sections (4) and (5), we s hal l, for a w hile, stick to Ma c L ane’s terminolo gy c onc erning horizontal and vertic al c omp osition of natur al tr ansformations (also called functorial morphisms ), in order to allow free use of [McL] as reference . This means that gro up o ids (viewed as diagrams or functors) ha ve here to be thought as w r itten horizo n tally (instea d o f vertically as ab ov e), he nc e the group oid functors or mor phisms (i.e. the arrows of Gp d ( D )) a s wr itten vertic al ly , though this is somewhat uncomfortable. The natura l transforma tions betw een endofunctor s make up ca teg ories denoted by E nd ( N + c ) and End ( N + ∗ c ), which are indeed double categories (and even more precisely 2-catego ries [McL]) when c o nsidering b oth ho rizontal a nd vertical comp o- sition. The identit y maps deﬁne canonically: • a c ont r avariant functor: N + c → N + ∗ c , λ 7→ λ ∗ , where λ ∗ is λ with source and tar get exchanged ; denoting by ∗ × the pro duct in N + ∗ c (i.e. the sum + in N + c ), we can write (for a n y pair of arrows λ , µ ): ( λ + µ ) ∗ = λ ∗ ∗ × µ ∗ ; • a c ovariant functor: End( N + c ) → E nd( N + ∗ c ) , Φ 7→ ∗ Φ ∗ = ∗ Φ , with ∗ Φ ( λ ∗ ) = (Φ( λ )) ∗ ; • a bijection End ( N + c ) → End ( N + ∗ c ) , ( ϕ : Φ · → Φ ′ ) 7→ ( ϕ ∗ : ∗ Φ · ← ∗ Φ ′ ) , with ϕ ∗ ( n ) = ( ϕ ( n )) ∗ , which is : – c ova riant with resp ect to horizontal comp osition laws ; – c ontr avariant with resp ect to vertic al comp osition laws. 8.5.2. Description of Gp d ( N + ∗ c ) . By the previous bijection: • N + ∗ c -group oids der ive c ovariantly from the endofunctor s of N + c pr eserving surje ct ions, inje ctions and pus hout s ; • morphisms of N + ∗ c -group oids derive c ontr avariantly from morphisms b e- t ween endofunctors o f the previous t yp e . GENERALIZED A TLASE S 27 Moreov er we kno w from the general descriptions of D -gro up o ids (6.2.3) that: • the endofunctors Γ = ∗ Φ of N + ∗ c deﬁning N + ∗ c -group oids are uniquely deter- mined b y the data: (Γ (0) , Γ (1) , ω Γ , α Γ , δ Γ ) , hence (resulting from the previous study) by the da ta in N + c : ( n 0 = Φ( · 0) , n 1 = Φ( · 1) , n 2 = Φ( · 2) , ω : n 0 ◭ n 1 , α : n 1 < ◭ n 0 , δ : n 2 < ◭ n 1 ) . 8.5.3. Some b asic examples of N + ∗ c -gr oup oids and morphisms. As ju st explained, such a group oid mor phism γ : Γ ← Γ ′ derives fro m a functor ia l morpism ϕ : Φ · → Φ ′ where Φ , Φ ′ preserve surjectio ns , injectio ns and pushouts. These deﬁne a 2-sub catego ry of End ( N + c ) denoted b y  . Those Φ ’s whic h preserve s ums too are of t ype : p × : · n 7→ p × · n, λ 7→ p × λ ( p times itera ted sum in N + c ). The asso ciated group oids , deno ted by p ∗ × are the b anal (but no t principal 59 ) N + ∗ c - group oids, in the sense of (7.2.3) 60 . F or p = 0 , we get the “ n ul l gr oup oid ”, denoted by 0 , asso ciated to the constant functor 0 : · n 7→ · 0, and, for p = 1, the “ unit gr oup oid ”, denoted b y 1 , ass o ciated to the iden tit y functor. F or p = 2, one has the “ squar e gr oup oid ” 61 , denoted b y  ∗ = 2 ∗ × , asso ciated to: 2 × : · n 7→ 2( · n ) = · n + · n, λ 7→ 2 λ = λ + λ . Among the non pr incipal ones (7.2.2), the simplest is denoted by △ ∗ : it is asso ciated to the “ shift functor ”: · 0 + : · n 7→ · 0 + · n, λ 7→ · 0 + λ where the last · 0 is understo o d as the iden tit y of the ob ject · 0 = 1. W e have also a N + ∗ c -group oid morphism: (1) δ ∗ 0 : △ ∗ − → 1 asso ciated, with the notations of (7.1 .1) and of [McL], to the the “ shift morphism ”, deﬁned by the family ( δ · n 0 ) n ∈ N (injections skipping the 0 in · 1 + · n ). As to  ∗ , asso ciated to 2 × , we hav e in N + c the copro duct morphisms ( n ∈ N ) : ( · n ) < co diag ( · n ) + ( · n ) < ι 2 < ι 1 ( · n ) which deﬁne morphisms of N + ∗ c -group oids denoted sugges tively by: (2) 1 ω >  ∗  2 >  1 > 1 . By iteration we ca n ev en get a c anonic al gr oup oid (  ∗ ( n ) ) n ∈ N in Gp d ( N + ∗ c ) 62 . The “ s ymm et ry gr oup oid ” Σ ∗ pro ceeds from the (inv olutive) “ symmet r y ma p ” Σ : N + c → N + c , derived by r eversing the order on the integers viewed a s ordinals 59 Since the terminal ob ject 0 has b een dropp ed. 60 While the banal (and pri ncipal) N + c -groupoi ds are p × p . 61 The terminology wi ll b ecome clear b elow in 9.3. 62 More precisely , with a suitable dipt yc h structure (see 8.4). This is i ndeed a double group oid, or b etter a “ gr oup oid-c o gr oup oid ”. 28 JEAN PRADINES ( − → n 7→ ← − n ) ; it is deﬁned by the identit y on the ob jects, a nd, on the generator s (with Mac Lane’s notations), b y ( n ∈ N + ): Σ : δ n j 7→ δ n n − j , σ n j 7→ σ n n − j , and the family of maps : ( ς n : n → n, j 7→ n − j ) ( n ∈ N + ) deﬁnes a natura l transfor- mation from iden tit y tow ards Σ, hence also a group oid morphism: (3) ς : Σ ∗ − → 1 . 9. Double functoriality of the definition of D -groupoids. 9.1. Bi v ariance of D -group oids and D -functors. The deﬁnition of a D -gr oup oid as a dipt yc h morphism G : N + ∗ c → D shows immedia tely (look at the adjoining di- N + ∗ c Γ > ↓ γ Γ ′ > N + ∗ c H > ↓ f G > D T > ↓ t T ′ > D ′ agram, where we go on sticking to Ma c Lane’s conven tions) this deﬁnition is: • functorial with resp ect to co mpo sition, on the left (target side ), with an exact diptyc h morphism T : D → D ′ , as well a s with resp ect to right (hor- izontal ) co mpo sition with a natural transformatio n t : T · → T ′ ;this means that any D -functor f : H → G g ives ris e to a D ′ -functor T ◦ f : T ◦ H → T ◦ G 63 and to a natural transformatio n: t ◦ f : T ◦ f · → T ′ ◦ f ; the la tter gives ris e to the following commutativ e square o f D ′ -group oids 64 : T H T f > T G T ′ H t ( H ) ∨ T ′ f > T ′ G t ( G ) ∨ ; actually Gp d b ehav es here lik e a functor and w e can deﬁne: Gp d ( T ) : Gp d ( D ) → Gp d ( D ′ ), Gp d ( t ) : Gp d ( T ) · → Gp d ( T ′ ) ; • functorial with res pect to horizo nt al comp osition o n the right (so urce side), with a N + ∗ c -group oid Γ, as well as with a group oid morphism γ : Γ → Γ ′ viewed as a natura l transfor mation b etw een endofunctors of N + ∗ c ; this means that any D -functor f : H → G gives ris e to a D - functor: Γ • f : Γ • H → Γ • G , where Γ • H = H ◦ Γ , Γ • G = G ◦ Γ , Γ • f = f ◦ Γ , and to a natural transformatio n γ • f = f ◦ γ : Γ • G → Γ ′• G ; 63 As i n [McL], the same notation is used f or a functor and the ident ity natural transformation associated to this functor, here T 64 W r itten i n lo ose notations, i .e. identifying groupoids and functors with their 1-l ev el part. GENERALIZED A TLASE S 29 this deﬁnes a (cov ariant) functor: Γ • : Gp d ( D ) → Gp d ( D ) , hence a c anonic al (v ertical) r epr esentation : Gp d ( N + ∗ c ) → Gp d ( D ) , but this representation dep ends in a c ontr avaria nt wa y up on Γ, with resp ect to the horizontal comp osition of N + ∗ c -group oids deﬁned ab ov e, since Γ • ◦ Γ ′• = ( Γ ′ ◦ Γ ) • , γ • ◦ γ ′• = ( γ ′ ◦ γ ) • . Note that, when going back to the generating endofunctors of N + c , we g et a doubly c ontra variant c anonic al r epr esentation of  (notation o f 8.5.3) into Gp d ( D ). W e give a few examples. 9.2. Exampl es for the left functorialit y . W e can either “for get” the structures on ob jects o f D , or “enrich” them. W e g ive examples of these opp osite directions . 9.2.1. Concr ete diptychs. Thinking to the basic exa mples of T op and Dif , we shall say D is “ c oncr ete ” if it c omes e quipped with an a djunction f rom E = Set to D deﬁned by the following a djoint pair [McL] of functors 65 : (discrete, forgetful) = ( ˙ , | | ) : E ˙ > < | | D . Then we ca n spe a k of the underly ing E -gro upo id, and make use of set-theoretical descriptions. 9.2.2. The tangent functor. Thinking now to the c ase when D = Dif , we can c o n- sider (see 3.3.6) the tangent functor: T : Dif − → VectB , which is equipp ed with tw o natural transformations : 0 · o > T · t > 0 , . Once one ha s chec ked it deﬁnes an exact diptyc h morphism, we can immediately transfer to the tang ent groupoids all the general constructions v a lid for genera l D -group oids (for instance constructions of ﬁbred pro ducts, and so on). 9.2.3. Double gr oup oids. The idea of deﬁning and studying the notion of double group oids as g roup oids in the categ ory of g roup oids is due to Ehresmann, who prov ed the equiv alence with the alternative description by means o f tw o ca tegory comp osition laws satisfying the “exchange la w” 66 . In the dipt ych setting we g et sev eral no tions depending on the c hoice for the dipt ych structur e on Gpd( D ) (see 8 .4). W e stress the point tha t, even in the purely se t-theoretical s e tting, t he choice we made of exactor s for g o od epis implies adding a cer tain surjectivity conditio n 67 which does not app ear in Ehresma nn’s deﬁnition, but was encountered by several 65 Assumed moreo v er to be faithful, to pr eserv e pro ducts, and to deﬁne exact diptyc h m or - phisms. An ob ject B of D is viewed as a “struct ure” on the “underlying set” | B | , and an arrow A → B of D is fully describ ed by the triple ( A, | f | , B ) . An y set E may b e endow ed with the “discrete structure” ˙ E . 66 In our fr amew ork this would result f rom 5.3. 67 If w e consider the square made up b y the sources and targets of both laws, this con dition means that three of the four edges may b e giv en arbitraril y . 30 JEAN PRADINES authors, mainly Ronnie Brown (ﬁlling condition), and seems useful to develop the theory beyond just deﬁnitions. Applying the general theor y of D -diptyc hs, for instance for getting ﬁbred pr o d- ucts, or quotients, or Mor ita equiv alences, g ives results which it would b e very hard to get by a dire c t study (whic h has never b een done), ev en in the purely algebra ic setting. 9.3. Exampl es for the right functoriali t y . As announced,the coherence of v ar - ious notations in tro duced above will a pp ea r just below (to be precise , we ﬁnd : △ ∗ • = △ ,  ∗ • =  ). The examples g iven in 8.5.3, allow to transfer to D -group oids some cla ssical set-theo r etic cons tructions, a nnounced ab ove. T aking for γ : Γ → Γ ′ : • formula (1) o f 8.5.3, we recov er the morphism: δ G : △ G − → G ; • formula (2), we get the canonical equiv alences: G →  G ⇒ G ; • formula (3), we get the dual gro upo id G ∗ = Σ G , and the inverse law ς G = ι G , which deﬁnes an inv olutive isomo rphism: ς G : G → G ∗ . 9.4. D − natural transformations, holom orphisms. Once  G is deﬁned, one can also deﬁne D - natur al tr ansformations b etw een functors from H to G as D - functors H →  G . The canonica l i -equiv ale nc e G →  G deﬁnes the ident ical transformatio n o f the iden tit y functor. Since G is a group o id, suc h natural transfo rmations a r e nece ssarily functorial isomorphisms. An isomo r phy class o f D - functors will be called a “ holomorphism ” (an a lternative terminology migh t b e “exomorphism”, since this notion g eneralizes the outer automorphisms of groups). Since the horizontal comp osition of natural transformations comm utes with the vertical one, it deﬁnes a compo sition b etw een holomor phisms, and this yie lds a quotient categ o ry of Gp d ( D ), whic h w e shall denote by Hol ( D ). An y D - functor f : H → G g e nerates the following commutativ e diagra m, in G K > p ( f ) >  G π 2 ∼ N pb H q ( f ) ∼ H f > G π 1 ∼ H which q ( f ) is an s -equiv a lence, a nd p ( f ) an exacto r. The pair ( p ( f ) , q ( f )) is called the “ holo gr aph ” of f . Moreov er the s - equiv ale nc e q ( f ) is split (8.1.1) since π 1 is. GENERALIZED A TLASE S 31 10. The butterfl y diagram. This section illus tr ates, in the case of orbital structures presen tly des crib ed, the use of diagrams in a diptyc h for transferr ing constructions in Set to constructions in Dif or other v arious categories , as w ell as the use of the v arious kinds of D - functors introduced a b ove. W e can be only v ery sketc h y . More precis e des criptions and results ma y b e found in [P4], where they a r e stated for the diﬀerentiable case, but written to b e ea sily tr ansferable to g eneral diptyc hs. Mor e deta ils a bo ut the diagrams used for pro ofs will be given elsewhere. 10.1. Generalized “structure” of the orbit space. 10.1.1. Algebr aic “stru ct ur e”. First, from a purely set-theoretic po int of view, any equiv alenc e b etw een t wo gro upo ids, in the general categor ical se nse [McL] 68 , pre- serves 69 the set-theoretic orbit space Q , but indeed it prese rves muc h more, since, to e ach or bit (or transitive comp onent), there is a (w ell deﬁned only up to isomor - phism) attac hed isotropy gro up, and this endo ws Q with a kind of an algebraic “structure”, in a (non s et-theoretical) gener alized sense. F or instance, the le aves of a foliation are marked by their holonomy gro ups, the orbits of a group(oid) action are marked by their ﬁxato r s. Suc h a “structure” is sometimes called a group sta ck. 10.1.2. D - “ structure ” of the orbit sp ac e. Then, in the D -fra mework, replacing alge- braic equiv alences by D -equiv a lences will mo reov er encapsulate in this g eneralized structure the memory of the D - structure as well. It turns out (though t his is by no means a priori ob vious) that it is enough to make use of s -equiv alences (8.3.2). Finally w e are led to the following: Deﬁnition: Tw o D -group oids H , G , are s aid to b e D -e quivalent if they are linked by a pair of s -equiv alence s : H ◭ q ∼ K p ∼ ◮ G . The fact that this is indeed an equiv alence relation is an ea sy co nsequence of the results stated in 8 .4, taking s -equiv alences as go o d epis in Gp d ( D ), and using ﬁbred pro ducts of go o d epis. A D -equiv alence class of D -group oids may b e calle d an orbital stru ctur e . An y representative of this equiv alence class is called an atlas o f the orbital structure. 10.2. In v erti ng equi v alences. 10.2.1. Mer omorphi sms. Note than in g e ne r al or bita l structur es ca nnot be taken as obje cts of a new categor y . Ho w ever one can deﬁne [P4 ] a new catego r y which s hall be deno ted her e b y M ero ( D ), with the same obje cts a s Gp d ( D ), and ar rows ca lled “ mer omorph isms ”, in whic h the s -equiv alenc e s (and indeed all the D -equiv alences) bec ome inv ertible, in other w ords are turned into isomo rphisms. In the topolog ical case, these isomo rphisms may b e identiﬁed with the Morita e quivalenc es . In this new catego ry the orbital structures now b ecome isomo rphy classes of D -group oids, though the ob jects s till remain D -g roup oids and not isomor ph y classes , so that the orbital “structures” ar e not c arrie d by actual sets , and, as such, r emain “virtual”. This means that Mero ( D ), is the universal solution for the proble m of frac- tions consisting in fo r mally inv erting the s - equiv ale nces, and indeed all the D - equiv alenc e s. 68 i.e. a functor which is ful l, faithful, and essentially sur j ectiv e, but p ossibly non-sur jectiv e. 69 More precisely , this means that it deﬁnes a bijection b etw een the tw o orbit spaces. 32 JEAN PRADINES This kind of problem alwa ys admits a general s olution [G-Z]: the a rrows are given b y equiv alence classes of diagr ams, the de s cription of whic h is simpler when the conditions for right ca lc ulus of fractions ar e satisﬁed. It is worth noticing that these ass umptions are not fulﬁlled here, while our con- struction is in a cer ta in sense muc h simpler, s ince , a s is the case in Arithmetic, ea ch fraction will admit of a simpliﬁed or irr e ducible canonical repr esentativ e ( p, q ). In the top ologica l case these repres ent atives may be identiﬁed w ith t he “ge ne r alized homomorphisms” describ ed by A. Haeﬂiger in [Ast 1 1 6] and attributed to G. Sk an- dalis, or the “ K -or iented morphis ms ” of M. Hilsum and G. Sk a nda lis. W e can say more. The ca nonical functor Gpd ( D ) Φ → Mero ( D ) a dmits of the following factorizatio n: Gp d ( D ) Φ 1 → Hol ( D ) Φ2 → Mero ( D ) (s e e 9.4) with Φ 1 full (i.e. here surjective) and Φ 2 faithful (or injective). It turns out that H o l ( D ) is the solution o f the problem of fractions fo r split (8.1.1) D -equiv alences. It is embedded in M e ro ( D ) by means of the hologra ph (9.4). 10.2.2. Description of fr actions. Let ( p, q ) denote a pa ir of exactors with the same source K : p : K → G, q : K → H . W e set R = Ker q , S = Ker p (see 8.2). Let ( p ′ , q ′ ) another pair p ′ : K ′ → G, q ′ : K ′ → H with the sa me G and H . Letting for a while G a nd H ﬁxed, we star t consider ing arrows k : ( p ′ , q ′ ) → ( p, q ) deﬁned as D -functors k : K ′ → K s uc h that the whole diagra m co mm utes. Let p/q denote the isomorphy clas s of ( p, q ), and call it a “ fr action ”. On the other hand w e say tw o pairs ( p i , q i )( i = 1 , 2 ) a re e quivalent if there exist t wo s -equiv alences k i : ( p, q ) → ( p i , q i ). This is indeed a n equiv alence relation, and the class of ( p, q ) will b e denoted b y pq − 1 . W e consider no w those pairs ( p , q ) satis fying the s ubs equent extra conditions, which turn out to b e preser ved by the previo us equiv alence: (1) q is an s -equiv alence; (2) p and q are “ c otra nsversal ”. The for mer condition implies that the kernel R = Ker q is principal (7.2.2). The la t- ter condition w ill b e expres sed by mea ns of the following (comm utativ e) “ butterﬂy diagr am ” g athering the previous data: S R K < i ◭ ◮ j > H v H ◭ ∼ q G . u ∨ exa p > Then the condition of c otransversality means that u or (this is indeed equiv alen t) v is an exactor (then v will b e an s -exactor ). When u (and v ) are a ctors, p and q are s a id to b e “ tr ansverse ”, a nd the fractio n p/q is called “ irr e ducible ” (or simpliﬁed). One can show (using the theory o f extens o rs) that the class pq − 1 owns a unique irr e ducible rep r esen t ative p/q . Our mer omorph isms (fro m H to G ) ar e then deﬁned as the classes pq − 1 or their irreducible represe ntatives p/ q . GENERALIZED A TLASE S 33 A “Morita e quivalenc e” is the sp ecial c a se when p is an s -equiv a lence to o. The butterﬂy diagram is then p erfe ctly symmetr ic . W e say that ( u, v ) is a pair of “ c on- jugate princip al actors ”. Each one determines the other one up to isomorphism. Using ir reducible repr esentativ es and forg etting the D - structures, one then re- cov ers easily the set-theore tica l par t of the des cription of Sk andalis-Haeﬂiger homo- morphisms (tw o commuting actions , one b eing principal). No w, in the diﬀerentiable case, the lo ca l triviality conditions are automatically encapsulated in the sur mer sion conditions (impo s ed to the go o d epis) b y means of the Go dement theor em. One of the immense adv antages of this pr esentation (apart from b eing deﬁned in many v arious frameworks), is that the use of no n irreducible representativ es a llows a very natural deﬁnition o f the comp osition of meromor phisms (note that in [Ast 116] this comp osition is deﬁned b y A. Haeﬂiger but in very special cas es, when one a rrow is a Mor ita equiv alence). This co mp os ition is deﬁned by means of the following diagr am (using the dipt yc h prop erties of Gp d ( D )) (8.4): L K ′ ◭ ∼ pb K exa > G ′′ mero > ◭ ∼ G ′ mero > ◭ ∼ exa > G . exa > (Of course there are many things to chec k to justify all our claims). 10.3. Example. Let us come bac k to the example of the spa ce o f lea ves o f a (reg- ular) foliation (2.4). W e invite the reader to lo ok at what happ ens when we take as go o d epis: (1) a ll the surmersio ns ; (2) the r etro connected ones; (3) the r etro discrete (or ´ etale ) o nes; (4) the pr op er ones. In the ﬁrst case, we ar e allowed to tak e as a n a tlas the holonomy g roup oid and the transverse holono my pseudog roups asso ciated to v arious tranv ersals as w ell, which all belo ng to the same Morita class. The sec ond choice is adapted to the sear ch for inv a r iants o f the Molino equiv- alence clas s o f a folia tion: the Morita class of the holonomy gro upo id is suc h an inv ar iant. The third choice is adapted to the use o f holo nomy pseudog roups and of v an Est S-atlases [Ast 116], and to the study of the eﬀect of cov e rings o n foliations. The fourth c hoice is adapted to the study of compact leav es of folia tions and of prop erties related to Reeb s ta bilit y theorem, as well a s to the study of or bifolds (or Satake manifolds). In fact it is useful to use v ario us c hoices sim ultaneously . 11. E pilogue. • W e ha ve no t given her e statements concerning dipt ych structures on the category Mero ( D ), since our present results are s till partial a nd demand some further chec ks to ensure them completely , especially concer ning the Go dement proper ty . It is clear fo r us that s uch t ypes of statements w ould be very useful, s inc e , for instance, gr o up o ids in s uch dipt ychs would be 34 JEAN PRADINES fascinating ob jects. An yw ay it seems clear for us that this category has to be explored more deeply . • W e know that our for mal c o nstruction f or the pr evious category of frac- tions seems to work p erfectly as well when r eplacing the catego ry of s - equiv alenc e s b y that of s -extensor s or b y v ar ious s ubca tegories o f the latter (adding for instance conditions of c onnectedness on th e isotr o py gr oups). W e ar e co nvinced that such ca teg ories, whic h are muc h less kno wn (no t to say totally unknown) than the prev ious one, are ba sic for the under s tanding of ho lo nomy o f foliatio ns with singular ities ( ˇ S tef an f ol iations ), and that these enlarged Mor ita cla sses certa inly encapsulate some dee p and hidden prop erties of the orbit spaces . Ackno wledgements. (1) I would like to thank warml y the organizers for th eir invitation to this Conference at Krynica. I was happy with th e friendly and stimula ting atmosphere which was reigning throughout this session. (2) I am indebted to Paul T aylor for t he ( certainly very a wkwa rd) use I made of his pack age “diagrams”, allow ing va rious styles of arro ws, which are very useful as condensed visual mnemonics for memorizing p roperties of maps an d fun ctors. References 1. [P1] J.Pradines: “Buil ding categories in whi c h a Go dement theorem is a v ai lable“, Cahiers T op. e t G´ eom. Diﬀ. , XVI-3 (1975), 301-306. 2. [P2] : “Quotien ts de group o ¨ ıdes diﬀ´ erentiables”, C.R.A.S., Paris , 303, 16 (1986), 817- 820. 3. [P3] : “How to deﬁne the graph of a singular foliation”, Cahiers T op. et G ´ eom. Diﬀ. , XVI-4 (1985), 339-380. 4. [P4] : “Morphism s b et w een s paces of leav es viewed as fr actions”, Cahiers T op. et G ´ eom. Diﬀ. , XX X-3 (1989), 229-246. 5. [P5] :“F euilletages: holonomie et graphes lo caux”, C.R.A.S., Paris , 298, 13 (1984), 297-300. 6. [P6] : “Remarque sur l e group o ¨ ıde cot angen t de W einstein-Dazord”, C.R.A.S, Paris , 306, (1988), 557-560. 7. [B] N. Bourbaki: V ari´ et´ es diﬀ ´ er entiel les et analytiques , Hermann, Paris, 1971. 8. [McL] S. Mac Lane: Cate gories for the Working Mathematician , Spri nger-V erlag, New Y ork, 1971. 9. [LALG] J.-P . Serr e: Lie Algebr as and Lie Gr oups ,W. A. Benjamin Inc., New Y ork, 1965. 10. [G-Z] P . Gabriel and M. Zisman: Calculus of fr actions and homotopy the ory , Ergebn. Math. 35, Spri nger, 1965. 11. [Ast. 116] Ast ´ erisque 116, Structur e tr ansverse des feuil letages , Soci´ et ´ e Mat h´ ematique de F rance Paris, 1984. A. Haeﬂiger: “Groupo ¨ ıdes d’ holonomie et classiﬁant s”, pp. 70-97. W. T. v an Est: “Rapp ort sur les S-atlas”, pp. 235-292. 26, rue Alexandre Ducos, F31500 Toulouse, France E-mail addr e ss : jpradines@wanad oo.fr

Lie Groupoids as generalized atlases

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