Categories of Modules with Differentials

JOURNAL OF ALGEBRA 185, 50᎐73Ž. 1996 ARTICLE NO. 0312

Paul Popescu

Department of Mathematics, Uniërsity of Craioä, 13, A.I. Cuza st., Craioä, 1100, Romania

Communicated by Walter Feit CORE Metadata, citation and similar papers at core.ac.uk Received August 1, 1994 Provided by Elsevier - Publisher Connector

1. INTRODUCTION

The definitions of the module with arrow Ž.module fleche´´ , the infinitesimal module, and the Lie pseudoalgebra, as used here, are considered inwx 7 . Moreover, we define the preinfinitesimal module, called inwx 1 ‘‘un pre-´ espace d’Elie´ Cartan regulier.’’´ Inwx 7 the Lie functor is constructed from the category of differentiable groupoids in the category of Lie algebroids, but it is inwx 2 that the first general and abstract treatment of the algebraic properties of Lie algebroids is made, giving a clear construction of the morphisms of Lie algebroids. We shall use it fully in this paper to define completely the categories M W A Ž.Ž.modules and arrows , P I M preinfinitesimal modules , IMŽ.Ž.infinitesimal modules , and L P A Lie pseudoalgebras ; we call these categories the categories of modules with differentials. We notice that in wx7 and wx 1 the objects of these categories and the subcategories of modules over a fixed algebra are considered. Inwx 3 the category of L P A of Lie pseudoalgebras is defined and some aspects related to the functional covariance or contravariance correspondence with categories constructed with additional structures on vector bundles are studied.

In Section 2 we give a brief description of the category MA , of modules over commutative k-algebrasŽ considered also inwx 3. , and we define a strict category of modules using a suitable functor. In Sections 3 and 4 we define the categories of modules with differentials. In Section 5 we study certain wide and strict categories of modules with differentials which are defined in this paper. It is proved that, under suitable conditions on the functor which define them, they admit direct products.

0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. MODULES WITH DIFFERENTIALS 51

Finally, in Section 6, we study the pullback, which extends the inductor defined inwx 8 , using the ideas from wx 2 . We define the pullback in categories of modules with differentials and in categories of vector bundles with differentialsŽ. where it is in fact a fibred product , extending the definition of the inductor given inwx 2 in the case of Lie algebroids. Using two papers of the authorwx 5, 6 , we consider thoroughly with some examples the study of differential modules, considering categories which we call categories of vector bundles with differentials. They are extensions of the Lie algebroidwx 7, 4, 2 and are very close to it. This paper is the first part of a study about categories with differentials. The second part will contain the study of actions, fibrations and ideals, using the same approach as inwx 2 for Lie algebroids and groupoids. We intend this paper to be useful to those who follow the study of differential geometry using algebraic tools and also to those who use geometric examples in algebra.

2. CATEGORIES OF MODULES

The largest category whose objects and morphisms we shall use in this paper is a contravariant category of modules MA. The modules are considered over commutative, associative and unitary algebras. The algebras are taken in their turn over a commutative and unitary ring k, fixed from now on. The category MA is considered for example inwx 3 , where it is ¤ denoted as Mod . The commutative, associative, and unitary algebras over k and their morphisms are the objects and the morphisms of a category Ak denoted simply by A. Ž. The objects of the category MA are coupled A, L , where A is an object of A and L is an A-module. The morphisms of two objects Ž AXX, L . X and Ž.A, L are couples Ž.␸, ␺ , where ␸: A ª A is a morphism of A and XX X Ž ␺:LªAmA Lis a morphism of A -modules. The tensor product is an A-module, considering AX as an A-module through ␸.. We denote such a Ž XX.Ž. morphism in MA by A , L ªŽ␸, ␺ . A, L . The composition law of two morphisms in M: ŽA , L . Ž.Ž.ŽA , L , A , L A , L .is 00ªŽ␸11,␺.11 11ªŽ␸22,␺.22 Ž.A,L Ž.A,L, where, if 00ªŽ␸1212(␸,␺)␺.

XL,␺Ž.X aXXA L, g01sÝiAm11 ig0mA 1 i ␺Ž.XXbiXYAL 2 isÝ␣mA22␣g1mA 2 ␣ 52 PAUL POPESCU then

Ž.Ž.␺ ␺ X a ␸ bi X Y A L .1Ž. 21) s Ýi1Ž.␣mA22␣g 0mA2 i,␣

Notice that if A00is an object of A then the category of A -modules is a subcategory M of M . It has as objects all the objects and the morphisms A 0 A of M which have the form Ž.Ž.A , L and id , ␺ , respectively. Taking into A 0 A 0 account the canonical isomorphism

A L L, a X aX Ž.2 0mAiAiii00l ÝÝm l ii for every A -module L, there is an isomorphism of categories M and the 0 A0 category of A0-modules. From now on we shall not make any distinction between them. Notice also that M is not a full subcategory of M . A0 A Consider now AX a subcategory of A. In the same manner as above, denote by MAX the subcategory of MA whose objects and morphisms are defined by couples in which the algebra is an object of AX and the morphism is a morphism of AX. Consider now C00an arbitrary category and F: C ª MAa functor; we ŽXŽX.. denote as C0 MA a category which has as objects couples M , FM with MX an object in C X, and as morphisms couples Ž␸X, FŽ␸X .. for ␸X morphism of C00; we say that C MAis a strict category of modules induced by F.We X notice also that F induces a forgetful functor F0: C ª A that generates a category of algebrasŽ. see Section 5 . An important example of a strict category of modules is given below. It is used frequently in the examples of this paper. ϱ EXAMPLE 2.1. Take as C0 the category V B of C -vector bundles which have finite dimensional fibers. We denote by ␰ s Ž.E, p, B such a vector bundle which has the total space E, the base space B and the projection p, all smoothŽ i.e., of class Cϱ.. The morphisms of two vector X Ž XX X.Ž.X bundles ␰ s E , p , B and ␰ are couples f00, f , where f : B ª B is a X smooth map and f: E ª E is an f0-morphism of vector bundles. There is induced

U X U f00: FŽ.B ª F ŽB ., f Ž.u s u( f0, a morphism of commutative, associative and unitary algebras and U X U Ž X . f : E ª fE0 which is uniquely determined by a F B -module morphism, denoted also by f U :

U X U f : ⌫Ž.␰ ª ⌫ Žf0␰ .. MODULES WITH DIFFERENTIALS 53

Notice that for a smooth manifold B we denote as FŽ.B the real algebra of smooth real functions on B and for the vector bundle ␰ s Ž.E,p,Bwe denote as ⌫ Ž.Ž.␰ the F B -module of sections on ␰. From the isomorphism of FŽ BX .-modules

XXUX FBŽ.mFŽB.⌫ Ž.␰ ª⌫ Žf0␰ ., umXªuX, Ž.UUXX Ž . where X g ⌫ fE0 is x ª x , Xf Ž x. , it follows that f can be also Ž X .Ž.Ž0 . considered taking values in F B mFŽ B. ⌫ ␰ seewx 2 for more details . For k s ޒ a contravariant functor F0: V B ª MA defined by the formula

U U F00Ž.␰ s Ž.F Ž.B , ⌫ Ž.␰ , Ff Ž0,f .s Žf0,f . is obtained. This defines a strict category of modules V B MA , used in the next examples of this paper.

3. CATEGORIES OF MODULES WITH ARROWS

In this section and in the following one we define some categories which we call categories of modules with differentials; they contain as subcategories those obtained fixing an algebraŽ seewx 7. . The construction is inspired by wx2 ; the examples are fromwx 5 , wx 6 , and wx 2 . Most of the notation fromwx 7 andwx 2 is used, pointing out the extensions given here. Let A be a commutative, associate, and unitary k-algebra. For the p A-modules L and M we denote as Hom AŽ.L, M the A-module of alter- nated A-linear p-forms from L to M and as DerŽ.A the A-module of derivations of A Ž.which is a Lie k-algebra . An A-module with arrow Ž.Žm.w.a. cf.wx 7 module fleche´´.is an A-module M Mtogether with an A-module morphism ␳ s ␳ : M ª DerŽ.A called an anchor. As an example, DerŽ.A and its identity morphism is an m.w.a. If M ␳ s 0 we say that the m.w.a. is tri¨ial.If Mis an A-m.w.a. we make the notation

M wxX,aMs␳Ž.Ž.Xa, XgM,agA.

DEFINITION 3.1wx 7 . A morphism of A-m.w.a.’s L and M is an element 1 Ž. ␻gHom A L, M such that

␻Ž.X,aMswxX,aL, Ž.᭙XgL,agA.3Ž.

For example, using the definition, Ž.A, M M ŽA, Der Ž..A is a ªŽidA, ␳ . morphism of m.w.a.’s. An extension of this definition is 54 PAUL POPESCU

XX DEFINITION 3.2. Let Ž A , L .Ž.and A, L be m.w.a.’s. We say that a Ž XX.Ž. module morphism A , L ªŽ␸, ␺ . A, L is a morphism of m.w.a. provided X X that for every X g L such that

X X X ␺ Ž.X s ÝuiAm X ig A m AL,4Ž. i we have, for every t g A,

XX X,␸Ž.tLXsÝuii␾Ž.wxX,tL.5Ž. i

The next lemma shows that this definition does not depend on the X ␺-decompositionŽ. 4 of ␺ ŽX .Ž. Compare withwx 2, Lemma 1.4 .. X LEMMA 3.1. Let L be an A-m.w.a., ␸: A ª A ak-algebra morphism. 1 XX XŽ . Then for e¨ery t g A there exists ⌽tAAg Hom A m L, A such that

⌽ aXXX a␸X,t tiAiž/ÝÝm s iŽ.wx iL ii

X X for e¨ery Ýiia m AX ig A m AL. Proof. It suffices to see that

XXXX f11:A=LªA, faŽ.,Xsa␸Ž.wxX,tL has the property that for any a g A

XX X X fa1Ž.,aX s a␸ Ž.Ž.wxaX, t LLs a␾ aXwx,t sa␸Ž.a␸ Ž.wxX,t L X sfa1Ž.␸Ž.a,X; thus it extends to ⌽t required in the lemma. Notice that using the isomorphismŽ. 2 it follows that Definition 3.2 extends Definition 3.1. XX As inwx 7 , we can associate with a pair of m.w.a.’s ŽA , L .Ž.and A, L , and X X a k-algebra morphism ␸: A ª A , a morphism of A-modules

X X X X P : aL1Ž.,AmALªaL1Ž., Der␸ Ž.A, A XX X P␺ŽXt .Ž.sX,␸ Ž.tLXyÝuii␸Ž.wxX,t i XX Ž. Ž X. for every t g A and X g L such that 4 holds. Der␸ A, A is defined below in Definition 5.1 The condition that Ž.␸, ␺ be a morphism of m.w.a. is P␺ s 0. MODULES WITH DIFFERENTIALS 55

Notice that composing two morphisms of m.w.a.’s as module morphisms according toŽ. 1 we obtain a morphism of m.w.a.’s. Thus we have the following:

PROPOSITION 3.1. The m.w.a.’s and their morphisms are the object and the morphisms of a category M W AA. Notice that, as in the case of categories of modules, we can define:

ⅷ Subcategories of M W A of the form M W A , for a fixed algebra A A 0 A0; these are the categories considered inwx 7 . X ⅷ Subcategories of M W AA of the form M W AA X , for a subcategory A of A.

ⅷ Strict categories of m.w.a.’s defined by a functor F: C0 ª M W AA. An example of such a category is given below.

EXAMPLE 3.1. We begin with some definitions fromwx 5 . Let ␪ s Ž.R,q,Mbe a vector bundle and ␶ M s ŽTM, p, M .be the tangent bundle of the base M.Ananchor Žor a tangent map inwx 5. is an id-morphism of vector bundles D: ␪ ª ␶ M. Such a couple Ž.␪, D is called a ¨ector bundle XX with arrow Žv.b.w.a., or a relative tangent space inwx 5.Ž of M.If ␪,D.is X XX X XX another v.b.w.a., ␪ s Ž R , q , M .Ž, then a morphism of v.b.w.a.’s ␪ , D . Ž. X ªŽ f, g . ␪,Dis a morphism of vector bundles ␪ and ␪, provided that X X D( g s ␶ f ( D Žwhere ␶ f: ␶ M ª ␶ M is the f-morphism of vector bundles induced by the differentials of f in every point of M .. There follows a category denoted as V BW A. Using the same definition as that of functor F in Example 2.1, there follows a functor F1: V BW A ª M W AA , where k s ޒ, and consequently a strict category of m.w.a.’s, denoted as V BW A M W AA. In the same manner we define some functors in the next section.

4. CATEGORIES OF PREINFINITESIMAL MODULES

A preinfinitesimal module in this paper is called ‘‘un pre-espace´ d’Elie´ Cartan regulier’’´ inwx 1 . It extends the infinitesimal module ofwx 7Ž see Definition 4.4 below. .

DEFINITION 4.1. A preinfinitesimal module Ž.p.i.m. is an m.w.a. ŽA, M . together with a bracket

wxи,иM:M=MªM 56 PAUL POPESCU provided that it is k-bilinear, antisymmetric, and

wxwxwxX,uY MMs X, uYquX,Y M, Ž.᭙X,YgM,ugA.

DEFINITION 4.2. A morphism of A-p.m. is a morphism of A-m.w.a. ␻:MªLŽ.cf. Definition 3.1 provided that

␻Ž.X,␻ Ž.YLs␻Ž.wxX,YM,Ž.᭙X,YgM.

Using an idea fromwx 2 and w 3, Definition 4.3x we extend this definition: XX DEFINITION 4.3. Let Ž A , L .Ž.and A, L be p.i.m.’s. We say that a Ž XX.Ž. morphism of m.w.a.’s A , L ªŽ␸, ␺ . A, L is a morphism of p.i.m.’s provided that

XX XX XX ␺XYX u XY X XY Ž.wx,LsÝÝij¨m A i, jq ,¨ jLm A j i,jj

XX yÝwY,uiAixLXmX Ž.6 i

X X X for every X , Y g L with ␺-decompositions

X X X X X X ␺ Ž.X s ÝÝuiAm X ig A m AL, ␺ Ž.Y s ¨ jAjm Y g A m AL. ij Ž.7

The fact that the right side ofŽ. 6 is well defined follows from the next two lemmas.Ž Compare withwx 2, Lemma 1.4 .. Ž XX.Ž. LEMMA 4.1. Consider A , L ªŽ␸, ␺ . A, L a morphism of m.w.a.’s, whereŽ. A, Lisap.i.m., and

X X X X X X g L , ␺ Ž.X s ÝuiAm X ig A m AL. i Then there is a morphism of AX-modules

X X ⌽: A mAAL ª A m L X X such that for e¨ery a g A and Y g Lweha¨e

XXX XX X ⌽Ž.Ž.amAiAiAYsÝau m wxwX,Y Lq X ,axLmY. i

The proof of this lemma can be made as in Lemma 3.1 using the properties of brackets and tensor products. MODULES WITH DIFFERENTIALS 57

Ž XX.Ž. LEMMA 4.2. Consider a morphism of m.w.a. A , L ªŽ␸, ␺ . A, L , where Ž.A,Lisap.i.m., and the map

X X X ␹ : L = L ª A mA L Ž.8

XX XX XX ␹XY u XY X X Y Ž.,sÝÝ Ž.ij¨m A i, j q ,¨jLm A j i,jjL XX yÝwY,uiAixLXmX i

XX X for arbitrary X , Y g L with ␺-decompositions gi¨en by Ž.7.Then ␹ is well defined, antisymmetric, and k-bilinear. Using the two lemmas above and a direct computation the following can be proved XX PROPOSITION 4.1. ConsiderŽ A , L.Ž. and A, L some p.i.m.’s and Ž XX.Ž. A,LªŽ␸, ␺ . A,L a morphism of m.w.a.’s. Then the map

X X X K : L = L ª A mA L XX XX XX XX X KXŽ.,Y s␺Ž.wX,YxLXy␹Ž.Ž.X,Y , ᭙X,YgL

2XX XŽ .Ž. belongs to Hom AAL, A m L , where ␹ is gi¨en by 8. As inwx 7 or wx 1 we say that K is the curvature operator; it associates with 2 XX XŽ . a morphism of m.w.a.’s of two p.i.m.’s an element of Hom AAL, A m L . This vanishes iff the morphism of m.w.a.’s is a morphism of p.i.m.’s. In the X X Ž. case where A s A and ␸ s idA , we can use the isomorphism 2 to obtain the curvature operator used inwx 7 :

XX XX XX X KXŽ.Ž.Ž.,Y s␺X ,␺Y Ly␺Ž.wxX,Y L.

For a p.i.m. Ž.A, L consider the morphism of m.w.a.’s

Ž.A, L L Ž.A, DerŽ.A ªŽidA, p . and denote by D the curvature associated with p L. Explicitly

LL DŽ.X,YspX Ž.Ž.,pYDerŽ.A yp(LXŽ.wx,YL.

According towx 7 we give the following

DEFINITION 4.4. A p.i.m. Ž.A, L is an infinitesimal module Ž.i.m. if D s 0. 58 PAUL POPESCU

This is inwx 1 ‘‘un pre-espace´´ d’Elie Cartan regulier, sans courbure.’’ If Ž.A,Lis a p.i.m. then we call the map

J: L = L = L ª L,

JŽ.X,Y,ZsX,wxY,ZLLLLLLqY, wZ,X xqZ, wX,Y x the Jacobi map.Itis k-linear in every argument and if Ž.A, L is an i.m., 3 Ž. then J g Hom A L, L . According towx 7 we give the following

DEFINITION 4.5. An i.m. Ž.A, L is a Lie pseudoalgebraŽ. L.p.a. if J s 0. Now we can state:

THEOREM 4.1. The p.i.m.’s and their morphisms are the objects and the morphisms of a category P I MA. It contains as a full subcategory the category I MA of infinitesimal modules which contains in its turn as a full category the category L P AA of Lie pseudoalgebras.

DEFINITION 4.6. We call one of the categories M W AA , P I MA , I MA , or L P AA a category of modules with differentials. Notice that as in the case of the categories of m.w.a.’s we can define

ⅷ Ž. Subcategories of P I MA I MA , L P AA , respectively of the form XXXXXXŽ. PIMA or P I MA I MA or I MA , L P AA or L P AA , respectively , where AX is a fixed algebra of A and A X is a subcategory of A; the categories X corresponding to a fixed A are the categories considered inwx 7 .

ⅷ Strict categories of p.i.m.’sŽ. i.m.’s, L.p.a.’s, respectively defined by Ž. functors F: C0 ª P I MA I MA , L P AA , respectively can be defined using the same definitions as for strict categories of modules or m.w.a.’s. We call each of the categories a strict category of modules with differentials on C0. Examples of such categories are given below, as in Example 3.1:

EXAMPLE 4.1. We begin with some definitions fromwx 6 . If Ž.␪, D is a v.b.w.a. of M ŽŽ␪ s R, t, M ., D: R ª ␶ M .then a bracket Žan almost Lie map inwx 6.Ž.Ž.Ž. is a map L: ⌫ ␪ = ⌫ ␪ ª ⌫ ␪ provided that it is ޒ-bilin- eare, antisymmetric, and LXŽ.Ž.Ž.Ž.,fY s DX f Y q fL X, Y . According to wx6 we say that Ž.␪, D, L is an almost Lie structure Ž. a.L.s. on M. The definition of a morphism of a.L.s.’s is given inwx 6 using local coordinates and is not entirely clear; thus we prefer here to extend equivalentlyw 2, Definition 1.3x from Lie algebroids to a.L.s.’s: MODULES WITH DIFFERENTIALS 59

If Ž␪ XXX, D , L .Žand ␪, D, L .are two a.L.s.’s on manifolds MXand M, then a morphism of them is a morphism Ž.f, g of v.b.w.a.’s provided that

X X X X X X X X p( L Ž.X , Y s ÝÝuij¨ Ž.LXŽ. i,Y j(f q DŽ.Ž.X ¨jŽ.Y j(f i,jj

XXX yÝDŽ.Ž.ŽYuiiX(f . i

XX X for every X , Y g ⌫Ž␪ .Žwith g-decompositions as inwx 2. :

X X X X U g ( X s ÝÝuiiŽ.X ( f , g (Y s ¨ jjŽ.X ( f g ⌫Ž.g ␪ . ij

Using the same arguments as inwx 2, Lemma 1.4 it can be shown that this definition does not depend on the g-decompositions. The analogy with Lie algebroids fromwx 2 goes on to the fact that the a.L.s.’s and their morphisms are the objects and the morphisms of a category, denoted AL as inwx 5 . For an a.L.s. Ž.␪, D, L we can consider the F Ž.M -bilinear map

D: ⌫Ž.␪ = ⌫ Ž.␪ ª X ŽM . DŽ.X,YsDX Ž.Ž.,DY yDLXŽ. Ž.,Y and the Jacobi map

J: ⌫Ž.␪ = ⌫ Ž.␪ = ⌫ Ž.␪ ª ⌫ Ž.␪ JŽ.X,Y,ZsLXŽ.Ž.Ž.,LY Ž.,Z qLY,LZ Ž.,X qLZ,LX Ž.,Y .

If D s 0 we say that the a.L.s. is a prealgebroidŽ. p.a. . If D s 0 and Js0 the a.L.s. is a Lie algebroidŽ.Ž L.a.wx 7 , wx 4 , wx 2 , wx 3 , etc.. ; inwx 6 it is called a holonomic structure. It is easy to see that there is an prealgebroid full subcategory P A of AL. The category L A, of Lie algebroids, constructed inwx 2 , is in its turn a full subcategory of P A. Now we define the functors: F2 : AL ª P I MA , F3: P A ª I MA , F3: L A ª L P AA , where k s ޒ, as the functors F01and F were defined in Examples 2.1 and 3.1. They define strict categories of modules with differentials on the categories AL , P A, and L A respectively: AL P I MA , P AI MA , and L AL P AA.

5. DIRECT PRODUCTS

Let F: C0 ª A be a contravariant functor, where A is as defined in Section 2. We call a contra¨ariant category of algebras a category denoted as C0 A, whose objects and morphisms are, as in the case of category of 60 PAUL POPESCU

modules, couples ŽM, FM Ž ..and Žf, Ff Ž .., respectively, where M is an

object and f is a morphism from C0. It is clear that a strict category of modules or of modules with differentials defines a category of algebras considering a forgetful functor, which takes values only in A.

DEFINITION 5.1. Consider, in a category of algebras C0 A, two objects Ž. Ž. M␣ ,A␣␣,AsFM ␣,␣s1, 2, such that there is their direct product Ž. Ž. M,A,AsFM, in the same category. Denote as ␸␣ : A␣ ª A the morphisms of algebras that correspond to the direct product. We say that the two objects have the partial differential property if there exists a morphism of A-modules

␸ : A DerŽ.A A DerŽ.A Der Ž.A Ž.mA121[ Ž.mA2ª

Žthe direct sum is one of A-modules and the tensor products are related to

␸␣ ., such that the following properties are satisfied: Ž.PD1 ␸ŽŽ1 X .0, .␸ Ža . 0, Ž.᭙ X Der ŽA ., a w mA1 [ 22Derx ŽA.s g 12g A2; Ž.PD2 ␸ŽŽ0 1 Y ..Ž., ␸ a 0, Ž.᭙ Y Der ŽA ., a A ; w[ mA2 11DerxŽA.s g 21g 1 Ž.ŽŽ . . Ž . Ž .Ž. PD3 w␸ 1 mA X [ 0,␸11Dera x ŽA.s␸11DerwxX,a ŽA., ᭙ Xg Ž. 1 1 Der A11, a g A 1; Ž.ŽŽ ..Ž. Ž .Ž. PD4 w␸ 0 [ 1 mA X , ␸22Dera xŽA.s ␸22DerwxY, a ŽA., ᭙ Y g Ž. 1 2 Der A22, a g A 2; Ž.PD5 From the universal property of a direct product it follows that Ž. Ž. Ž . for every object M, B , B s FM in C0 A and morphisms g␣␣, ␺ , Ž. Ž . Ž . ␺␣ sFg␣␣, from M , A␣in M, B , ␣ s 1, 2, there is a morphism of k-algebras ␺ : A ª B such that the diagram

␸␣

AA6 6␣ 6 ␺␣ ␺ B

commutes in A. Then, given a ␺-derivation ⌽: A ª B Ži.e., a k-linear map ⌽:AªBprovided that ⌽Ža и b .s ␺ Ž.a и⌽ Ž.b q⌽ Ž.a и␺ Ž.Ž.b ᭙ a,bg .ŽŽ.. Awhich has the property ⌽ ␸␣ a␣␣s 0 for every a g A␣, ␣ s 1, 2, it follows that ⌽ s 0.Ž We denote the A-module of ␺-derivations as Der␺ Ž..A, B . Remark 5.1. The conditionŽ. PD5 means that every ␺-derivation ⌽ is

uniquely known from its actions on ␸11Ž.A and ␸ 2 Ž.A 2. Particularly, Ž. taking B s A and ␺␣ s ␸ it follows that every derivation from Der A is uniquely known from its action on these subalgebras of A. The conditions MODULES WITH DIFFERENTIALS 61

Ž.Ž.PD1 ᎐ PD4 show the action of ␸ on the generators of the A-module ŽA Der ŽA .. ŽA Der ŽA .. The conditionŽ. PD5 gives the mA121 [ mA 2 uniqueness of this action.

EXAMPLE 5.1. Take as C0 the category Mann of smooth manifolds and the contravariant functor F: Mann ª AR which associate with every manifold M the ޒ-algebra of functions FŽ.M and to every smooth map of U U manifolds f: M ª N the ޒ-algebras morphism f : FNŽ.ªFM Ž.,f Ž.␸ Ž. s␸(fsee Section 2 . Consider the category of algebras Mann AR defined by this functor. The direct product of two objects ŽŽ..M11, F M and ŽM22, F ŽM ..is Ž M 1212= M , FŽM = M .. and the canonical projections are Ž. Ž. ␸␣ sFp␣ . It is well known that the F M -module of derivations on real functions FŽ.M of a smooth manifold M is the FŽ.M -module of vector fields X Ž.M . We can take ␸ as the canonical isomorphism

␸ : X Ž.Ž.Ž.Ž.M = M F M = M X M F M = M 12ª Ž. 12mFŽM1. 212[ Ž XŽ.Ž.M.9 mFŽM2. 1.

It is routine to prove that every two objects of the category of algebras

Mann AR have the partial differential property. For checkingŽ. PD5 it suffices to use the following obvious lemma:

LEMMA 5.1. Let u: M ª N be a smooth map of smooth manifolds, and let ⌽: FŽ.N ª F Ž.Mbeanޒ-linear map which has the property

⌽Ž.f и g s ⌽ Ž.Ž.Ž.Ž.f и g (u q f (u и⌽ g .

Then, in a coordinate system around e¨ery point of N, the relation

n Ѩ f i ⌽Ž.f s Ý⌽ Žx . u Ž.10 Ѩxi( is1

i is ¨alid, where x are the coordinate functions.

Returning to the general case, consider a category of algebras C0 A defined by a contravariant functor F: C0 ª A. We can consider a category which has as objects and morphisms triplets ŽŽ..ŽŽ..M, FM,Land f, Ff,␺, where M is an object and f is a morphism of C0 such that ŽŽFM .,L .is an object and ŽŽ.Ff,␺ .is a morphism of m.w.a.Ž p.i.m., i.m., or L.p.a., respectively.Ž . We call such categories wide categories of m.w.a. p.i.m., i.m., . 0 Ž 00 or L.p.a., respectively , denoted as C0 M W AA or C0P I MA, C0I MA,or 0 . C0 LPAA , respectively and called together wide categories of differential modules. 62 PAUL POPESCU

0 PROPOSITION 5.1. Let C0 M W AA be a wide category of m.w.a.’s defined by a contra¨ariant functor F: C0 ª A. Suppose that there exists the partial differential property of the objectsŽ M11, F Ž M .. and Ž M 22, FM Ž ...Then in the 0 category C0 M W AA there exists the direct product of two objects of the form ŽŽ..ŽŽ..M111,FM ,L and M 222, FM ,L . Ž.Ž.Ž. Proof. Consider L s A mA L1 [ A mA L2 , where M, A , A s Ž. 12Ž. FM, is the direct product of the objects M␣ , A␣ , ␣ s 1, 2; consider Ž. also the morphisms of k-algebras ␸␣ : A␣␣␣ª A, ␸ s Ff , and ␸ given L by the conditionŽ. PD5 from Definition 5.1. We define p : L ª DerŽ.A using the formula

L L1 L 2 ␳ Ž.X s ␸ ÝÝaiAm p Ž.X i [ b␣mAp Ž.Y␣ , ž/ž/12ž/ i ␣ where X Ý Ž.Ž.a X Ý b Y L. It is easy to see that Ž.A, L s iiAm 12 i[ ␣␣mA ␣g Ž. 0 becomes an m.w.a. and L˜s M, A, L is an object in C0 M W AA. Using the canonical projections p : L A L then Ž.␸ , p are morphisms ␣ ª mA␣␣␣␣ Ž. of m.w.a. and thus ˜p␣ s f␣␣␣, ␸ , p , ␣ s 1, 2, are morphisms in the 0 category C0 M W AA. These morphisms are those of a direct product. Indeed, consider another object N˜s ŽŽ.N, FN sB,P .in the category 0 Ž. C0 MWAA and, in the same category, the morphisms m˜ ␣␣␣␣s f , ␸ , ␺ , Ž. ␸␣ sFf␣␣,␣s1, 2, from N˜˜to M . Then there is a morphism ␩˜ s Ž.˜˜ Ž. ˜ f,␸˜˜,␺,␸sFf from N to M such that m˜˜˜␣s p␣(␩. This can be constructed as follows: let Ž.f, ␸˜ be constructed from the definition of the Ž.Ž. Ž. direct product of M11, A and M 2, A 2and ␺˜˜be defined as ␺ Z s Ž. Ž. ␺12Z[␺Z. For Z g P,

␺˜Ž.ZaXbYBLBL. sÝÝŽ.Ž.Ž.Ž.iAm1212 i[␣mA ␣gmA 1[mA 2 i␣ Ž.11

We define ⌽: A ª B using the formula

⌽Ž.a Z, ␸ Ž.a a ␸␸1 X 0,a s ˜˜Py ÝiAž/Ž.Ž.m1i[ Der A i

b ␸␸0 1 Y ,a . y Ý␣˜ž/Ž.[Ž.mA2␣Der A ␣

The propertiesŽ.Ž. PD1 ᎐ PD5 ensure that ⌽ is a null ␸˜-derivation; thus Ž.␸˜,␺˜ is a morphism of m.w.a. MODULES WITH DIFFERENTIALS 63

Using the above proof we can state

PROPOSITION 5.2. Let C0 M W AA be a strict category of m.w.a. defined by X a contra¨ariant functor F: C0 ª M W AA. Consider F : C0 ª A the forgetful Ž.Ž . functor and the corresponding category of algebras. Let F M␣ s A␣␣, L , Ž.Ž. ␣s1, 2 be two m.w.a.’s such that M11, A and M 2, Aha 2 ¨e the partial differential property in the category of algebras defined by FX and denote as Ž.M,A the direct product of these objects. Denote as F Ž.Ž. M s A, L the direct product of the objectsŽ M11, F Ž M .. and Ž M 22, F Ž M .. in the strict Ž. category C0 M W AA. Then M, A, L is the direct product of the objects Ž.Ž. 0 M111,A,L and M 2, A 22, L in the wide category C0M W AAiff

LAL AL. sŽ.Ž.mA121[mA2

With the notations from Proposition 5.2, we say that ŽŽ..M, FM s ŽŽM,A,L ..Žand M, A, L .are equal, as triplets.

PROPOSITION 5.3. The categories of AL S, AL G, and AL admit finite products. The proof of Proposition 5.3 is almost the same as that of the first part ofwx 2, Proposition 1.10 . Taking into account the isomorphismŽ. 9 it follows 0 COROLLARY 5.1. The categories RTS M W AA and RTS M W AA admit finite products which are equal, as triplets. Propositions 5.1 and 5.2 can be extended for strict and wide categories of p.i.m.s, i.m.’s, and L.p.a.’s; Corollary 5.1 has an analogous extension for a.L.s.’s, p.a.’s, and L.a.’s:

PROPOSITION 5.4. Substituting in Proposition 5.1 the wide category 0 0 0 0 C0 M W AA with one of the wide categories C0 P I MA , C0 I MA , or C0 L P AA , the conclusion is also ¨alid. Proof. Using the proof of Proposition 5.1 it remains to define only the bracket on L, using the brackets on L12and L . It is, for

XaXbY sÝÝŽ.Ž.iAm12 i[␣mA ␣ i␣ and

X X aX X X bX Y X L, s ÝÝŽ.Ž.jAm 12 j[ ␤mA ␤g j␤ 64 PAUL POPESCU defined as

XXX XX wxX,XLsÝÝaaijm AX i,X jLq X,a jm AX j ž111L i,jj

yÝY,aiAiLmX wx1 / i

XX XX [ÝÝaa␣␤mAY␣␤,Y q X,b␤ mAY␤ ž22L2L ␣,␤␤

yÝY,b␣LmAY␣.12Ž. wx2 / ␣

By a straightforward computation, usingŽ.Ž. PD1 ᎐ PD5 and maps Žp11, ␸ ., Ž.p22,␸and ␺ from the proof of Proposition 5.1, it follows that these maps are morphisms of p.i.m.Ž. i.m., L.p.a., respectively . In a similar way we can state

PROPOSITION 5.5. Substituting in Proposition 5.2 the category M W AA with one of the categories P I MA , I MA , or L P AA , the conclusion is also ¨alid.

COROLLARY 5.2. Substituting in Corollary 5.1 the category M W AA with one of the categories P I MA , I MA , and L P AA and the category RTS with one of the categories AL , P A, or L A, respecti¨ely, the conclusion is also ¨alid.

6. THE PULLBACKS

In this section we study pullbacks of modules and vector bundles with differentials. This extends the inductor construction ofwx 8 and wx 2 , from an inverse image of a Lie algebroid over a smooth map to some pullback diagrams of modules and vector bundles with differentials. It is proved that the pullback object constructed are fibred products always in the vector bundle case and under some supplementary conditions in the module case. Let

X X YY ŽA , L .ªŽ␸XX,␺.Ž Ž.Ž.A,L,A,Lª␸YY,␺. Ž.A,L Ž.13

Ž. XY be morphisms in MA modules and let ␸: A ª A be a morphism in A YX Ž.Ž.algebras as in Section 2 such that ␸ s ␸ (␸ . Consider the morphism MODULES WITH DIFFERENTIALS 65 of AY-modules obtained from the composition

Y X Y X Y A mAAAAX L ª A m X Ž.A m L ª A m L

YXYX YX aXXaXaXa␸Ž.aX, mAAiªmž/ÝÝmAiª imAi ii XŽX.X ŽX. where ␺ X s Ýiia m AX iand denote it by 1 m ␸, ␺ . Consider the AX-module

U Y Y X Y Y X L X Ž.Ž.a X X L A X L s ½[ ÝiAm ig [ mA r i

Y Y X Y X ␺Ž.X Ž.1 Ž␸,␺ . Ža XX . sm ž/ÝiAm i 5 i

Ž.where the direct sum is in the category of A-modules and the canonical projections

Y U Y U Y X a : L ª L , ␹ : L ª A mA L , which induce the morphisms of modules:

Y U Y Y Y U X X Ž.A , L Y Ž.Ž.Ž.Ž.A , L , A , L A , L .14 ªŽidAY, a .Žª ␸,␹.

Ž. DEFINITION 6.1. Consider two morphisms 13 of modules in MA such X Y Y X that there is a morphism ␸: A ª A of algebras in A with ␸ s ␸ (␸ . We say that the module Ž AY, LU . constructed above is the pullback module of the morphisms, related to ␸.

PROPOSITION 6.1. Let two morphisms Ž.13 be in a category of modules with differentials and consider their pullbackŽ AY, LU . in the category of X Y modules MA , related to ␸: A ª A . If ␸X is a surjection, then there is a natural structure of module with differential, of the same kind, onŽ AY, LU .Ž. such that 14 are morphisms of modules with differentials. Proof. First, we prove the assertion in the case when the objects and Ž. the morphisms in 13 are in the category M W AA. LU U Y Define ␳ : L ª DerŽ A . as

def LLU YYXY Y YYXU ␳XŽ.Ž.Ž.Ž.aXX␳X,᭙XaXXL ž/[ÝÝiAm i s[iAm ig ii Ž.15 66 PAUL POPESCU

U so Ž LU, ␳ L .Žis a m.w.a. and the first arrow in 14. is a morphism of m.w.a. To check that the second arrow inŽ. 14 is a morphism of m.w.a., we use the surjectivity of ␸X: Consider

U Y Y X U Y Y Y X X s X [ ÝÝŽ.aiAm X X ig L , ␺ Ž.Ž.X s aiAm X X i, ii XXXX ␺Ž.XiisÝ Žb␣mAX␣ .gAmAL ␣ and

X X X X t g A , t s ␸ Ž.t , t g A.

Y YX Ž .ŽX. We have ␹ X s ÝiiAa m X iand

UXYXYY X,␸Ž.tLLLUsX,␸Ž.tYYsX,␸Ž.t YXY sÝaii␸Ž.Žb␣␣␸X,t .L i,␣

YXX YUXX a␸b␸X,t aX,␸Ž.tX sÝÝiiž/␣␣wxLs ÝiiŽ.L ii␣ YXX sÝaii␸Ž.wX,txLX, i which proves that Ž.␸, ␹ is a morphism of m.w.a. In the case of preinfinitesimal modules, we define the bracket on LU as

YYXYYX X[ÝÝŽ.aiAmXXX i,Y[Ž.bjAmY j ijLU

def YY XX XX YY X X,YY ab XXX,Y XYX ,b Y swxL[žÝÝijm A i jLLq jm A j i,jj

YY X Y,aY XX. yÝwiAixLm / i

It is easy to check that the bracket is well defined; this means that Y YY X Ž Y .ŽŽ.. ␺ w X ,Y xL is the same thing as 1 m ␸, ␺ to the right side of the direct sum above. It remains to prove the assertions for infinitesimal modules and Lie pseudoalgebra. To do this we prove a lemma, which improves the conditions under morphismsŽ. 14 . MODULES WITH DIFFERENTIALS 67

LEMMA 6.1. Let two morphisms Ž.14 be in the category P I M, such that Ž Y U . there exists the pullback A , L in the module category MA and suppose that

LLYYYY LYYY Y pXŽ.,pY Ž.DerŽ.AYspXŽ.wx,YL

Y Y Y U for eëry X , Y g L which are inölëd as components of elements from L ŽŽin particular it is always true if AYY, L.. is an infinitesimal module . Then Ž AY, LU . is an infinitesimal module. Y YYY In the same hypothesis as aboë, supposing moreoër that JŽ X , Y , Z . Y Y Y Y s 0 for eëry X , Y , Z g L which are inölëd as components of elements from LU ŽŽ in particular it is always true if AYY, L.. is a Lie pseudoalgebra , thenŽ AY, LU . is a Lie pseudoalgebra. Proof. We can use the following identities, which are valid for mor- phismsŽ. 14 of preinfinitesimal modules:

LLUUUU LUUU U pXŽ.,pY Ž.DerŽ.AYypXŽ.wx,YL

LLYYYY LYYY Y spXŽ.,pY Ž.DerŽ.AYypXŽ.wx,YLŽ.16

Ž.with the notations used in the proof of Proposition 6.1 ; taking, moreover, U Y YXU Ž X. ZsZ[ÝkkAcmZ kgL, we have

U U U U Y Y Y Y YY X X X X JŽ.Ž.X,Y,ZsJX,Y,Z [ÝŽ.Žaijkbc m AX J X i,Y j,Z k .. i,j,k

DEFINITION 6.2. For two morphismsŽ. 13 in a category of modules with differentials with ␸X a surjection, we call the structure of modules with differentials induced by Proposition 6.1 on the pullback module related to X ␸: A ª A the pullback of modules with differentials related to ␸. From Proposition 6.1 we can state:

COROLLARY 6.1. Let two morphisms Ž.13 be in a category of modules X with differentials. Then their pullback related to ␸: A ª A exists in that category iff it exists in the category M W AA. We make now a Lie pseudoalgebra construction which corresponds to the Lie algebroid pullback construction fromwx 8 and wx 2 . Our construction works as well for prealgebroids. Consider the full category DerŽ.A of L P A which has as objects ŽA, Der ŽA ..with A an algebra from A. Ž Analogously, a full subcategory of L P A, denoted as DerŽ AX . can be associated with a subcategory AX of A.. Unfortunately, this is not a functorial association, since there is no Lie 68 PAUL POPESCU pseudoalgebra morphism from DerŽ AX .Ž.in Der A to correspond to an X algebra morphism ␸: A ª A . Let Ž.A, L be a Lie pseudoalgebra and X X Ž Ž ..U Ž Ž .. Ž.Ž. A , Der A ªŽ␸, ␸ . A, Der A be a morphism in Der A . Then A, L L L ŽA, Der ŽA .. Žwhere ␳ is the anchor of L. is a morphism of Lie ªŽidA, ␳ . pseudoalgebras. From Proposition 6.1 it follows that there exists the Lie pseudoalgebra pullback Ž AY, LU .. Let us return to the general case. It is routine to prove the following

PROPOSITION 6.2. If the diagram

Ž.YY ␸,␺6 YY 11 XX Ž.A,L1 Ž.A,L Ž.17 Y

Ž.id Y, ␺ XX

A 0 Ž.␸,␺ 6 YY 6

Ž.␸,␺6 Ž.AYY,L Ž.A,L is commutatië in the category P I M and the morphisms Ž.13 haëa Ž YY. pullback, then there is a unique morphism of p.i.m.’sA,L1ªŽid Y,␺. Y U A ŽA , L. that makes commutatië the diagrams

YY Ž.A,L1

Y YY

Ž.idAY, ␺ 0 Ž.␸11, ␺

ŽidAY, ␺ . 6

Ž. 6 6 6 idAЉ , aЉ Ž.␸ , ␹

Ž.AЉ , LЉ 6 Ž.AЉ , L* Ž.AЈ, LЈ in the category P I M.

THEOREM 6.1. In a category of modules with differentials, if there exist the X Y fibred product and the pullback related to a ␸: A ª A of two morphisms Ž.13 and the fibred product is an AY-module, then the fibred product and the pullback are isomorphic. Proof. Using Proposition 6.2 and the universal property of fibred product, the assertion follows. It is obvious that in every wide category of differential modules the pullback of two morphisms

XXX YYY

Ž f,␸,␺.Ž6 f,␸,␺.6 ŽMXXX,A,LM . Ž.Ž.,A,L,MYYY,A,LM Ž.,A,L Ž.18

X X Y Y can be defined, where A s FMŽ.., AsFMŽ ., AsFMŽ ., and X F: C0 ª A, provided that there exists ␸ and ␸ is a surjection. This pullback is Ž MY, Ž AY, LU ... In a strict category of differential modules MODULES WITH DIFFERENTIALS 69 defined by a functor F, the pullback of two morphisms of the formŽ. 18 as XX X YY Y triplets, where Ž.Ž..ŽA, L s FM , A,L.ŽsFM.Ž, and A , L .Žs FM., U Ž U. can be defined only if there is an object M of C0 such that FM s Y U XY X Y Ž . Ž.ŽY . A,Land the morphisms g and g in C0 such that Fg sidA , a , Y FgŽ .Žs␸,␹ .. We study now the case in which the pullback is defined for categories of vector bundles with differentials Ž.RTS , AL , P A , L A . This is inspired bywx 2 , where the Pradines inductor for Lie algebroids is studied, working X XX X Y as well for prealgebroids. Let ␰ s Ž.E, p, B , ␰ s ŽE , p , B ., and ␰ s YY Y YX X YY Ž E,p,B.be vector bundles, ␸˜˜: B ª B , ␸: B ª B, and ␸ ˜: B ª B YX XX X be smooth maps such that ␸˜˜˜s ␸ (␸; let ␺˜: E ª E be a ␸˜-morphism XY Y X ˜ Ž XX and ␺ : E ª E be a ␸˜ -morphism of vector bundles we write ␰ ªŽ␸ , ␺ . ␰ Y YY . and ␰ ªŽ␸ , ␺ . ␰ . Consider the set

U XY X Y XX YY EsÄ4Ž.e,egE=E:␺˜ Ž.Ž.es␺˜e

Y U X U Y and let b : E ª E and ␹ : E ª E be the canonical projections. DEFINITION 6.3. We say that the diagram

Ž.␸˜,␹6 ␰␰U X Ž.19

Y XX

Ž.idBY, b Ž.␸˜,␺˜

6 6 YY

Ž.␸˜,␺˜6 ␰Y ␰

U is a ¨ector bundle pullback induction provided that ␰ is a vector bundle pullback and ␸˜X is an injective immersion; i.e., U UU Y U Y Ž.PB1 ␰ s ŽE , p , B ., where p s p ( ␹ is a vector bundle; Ž.PB2 ␳˜XYis an injective immersion and b is a morphism of vector bundles over BY ; UU UX Ž.PB3 ␹ is a ␸˜-morphism of vector bundles; denote by ␹ : ␰ ª ␸␰ the morphism of vector bundles over BY induced by ␹. The diagramŽ. 19 will be used in the category of vector bundles with differentials and in this case it extends the Pradines inductorwx 2

␹ 6 AAY Ž.20

aX a

6 6

␶f 6 TBY TB 70 PAUL POPESCU

YY where Ž A, a, wxи, и .is a Lie algebroid, f: B ª B is a smooth map, and A is required to be a vector bundle pullback. As inwx 2 it is necessary to think of ␰ U also as a pullback

␹ 6 ␰␸␰UU˜X

b Ž.Ž.␸␺˜XXU˜ 6 YU 6

Ž.␺˜ 6 ␰Y Ž.␸␰˜YU of vector bundles over BY and thus the sections of ␰ U have the form YYYUX X[C, where X g ⌫Ž␰ .Žand C g ⌫ ␸␰˜ .so that

YYU XX Ž.␺˜˜Ž.Xs␸␺˜ Ž. Ž.C.21Ž.

Taking into account the arguments and the notations used inwx 2 we have

Y X C s ÝuiiŽ.X (␸˜ i or Y X C s Ý ui mFŽ BX . Xi i Y Ž Y . X Ž X . for uiig F B , X g ⌫ ␰ . We can transcribe these relations using the Ž. Ž.˜ functor F0: V B ª MA defined in Example 2.1: Denote ␸, ␺ s F0 ␸˜, ␺ , Ž.Ž.Ž.Ž.XX XXYY YY ␸,␺sF00␸˜˜,␺˜,␸,␺sF␸,␺˜. With the notations used be- fore in this section, the relationŽ. 21 becomes

Y Y X Y X ␺ Ž.X Ž.1 Ž␸, ␺ . u X X s m ž/ÝimFŽB.i i or Y Y Y X ␺ Ž.X s Ýuii␸ Ž.¨ ␣mFŽB.X␣, i,␣ XŽX. X where ␺ Xi s Ý␣ï␣ mFŽ B. X␣ . In the sequel we consider the vector bundles with differentials. X Y THEOREM 6.2. Let ␰ , ␰ , and ␰ be ëctor bundles with differentials XX YY of the same kind ŽŽ␰ , a ., Ž ␰ , a .Ž, ␰ , afor. ¨.b.w.a.or Ž.Ž␰,a,L,␰XX,a,L X.Ž,␰YY,a,L Y.. for a.L.s., p.a., and L.a. and suppose that diagram Ž.19 is a ëctor bundle pullback induction and the morphisms X XYY Ž.Ž␸˜˜,␺˜ and ␸ , ␺˜ .are of ëctor bundles with differentials. Then there is a UUU canonical structure of ëctor bundles with differentials on ␰␰ŽŽ ,a. for UU U ¨.b.w.a.or Ž ␰ , a , L.. for a.L.s., p.a., and L.a. such that the morphisms MODULES WITH DIFFERENTIALS 71 in diagram Ž.19 are of ëctor bundles with differentials, and this structure on U ␰ is a fibred product in the category of ëctor bundles with differentials considered Ž.RTS , AL , P A, or L A, respectiëly . YY UU UY YYŽ . Proof. For b g B , a vector Xbbg E from the fibre of E in b can UYX YYXXXXXX YX Ž. be written as X s X [ X , where X g Ebb, X g E , b g B , ␸˜ b s YY YU YXUXX YY XX ␸˜˜Ž.band Ž.Ž.Ž.Ž.Ž.␺˜X s ␸␺˜X. Thus a(␺˜ã(␺ Ž.X s a(␺˜ Ž.X YYY and using the properties of v.b.w.a. morphisms it follows that ␶␸˜(aXŽ. XXX YX YY s␶␸˜˜˜˜˜(aXŽ.; but ␸ s ␸ (␸ and ␸ is an immersion, thus ␶␸ ˜(aXŽ. XX UUUUYY saXŽ.or, defining an anchor a on E as aXŽ.saXŽ., we have U U X U ␶␸˜(aXŽ . sa(␹Ž X.Ž. The commutativity of diagram 19. follows in the category RTS. In the case of the categories AL , P A, and L A we can define the Lie map of two sections in ⌫Ž ␰ U . as

U Y Y X Y Y X L X Ž.Ž.u X X , Y X Y ž/[ ÝÝimFŽB.ij[ ¨ mFŽB.j ij

def YYY sLXŽ.,Y

YY Y Y Y Y ÝÝŽ.uij¨mFŽBX.LXŽ.ij,Y q Ž.aŽ.Ž.X ¨ jmFŽBX.Yj i,jj . [YY Y Y yÝŽ.aYŽ.Ž. uimFŽBX.Xi 0i

It is well defined, as can be seen using the same arguments as for Lie algebroids inwx 2 . In the case of prealgebroids and Lie algebroids we can use the following identities valid for an a.L.s.: For

Y Y X Y Y X U s X [ ÝÝŽ.uimFŽBX .Xij, V s Y [ Ž.¨ mFŽBX .Yj ij

YYXU WsZ[ÝŽ.wkmFŽBX.Zkg⌫Ž.␪Ž.22 k we have

UU UU aUŽ.,aV Ž.yaLUŽ. Ž,V . Y Y YY YY YY EYYY saXŽ.Ž.Ž.,aY yaLXŽ.,Y Ž.sD Ž.X,Y U YYYY YYYXXXX JŽ.ŽU,V,WsJX,Y,Z .[ÝŽ.uij¨w kmFŽBX.JŽ.Xij,Y,Z k. i,j,k

It remains to show the property concerning the fibred product. For this Ž. Ž take a vector bundle ␰ 1111s E , p , M with a differential structure v.b.w.a., 72 PAUL POPESCU a.L.s., p.a., or L.a.. and a commutative diagram

Ž.X ˜X ␸˜11,␺6 ␰␰1 Ј

Ž.␸˜11,␺˜ Ž.␸˜Ј,␺˜Ј

6 6

Ž.␸˜Љ,␺˜Љ6 ␰Љ ␰ with morphisms in the differential category considered. It is easy to construct a vector bundle morphism Ž.␸˜00, ␺˜ which is as well a morphism in that differential category and makes the diagram

␰ 1 YY Ž.␸˜11,␺˜ XX

Ž.␸˜11,␺˜

Ž␸˜0 ␺˜0.

6 6 Ž. idMЉ , bЉ Ž.␸˜Ј, ␰ 6 X ␰ Љ 6 ␰ * ␰ commutative in the differential category considered. The fact that the pullback is a fibred product is a universal property that extendswx 2, Proposition 1.8 . We notice that the inductor defined in wx 2 is good for prealgebroids andwx 2, Proposition 1.8 holds without any modifica- tion. From the proof of Theorem 6.2 we can state

COROLLARY 6.2. Consider some a.L.s.’s as in Theorem 6.2 Y YY 1. Suppose that for eëry U and V as in Ž.22 we haë D ŽX , Y .s 0. Then a prealgebroid structure Ž ␰ UUU, D , L. is induced. Y YY 2. Suppose that for eëry U, V, WasinŽ.22 we haë D ŽX , Y .s 0, YYY YY Y X XXX Ž .ŽX Ž .. J X , Y , Z s 0 and Ýi, j, kijku ¨ w mFŽB.J Xij, Y , Z ks 0. Then a Lie algebroid structure Ž␰ UUU, D , L. is induced.

ACKNOWLEDGMENTS

The author thanks the referee, who suggested some new facts, especially in the final section.

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