This is in response to Andre Joyal's message of Jul 13, and in enlargement of Ross Street's of last week. I have been away for the last three weeks. As Ross already intimated, quite a lot of work has been done on the infinitesimal theory associated with Lie groupoids since Pradines' notes of the 1960s. I just add a few comments. (1) The object which Joyal describes:
It is a pair (A,D) where A is a commutative R-algebra and D is a Lie algebra (over R) such that
1) D is acting on A (as derivations): X(fg) = X(f)g + fX(g) , [X,Y](f) = X(Y(f)) - Y(X(f)) 2) D is equipped with an A-module structure such that (fX)(g) = fX(g) and [X,fY] = X(f)Y + f[X,Y] (for any f,g in A and X,Y in D)
was defined abstractly by Herz in 1953 and called a Lie pseudo-algebra. It has since been reinvented by a very large number of people, most of whom also invented a new name; there are some 14 different terminologies in the literature. Pradines was unique in defining the narrower concept of Lie algebroid: a vector bundle $A$ on base $M$ (the set of identities when the Lie algebroid comes from a Lie groupoid) together with a vector bundle morphism $a\colon A \to TM$ and a bracket $[\ ,\ ]$ of global sections of $A$ which makes the real vector space of global sections into a real Lie algebra and obeys $a[X,Y] = [aX,aY]$ and $[X,fY] = f[X,Y] + a(X)(f)Y$ for global sections $X,Y$ of $A$ and smooth functions $f$ on $M$. This is the object which matters for the Lie theory of Lie groupoids. Note that the partiality of the multiplication for groupoids gets transformed into the module structure for the global sections and the map $a$, rather than into a "partial bracket". It can be interesting to think of the associated Lie pseudo-algebra as an infinite-dimensional Lie algebra associated to the infinite-dimensional group of admissible sections (H in Joyal's notation). But so far as I know there is no sufficently developed general theory that one could then apply. Rather, the Lie groupoid structure enables one to do directly in this case (most of) what one wishes a general theory of infinite-dimensional Lie groups could do. (For a groupoid which comes from a principal bundle, the group H of admissible sections is the group of gauge transformations.) The Lie groupoid/Lie algebroid formalism allows one (amongst many other things) to pretend that any manifold is a Lie group, any (sufficiently smooth) equivalence relation arises from a smooth Lie group action, and so on. (2) Pradines' notes of the 1960s sketched a Lie theory for Lie groupoids and Lie algebroids, but gave very few details. In the 1987 book which Ross mentioned I gave a full account for locally trivial Lie groupoids and transitive Lie algebroids (= the map $a\colon A\to TM$ is surjective). In this case, one can use methods quite different from those which Pradines sketched. In the general case quite a lot is known but there is no systematic account and some things are still open. Since my book was written Weinstein and, independently, Karasev, have created an extensive theory of symplectic groupoids, and special techniques are available in this case also. Symplectic groupoids are very far from being locally trivial. (Terminology: In my book, a Lie groupoid is always taken to be locally trivial, and the general concept is called a differentiable groupoid.) (3) Functoriality can be achieved without any dualization. Philip Higgins and myself (J. Alg. 129, 1990, 194--230) developed a means of working with general morphisms of Lie algebroids. The basic categorical constructions with groupoids (equivalence between actions and action morphisms, semi-direct products, quotients, etc) then also hold for abstract Lie algebroids. It is a curious fact that, although there is a very natural definition of morphism for Lie pseudo-algebras, it does not correspond to the infinitesimal map induced by a morphism of Lie groupoids. This is another reason for distinguishing between Lie algebroids and Lie pseudo-algebras. The dual of a Lie algebroid has a natural Poisson structure, extending the linear Poisson structure on the dual of a Lie algebra (Courant, Trans AMS, 319, 1990, 631--661). One knows that a linear map between Lie algebras is a morphism iff its dual is a Poisson map. Extending this to Lie algebroids runs into the problem of how to dualize a base-changing morphism of vector bundles. Higgins and I found a way around this ("Duality for base-changing morphisms...", Math. Proc. Camb. Phil. Soc. to appear very shortly). (4)
It is easy to see that we have a contravariant functor
{Lie groupoids}--->{de Rham complexes}
and it follows from the work of Pradine that it has a left adjoint Exp (the exponential functor)
Exp:{de Rham complexes}--->{Lie groupoids}
when it is restricted to complexes (A,F,D) in which A is an algebra of smooth functions on a manifold.
The functor Exp is full and faithful. Its image should consist of some kind of simply connected Lie groupoids. I guess (I should really read Pradine ...) these groupoids are those for which the fibers of d0 are simply connected. ??
Not all Lie algebroids arise as the Lie algebroids of Lie groupoids, even in the transitive case. Counterexamples were given by Almeida and Molino (CRAS (Paris), 300, 1985, 13--15). For transitive Lie algebroids I gave a cohomological obstruction to integrability; this is a kind of nonabelian first Chern class. See Chapter 5 of the book already mentioned for the case where the base is simply-connected and Cahiers 28, 1987, 29--52, for the general case. A general framework for thinking of this obstruction is given in JPAA 58, 1989, 181--208. (5) There is also a Lie theory associated with double Lie groupoids. Here a double groupoid is a groupoid object in the category of groupoids: it is a double Lie groupoid if it has a smooth structure making all four groupoid structures Lie and such that the map which sends any square to (say) its right side and its bottom side is a surjective submersion. Double Lie groupoids arise by natural games with ordinary groupoids, and in homotopy theory, but also in the integration of Poisson Lie groups (Lu and Weinstein, CRAS (Paris), 309, 1989, 951--954) and Poisson groupoids. Taking the infinitesimal object associated to a double Lie groupoid is a two-step process; the first step yields an LA-groupoid, that is, a groupoid object in the category of Lie algebroids. These are of interest in themselves: the cotangent bundle of a Poisson Lie groupoid is an LA-groupoid, for example. See Adv. Math. 94, 1992, 180--239. QUESTION: What is the infinitesimal object corresponding to a (Lie) 2-category? Is there a Lie theory here also? Lastly, to revert to Jim Stasheff's original question, given all the above and given that there is also a theory of Lie semigroups (Hilgert-Hofmann-Lawson), it should be straightforward to define an infinitesimal object associated to a "Lie category". How far one could get with a Lie theory is another matter entirely. My impression of the semigroup theory is that the absence of inverses causes plenty of difficulties there already. Kirill Mackenzie +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++