I would like to comment on the considerations raised in the recent messages of Joyal, Street, Mackenzie, and Brown, but from a somewhat different perspective. Although I accept the fact that the notion of Lie algebroid is what matters for the Lie theory of Lie groupoids, I think there are a couple of reasons why the general concept of Lie pseudo-algebra should not be dismissed. The first is that examples arise from a variety of sources in algebra. A fundamental one comes from any k-algebra A (k a field, say) with commutative subalgebra K. Namely, let A+ be the "differential normalizer" of K in A: The set of all u in A with ux - xu in K for all x in K. Then (K,A+) is a Lie pseudo-algebra over k; moreover, for fixed K/k, the functor A ------> (K,A+) has a left adjoint (K,L) ------> U(K.L), the universal enveloping algebra of a Lie pseudo- algebra constructed by Rinehart [TAMS 108 (1962)]. (Actually, this is not quite the whole story, since in general one must consider algebra maps K ----> A that are not necessarily embeddings). Another important example arises from a Lie k-algebra L acting on a commutative k-algebra K by derivations: (K,K#L) is then a Lie pseudo-algebra, where K#L is the tensor product of K with L over k with a suitable twisted bracket operation. My other reason to keep Lie pseudo-algebras in mind is that they should form part of a larger theory of groupoid schemes and formal groupoids. Although some years ago I explored and used some special cases of these notions (see below), I am not aware of any systematic development of them in the literature. That is, with the exception of the appendix on groupoid schemes and Hopf algebroids in Ravenel's 1986 book on stable homotopy. ;I would be interested in hearing of other references. (In this regard I should also mention Huebschmann's paper on Poisson algebras [Crelle 408 (1990)], in w;hich Lie pseudo-algebras play an important role). As is suggested by the examples discussed above, Lie pseudo-algebras have a long history in the Galois theory and Brauer group literature. Jacobson's Galois correspondence for purely inseparable extensions of exponent one is between intermediate fields in such an extension and restricted Lie pseudo-subalgebras of (K,L), with L the restricted Lie pseudo-algebra of k-derivations of K [Amer. J. Math. 66 (1944)]. Hochschild classified Brauer classes of central simple k-algebras split by K (with K/k as above) in terms of extensions of restricted Lie pseudo-algebras [TAMS 76 (1954) and 79 (1955)]. In my paper [Amer. J. Math 98 (1976)] I generalized Jacobson's theorem to modular purely inseparable extensions of arbitrary exponent by a method that rests on the notion of the group scheme G# of admissible sections of a finite groupoid scheme. Although at the time it seemed clear to me that an analogous theory of Lie groupoids should exist, until 1988 I was unaware of the work that had been done in that area, and I am indebted to Ronnie Brown for informing me of it. It is not surprising that the group of admissible sections should play a role in Galois theory, since it provides a very natural definition of the automorphism group (or "group of symmetries") of a collection of objects, in a manner which takes into account not only the symnmetries of the individual objects but also their isomorphisms with each other. The literature described above also suggests the appropriate groupoid analogue of the cocommutative Hopf algebra that arises as the continuous dual of the algebra of representative functions of a formal group, or the Hopf-Sweedler dual of the algebra of representative functions of a group scheme. In my view, this analogue should be Sweedler's notion of a xK-bialgebra [Groups of simple algebras, Publ. IHES no. 44 (1975)] (although presumably with some sort of antipode). To oversimplify a bit, a xK-bialgebra is a k-algebra A which is also a K-coalgebra, with suitable axioms linking the two structures (the module of primitive elements of A then yields a Lie pseudo-algebra for K/k). These algebras are a ubiquitous as Lie pseudo-algebras: In addition to the universal enveloping algebras mentioned above, they include the classical smash product K#G (G a group acting on K by k-algebra automorphisms) and the algebra of differential operators on K. But logically (if not historically) the paradigm here is the k-algebra A = kC of a small category C, with K the k-algebra of k-valued functions on the set X of objects of C; the coalgebra structure on A is defined in a manner entirely analogous to that of a group algebra (it also satisfies a similar universal property). Finally, the grouplike elements of A constitute the monoid of admissible sections of C. Steve Chase +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++