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Reduced smooth stacks?

An arbitrary Lie groupoid gives rise to a groupoid of germs of local diffeomorphisms over its base manifold, known as its effect. The effect of any bundle of Lie groups is trivial. All quotients of a given Lie groupoid determine the same effect. It is natural to regard the effects of any two Morita equivalent Lie groupoids as being equivalent. In this paper we shall describe a systematic way of comparing the effects of different Lie groupoids. In particular, we shall rigorously define what it means for two arbitrary Lie groupoids to give rise to equivalent effects. For effective orbifold groupoids, the new notion of equivalence turns out to coincide with the traditional notion of Morita equivalence. Our analysis is relevant to the presentation theory of proper smooth stacks.


Introduction
The presentation theorem is a classical result in the theory of orbifolds which essentially dates back to the original papers on "V-manifolds" by Satake and others [23,12]. (A detailed exposition is given in [20].) In modern language [21,18], this theorem states that any effective orbifold groupoid 1 is Morita equivalent to the translation groupoid associated to some compact Lie group action on a smooth manifold; of course, the action in question will be effective and have discrete stabilizers. Thus, when regarded as a smooth (Deligne-Mumford) stack [17,15], any effective orbifold is isomorphic to a stack of the form [M/G], where G is a compact Lie group and M is a smooth manifold on which G operates smoothly and with discrete stabilizers. The importance of this result nowadays lies principally in the fact that it enables one to reduce the computation of many topological and cohomological invariants of orbifolds to a better understood special case [1]. A number of popular research topics in orbifold theory are, in a way or another, related to the presentation theorem. For instance, a longstanding conjecture affirms that any, say, connected, smooth orbifold stack (effective or not) is of the form [M/G], for G, M as above. (The reader is referred to [9] for an exhaustive discussion and partial results on this conjecture.) For various purposes, for example, for computing Chen-Ruan orbifold cohomology [5], classifying smooth symplectic resolutions of (possibly singular) affine Poisson varieties [8], or building models of conformal field theories on singular spaces [4], it is important to understand under what conditions an orbifold stack is isomorphic to a "global quotient", that is to say, to a stack of the form [V/G], where G is a finite group of smooth automorphisms of a non-singular manifold V. The presentation theorem itself proves to be a useful tool in the study of this kind of problem [1]. The presentation theorem also has application in the study of asymptotic spectral properties of elliptic operators on orbifolds [13]. The present paper originates from the author's endeavor to extend the presentation theorem beyond the scope of orbifold theory [27] with the intent of gaining a better geometric understanding of general proper smooth stacks and, possibly, laying the foundations for a classification theory of such objects along the lines of [19].
It turns out that presentation results are by no means special to orbifolds. It is a wellknown fact (for a proof of which we refer the reader to Section 5 of [25]) that any Lie groupoid which admits faithful representations is Morita equivalent to the translation groupoid associated to a smooth action of some Lie group on a smooth manifold; when the groupoid is proper, the Lie group can be taken to be compact. The presentation theorem for effective orbifolds is then simply a corollary of this fact and of the fact that any effective orbifold groupoid admits a canonical faithful representation on the tangent bundle of its base manifold. The theorem is thus substantially a result in the representation theory of Lie groupoids. Even though Lie groupoids normally do not admit faithful representations [26,11], in view of recent results obtained by the author [27] it seems plausible that the more "effectively" a given proper Lie groupoid acts on its own base the larger is the number of "interesting" representations the groupoid possesses. If we agree to call a Lie groupoid effective when the only isotropic arrows that have trivial infinitesimal effect (compare Section 1 below) are the identities, one may conjecture that any effective proper Lie groupoid admits faithful representations. Although at first sight this assertion might look like a good candidate for a generalized presentation theorem, it does not take long to realize that in the non-étale case the notion of effective groupoid which we have just introduced is so restrictive that the conjectured result, even if true, would be of little practical utility. [By way of example consider the translation groupoid Γ = SO(3) ⋉ R 3 ⇒ R 3 associated to the canonical action of the special orthogonal group SO(3) on three-dimensional euclidean space. Since this action is faithful, one would like to say that Γ acts effectively on its base R 3 . On the other hand, outside the origin all the isotropic arrows of Γ have trivial effect.] The approach we propose instead is the following.
Our starting point is the observation that an arbitrary orbifold groupoid can be pre-sented as an extension of some effective orbifold groupoid by a bundle of finite groups in a canonical fashion: in fact, the ineffective isotropic arrows of an arbitrary orbifold groupoid Γ form a bundle of finite groups K over the base manifold M of Γ, and the quotient groupoid Γ/K ⇒ M is an effective orbifold groupoid. (Compare e.g. [9, §2.2].) In general, for an arbitrary short exact sequence of Lie groupoid homomorphisms 1 → K ֒→ Γ ։ Γ ′ → 1 presenting a given Lie groupoid Γ ⇒ M as an extension of some other Lie groupoid Γ ′ ⇒ M by a bundle of Lie groups K, the kernel of the homomorphism Γ ։ Γ ′ consists of ineffective arrows. Heuristically speaking, we may express the circumstance that the two Lie groupoids Γ and Γ ′ induce the same "action" by germs of local diffeomorphisms on their base manifold M by saying that they give rise to the same "effective transversal geometry". Thus, we may liberally rephrase the classical presentation theorem for orbifolds by saying that an arbitrary orbifold groupoid gives rise to the same "effective transversal geometry" as the translation groupoid associated to some compact Lie group that acts on a smooth manifold effectively and with discrete stabilizers. Much of the above picture naturally generalizes from orbifold groupoids to arbitrary proper Lie groupoids, in the following manner. Suppose a proper Lie groupoid Γ fits in a short exact sequence of Lie groupoid homomorphisms 1 → K ֒→ Γ ։ Γ ′ → 1 where Γ ′ is a faithfully representable proper Lie groupoid and K is a bundle of compact Lie groups. By one of the preceding remarks, if we pick a faithful representation of Γ ′ and pull it back along the homomorphism Γ ։ Γ ′ to a representation of Γ, we obtain what in [26] is called an effective representation, i.e. one whose kernel consists of ineffective arrows. Conversely, it follows from results proven in [26, §4] that an arbitrary proper Lie groupoid which is effectively representable fits in a short exact sequence of the preceding form. On the basis of these considerations, we argue that any positive result about the existence of effective representations ought to be regarded as a presentation theorem of a generalized kind. Allowing ourselves a certain freedom of speech, we might further argue that the primary goal of the presentation theory of proper smooth stacks is to give an explicit characterization of those proper Lie groupoids which give rise to the same "effective transversal geometries" as effectively representable proper Lie groupoids or, equivalently, as the translation groupoids associated to compact Lie group actions on smooth manifolds.
The present article is intended to put the heuristic statements involved in the above speculations on a firm mathematical basis. To this end, we are going to propose a conceptual framework for the study of what we shall call "generalized reduced orbifolds" or, better, "reduced smooth stacks". The basic idea behind our theory is that, instead of trying to define what it means for a smooth stack to be "reduced", one should try to rigorously make sense of the assertion that two smooth stacks share the same "effective transversal geometry". In the spirit of F. Klein's Erlangen program, in order to define when two Lie groupoids (regarded as smooth stacks) give rise to the same "effective transversal geometry" without having to say what the "effective transversal geometry" associated with a Lie groupoid is, we are going to highlight a certain class of Lie groupoid homomorphisms which in a plausible sense preserve all of the groupoid effective transversal structure and then, by a standard categorical localization procedure, declare the members of that class to be isomorphisms. By definition, two Lie groupoids give rise to the same "effective transversal geometry" if they turn out to be isomorphic when regarded as objects of the resulting localized category. Of course there are some difficulties which we have to face if we want to make sense of these ideas in a truly satisfactory way: our localized category should admit a calculus of fractions [7]. This is required, for instance, in order to show that the notion of reduced smooth stack is indeed a generalization of the notion of reduced orbifold (we want the category of reduced orbifolds to imbed into that of reduced smooth stacks as a full subcategory).
We shall now give a section-by-section description of our article and, with it, some information about the above-mentioned localized category. The first two sections are essentially preparatory. In Section 1 we review some background notions, such as the notion of effect of an arrow or the notion of ineffective isotropy group, and study how these behave with respect to homomorphisms. In Section 2 we make a preliminary study of the homomorphisms that will be formally inverted in the process of constructing our localized category, in particular, we explain in what sense these homomorphisms preserve the effective transversal structure of Lie groupoids. In Section 3, we describe our prototype category of reduced smooth stacks. At the outset there is the category LGpd · with objects all Lie groupoids and with morphisms all Lie groupoid homomorphisms that carry ineffective isotropic arrows into ineffective isotropic arrows. On the morphisms of this category, we introduce an equivalence relation, which we call natural congruence, which identifies two homomorphisms when there exists some smooth transformation between them that is natural "modulo ineffective isotropy". Next, we form the category LGpd · / . ≡ with objects all Lie groupoids and with morphisms all natural congruence classes of homomorphisms in LGpd · . We show that the homomorphisms studied in Section 2, which automatically belong to LGpd · , project down under the quotient functor LGpd · → LGpd · / . ≡ to a class of morphisms E which admits a calculus of right fractions. The localized category LGpd · / . ≡ [E −1 ] is our prototype category of reduced smooth stacks. We call effective equivalence a homomorphism in LGpd · whose image under the canonical functor LGpd · → LGpd · / . ≡ [E −1 ] is an isomorphism. In Section 4 we show that any effective equivalence induces a homeomorphism at orbit space level and preserves the effective infinitesimal transversal structure at every base point. Moreover, we show that the category of reduced orbifolds imbeds canonically into LGpd · / . ≡ [E −1 ]. Finally, in Appendix A, we point out a few consequences of an assumption we make on Lie groupoids, namely, second countability, which motivate some basic definitions we give in Sections 2-3. Among those consequences there are the following two statements: (1) Any fully faithful homomorphism of Lie groupoids that at base level covers the identity map is an isomorphism. (2) Any Lie groupoid with just one orbit is transitive.
Apart from the formulation of generalized presentation results and the related study of effective representations, which provided the original motivation for our analysis, there is another context where the ideas outlined in the present article are likely to find application, namely, the theory of Riemannian metrics on Lie groupoids propounded recently in [3]. (We are indebted to M. del Hoyo for drawing our attention to this potential utilization of our theory.) The existence of such metrics is a property of Lie groupoids which presumably only depends on the underlying "reduced smooth stacks" and, therefore, is invariant under effective equivalence. Our theory may even provide a convenient framework for the analysis of other (e.g. symplectic or Poisson) "trans-versely invariant" geometric structures on Lie groupoids. The problem of how to give a systematic treatment of such structures is relevant e.g. to Poisson geometry [6].
We consider the constructions described in this article to be simply a first step towards a full-fledged theory of reduced smooth stacks. In particular, the localized category LGpd · / . ≡ [E −1 ] should be regarded simply as a "minimal working model" which, albeit already satisfactory from the point of view of our original objectives, may lend itself to further development. Just to mention one possibility, we have contented ourselves with only formally inverting those homomorphisms (among those that preserve the effective transversal structure) for which the standard weak pullback construction suffices to establish the existence of a calculus of fractions. Nothing however excludes that, by suitably modifying that construction, one might be able to invert a larger class of homomorphisms. Ideally, we would like effective equivalences to admit an explicit characterization (at best, one stable under natural congruence), like weak equivalences. We have also deliberately ignored any 2-categorical aspect in our exposition. However, as argued in [15], higher-level information ought to be taken into account in order to obtain a fully satisfactory theory. The appropriate setting for a theory of "reduced smooth stacks" is, most probably, that of bicategories of fractions [22]. All these aspects, along with those mentioned in the previous paragraph, shall be addressed elsewhere.

Overall conventions about terminology and notation
Throughout the article the name groupoid will designate a small category in which all arrows are invertible. A generic groupoid Γ will be written Γ (1) ⇒ Γ (0) when there is need to specify its set of objects Γ (0) (also called the base of Γ) and its set of arrows Γ (1) (itself often written Γ by abuse of notation) individually. The structure maps of a groupoid Γ will be denoted s Γ (source), t Γ (target), m Γ (composition law), u Γ (unit) and i Γ (inverse), omitting the superscript 'Γ' whenever there is no risk of ambiguity. For every pair of objects x, y ∈ Γ (0) the set of all arrows of source x respectively target y will be indicated by Γ x = Γ(x, −) respectively Γ y = Γ(−, y); moreover Γ x y = Γ(x, y) will denote the intersection Γ x ∩ Γ y . The following standard abbreviations will be used systematically: sg for s Γ (g); tg for t Γ (g); g ′ g for m Γ (g ′ , g) when sg ′ = tg; 1 x or just x for u Γ (x); g −1 for i Γ (g).
By a differentiable manifold we mean a (non-empty) locally compact manifold of class C ∞ . For each point x of a differentiable manifold X there is some local chart ϕ : U ≈ → R n of class C ∞ centered at x = ϕ −1 (0) ∈ U. The integer n ∈ N, which does not depend on the choice of ϕ, is called the local dimension of X at x and indicated by dim x X. The function dim X : X → N is locally constant over X. When it is (overall) constant, we say X is of constant dimension. By a smooth manifold we mean a differentiable manifold of constant dimension whose topology is Hausdorff and possesses a countable basis of open sets.
A differentiable groupoid will be a groupoid Γ ⇒ X in which X and Γ are differentiable manifolds, s Γ and t Γ are submersive differentiable maps, and the other groupoid structure maps (namely m Γ , u Γ and i Γ ) are differentiable. A homomorphism (of differentiable groupoids) will be a differentiable functor. The term Lie groupoid will be regarded as synonymous with smooth groupoid; the latter term indicates a differentiable groupoid whose manifold of objects and whose manifold of arrows are both smooth.

Ineffective isotropy
The purpose of this section is to provide a self-contained introduction to some concepts which lie at the heart of the theory expounded in Sections 2 to 4. There is not much claim to originality to be made here. In view of the fundamental role played by the notions to be discussed below, the reader will forgive us if, occasionally, we give full proofs of well-known facts. We will start by reviewing the basic structure theory of differentiable groupoids, in particular, the notion of orbit; in combination with Godement's theorem [24], the arguments given in [20] are essentially still valid in the present, more general context.
Let Γ ⇒ X be an arbitrary differentiable groupoid. For each pair of base points x, y ∈ X the subset Γ x y = Γ(x, y) ⊂ Γ is a differentiable submanifold of Γ. In particular, the isotropy group Γ x x = Γ(x, x) has a canonical differentiable group structure. The source fiber Γ x = Γ(x, −) = s −1 (x) ⊂ Γ is also a differentiable submanifold of Γ. The composition of arrows restricts to a differentiable action of the group G x = Γ x x on the manifold Γ x from the right. This action is free and has the property that on the quotient set Γ x /G x there exists a unique differentiable structure relative to which the canonical projection pr Γ x : Γ x ։ Γ x /G x becomes a submersion. The differentiable manifold obtained in this way shall be denoted by O Γ x and referred to as the orbit of Γ (Γ-orbit) through x. The action of the group G x on the manifold Γ x makes the differentiable fibration pr Γ x is injectively immersed into the groupoid base X in a canonical fashion. Namely, there is a unique map in Γ x : O Γ x → X such that the composition in Γ x • pr Γ x equals t x : Γ x → X, the restriction of the target map to the source fiber, and this map is necessarily differentiable, injective and immersive. The longitudinal tangent space at x is defined to be the image of the tangent linear map T [x] in Γ x : (the base point x and the corresponding unit arrow 1 x being identified notationally). The transversal tangent space at x is defined to be the quotient vector space Intuitively, the former space consists of all those vectors in T x X that are "tangent" to the orbit O Γ x , whereas the latter consists of those that are "perpendicular" to it. We observe that for every arrow g ∈ Γ im(T [tg] in tg ) = im(T [g] in sg ). ( To see this, notice that the right-translation map h → hg is a diffeomorphism of Γ tg onto Γ sg which is equivariant relative to the group isomorphism Γ tg tg ∼ → Γ sg sg given by conjugation by g −1 and therefore descends to a well-defined diffeomorphism between the orbits O tg and O sg which intertwines the maps in tg and in sg .
Let an arrow g ∈ Γ be given. Put x = sg and x ′ = tg. Consider an arbitrary local differentiable section γ : U ֒→ Γ to the source map s : • in x [such a map exists as a corollary to the local triviality of the orbit fibration pr x : Γ x ։ O x ]. As a consequence, T x (t • γ) must induce a well-defined linear map between the transversal tangent spaces T ⌢ x Γ (0) and T ⌢ x ′ Γ (0) ; we let this map be indicated by ε γ x provisionally.
does not depend on the choice of a local source section γ through the given arrow g ∈ Γ(x, x ′ ).

2] differ from one another by a linear map taking values in T ⌢ ↔
x ′ Γ (0) and hence induce the same map between the transversal tangent spaces Both identities are an immediate consequence of the definitions.

Lemma 1.2.
For any base point x ∈ X, the correspondence that to each arrow g ∈ Γ x x associates its effect ε(g) gives rise to a differentiable group homomorphism of Γ x x into GL(T ⌢ x Γ (0) ) which shall be indicated by ε x .
Proof. There is only to check the differentiability of the correspondence ε x . Put G = Γ x x for brevity. Consider an arbitrary local C ∞ source trivialization ϕ : where Ω ⊂ Γ and U ⊂ X are open subsets]. Put A = G ∩ Ω and let ψ : A × U → X be the map given by ψ(a, u) = tϕ −1 u, f (a) . Clearly ψ is of class C ∞ and the same is true of its second partial D 2 ψ(−, x) : A × T x X → T x X. From the definitions it follows that if we fix any vector-space basis 1 , . . . , m of T x X so that the last m − r vectors span the longitudinal subspace T ⌢ ↔ x Γ (0) then the top left r × r minor in the matrix representing D 2 ψ(a, x) ∈ End(T x X) will be the matrix representing the effect of a relative to the induced basis¯ 1 , . . . ,¯ r of the transversal tangent space T ⌢ x Γ (0) . We shall refer to the closed subgroup as the ineffective isotropy group of Γ at x. We proceed to study the behavior of ineffective isotropy under homomorphisms. Let φ : Γ → ∆ be an arbitrary homomorphism of differentiable groupoids. Let X denote the base manifold of Γ and Y that of ∆. Also let f : X → Y denote the base map induced by φ. For each base point where φ x : Γ x → ∆ f (x) indicates the map induced by φ between the source fibers. This map is necessarily differentiable (because pr Γ x is a surjective submersion and therefore admits local sections). Alternatively, O φ x can be characterized through the equation which makes it evident that the tangent linear map T x f : . More explicitly, the following identity holds for every arrow g ∈ Γ x x : Proof. Let us set y = f (x) and h = φ(g) for brevity. Let γ be any local source section of class C ∞ through g and let δ be any similar section through h. We have

Proposition 1.4. The following implications hold for any homomorphism of differentiable groupoids
Proof. The last assertion follows from the other two, which in turn are straightforward consequences of the identity (3).
We shall call a homomorphism of differentiable groupoids φ : Γ → ∆ transversal whenever the following map is a submersion (notations as above).
Proof. As before, let f : X → Y denote the map induced by φ on the groupoid bases. Also set y = f (x). By the transversality hypothesis on φ, over some open neighborhood V of y in Y it will be possible to find a C ∞ section through (x, 1 y ) ∈ X f × t ∆ to the map (4). This section will be of the form (a, δ) with a : V → X a C ∞ map and δ : Recall that a homomorphism of differentiable groupoids φ : Γ → ∆ is said to be a weak equivalence if the associated map (4) is a surjective submersion (so that in particular φ is transversal) and the square diagram (in the above notations) is a pullback within the category of differentiable manifolds. Proposition 1.6. Let φ : Γ → ∆ be a weak equivalence of differentiable groupoids.
followed by a; this can be seen by exploiting the availability of local C ∞ sections to the projection pr y : ∆ y ։ O y together with the pullback universal property of the diagram (5). The existence of such a map implies that T y a : which we may call say α. From the proof of Proposition 1.5 it follows that α must be a right inverse for T ⌢ x φ (0) . We contend that α is also a left inverse for T ⌢ x φ (0) . To see this, choose any open neighborhood U of x in X so that f (U) ⊂ V and let f U : U → V denote the map induced by f by restriction. The pullback universal property of the diagram (5) applied to the maps δ • f U and (id, a • f U ) yields a local source section γ : Combining the above with Proposition 1.4(c), we obtain the following notable property of weak equivalences, which appears already in [26] We conclude the section with a remark about natural transformations which will be needed only later in Section 4 on a single occasion. Recall that a natural transformation τ between two homomorphisms of differentiable groupoids φ, ψ : Γ → ∆, in mathematical notation 'τ : φ ⇒ ψ', is a map of class C ∞ from the base X of Γ into the arrows of ∆ which to each point x ∈ X assigns an arrow τ(x) ∈ ∆(φx, ψx) in such a way as to yield an ordinary natural transformation of (abstract) functors between φ and ψ.
Proof. Pick any local s ∆ -section δ through τ(x) = δ(φx). Then will be a linear map taking values in the longitudinal tangent space T ⌢ ↔ ψx ∆ (0) , because the parenthesized linear map takes values in ker T s.

Completely transversal and full homomorphisms
We open the present section-which like the previous one is preparatory-by reviewing some standard basic constructions. Then, we proceed to establish some seemingly not so well-known results, notably Proposition 2.3 and Lemma 2.8, which motivate and underlie the theory discussed in the subsequent sections.
Let Γ ⇒ X, ∆ ⇒ Y be differentiable groupoids. We remind the reader that we call a homomorphism φ : Γ → ∆ transversal if the associated map s • pr 2 : X φ × t ∆ → Y is submersive. We call φ completely transversal if the same map is, moreover, surjective. 2

Lemma 2.1. The composition of two (completely) transversal homomorphisms of differentiable groupoids is also (completely) transversal.
Proof. Let φ : Γ → ∆ and ψ : ∆ → Σ be arbitrary homomorphisms of differentiable groupoids. Let X, Y, Z denote the base manifolds of Γ, ∆, Σ respectively. We have the following commutative diagram of C ∞ maps in which the top horizontal map is a surjective submersion. The claim about the composition ψ • φ is obvious now.
Let Γ ⇒ X, ∆ ⇒ Y and Σ ⇒ B be differentiable groupoids. Let φ : Γ → Σ and ψ : ∆ → Σ be differentiable groupoid homomorphisms. Assume that φ is transversal. Then we can form the weak pullback of φ along ψ whose construction we proceed to recall. Because of the transversality of φ, the following turn out to be differentiable manifolds: Regarding Z as the base manifold and Π as the manifold of arrows, one declares the map s × id × s given by (h, k, g) → (sh, k, sg) [which is clearly a surjective submersion] to be the source, and the map (h, k, g) → th, ψ(h)kφ(g) −1 , tg [which is obviously C ∞ ] to be the target. The composition law, unit and inversion map are respectively given by they are obviously all C ∞ and make Π ⇒ Z into a differentiable groupoid which we agree to indicate by ∆ ψ ⊓ φ Γ. (7), the homomorphism pr ∆ given by (h, k, g) → h is transversal and in fact submersive at the level of groupoid bases. Moreover, if φ is completely transversal then pr ∆ is onto at the level of bases and thus a fortiori also completely transversal.

Lemma 2.2. In the weakly commutative diagram
Let ∆ ⇒ Y be a differentiable groupoid. Let f : X → Y be a differentiable map which is essentially submersive in the sense that the associated map s•pr 2 : whose groupoid structure is uniquely determined by the requirement that the obvious 'projection' π : f * ∆ → ∆ ought to be a homomorphism of groupoids. The 'projection' π is in fact a homomorphism of differentiable groupoids, and a weak equivalence as soon as f is also essentially surjective (i.e. as soon as s • pr 2 is also surjective).
there exist an open cover c :

a surjective map which on each connected component of U ′ restricts to a diffeomorphism onto an open subset of U) and a C
Proof. Since the map f : X → Y is essentially submersive, we can form the pullback groupoid f * ∆ ⇒ X, as in (8). In the case under consideration this is a Lie groupoid. Its 'projection' π : f * ∆ → ∆ is a homomorphism of Lie groupoids which covers the map f : X → Y at base level. By the evident C ∞ -pullback universal property of this homomorphism, φ will factor through f * ∆ as φ = π • φ ′ for a unique homomorphism φ ′ : Γ → f * ∆ covering the identity on the common base X of Γ and f * ∆.
Fullness of φ plainly entails fullness of φ ′ , so that φ ′ is a surjective homomorphism of Lie groupoids which covers the identity over X. It follows from Proposition A.2 in the appendix that φ ′ is actually an epimorphism of Lie groupoids viz. that the map which φ ′ induces between the manifolds of arrows is a surjective submersion. Now, suppose we are assigned maps h and (x, x ′ ) as in the universal problem (9). By the C ∞ -pullback universal property of the square (10), there will be a unique C ∞ map h ′ : U → f * ∆ satisfying the conditions π • h ′ = h and (s, t) • h ′ = (x, x ′ ). We are thus reduced to the simpler problem depicted below, where the outer square commutes by definition of h ′ .
Consider any point u ∈ U. Since φ ′ : Γ ։ f * ∆ is a surjective submersion, we can find a local C ∞ section g ′ u : V u → Γ to φ ′ defined around h ′ (u). We can then select an open neighborhood W u of u in U so that h ′ (W u ) ⊂ V u and define a map g u on W u by setting g u = g ′ u • h ′ W u . The coproduct map g = g u [defined on the disjoint union U ′ = W u of all the open sets W u as u is let vary over U] will be a solution to the problem schematized in (11).

Corollary 2.4. Let φ : Γ → ∆ be a completely transversal homomorphism of Lie groupoids which is both full and faithful (as an abstract functor). Then φ must be a weak equivalence.
Of course, the property of C ∞ -fullness defined with Proposition 2.3 makes sense for homomorphisms between arbitrary differentiable groupoids. Any C ∞ -full homomorphism of differentiable groupoids is full.
Lemma 2.5. The composition of any two C ∞ -full homomorphisms of differentiable groupoids is itself C ∞ -full. Lemma 2.6. In the weak pullback diagram (7), the homomorphism pr ∆ must be C ∞ -full whenever so is φ (of course assuming this is transversal).
Now, since φ is C ∞ -full, it will be possible to find an open cover c : The map (h • c, k • c, g) : U ′ → ∆ ψ•s × t Σ s × φ•s Γ will be a solution for the original universal problem expressing the C ∞ -fullness of the homomorphism pr ∆ .
Remark. Making U = { * } in the preceding proof, the same reasoning shows that pr ∆ must be full whenever so is φ.
The above lemmas suggest that one might be able to build a reasonable category of fractions by localizing differentiable groupoids at their completely transversal, C ∞ -full homomorphisms. This is indeed so, as we will see shortly. Before proceeding further, however, we must convince ourselves that such homomorphisms preserve the "effective transversal geometry" of differentiable groupoids. This is the goal of the next couple of lemmas. Recall that any differentiable groupoid Γ ⇒ X gives rise to an associated orbit space X/Γ; this is the quotient of X by the equivalence relation that identifies any two points which can be connected by an arrow in Γ, topologized with the finest topology that makes the quotient projection X → X/Γ continuous.
x is a closed normal subgroup of the differentiable group Γ x x , the quotient group Γ x x will inherit a canonical differentiable group structure. We shall refer to the effective Γ x x space T ⌢ x Γ (0) as the effective first order approximation of Γ at x or as the effective infinitesimal model for Γ at x.
A homomorphism of differentiable groupoids φ : Γ → ∆ shall be called faithfully transversal if (it is transversal and) for every point x in the base of Γ the linear map is injective (hence bijective by Proposition 1.5). By Corollary 1.4, any transversal φ : Γ → ∆ must induce a differentiable group homomorphism φ x x : Γ x x → ∆ φx φx at every x, which will be injective as soon as φ is faithfully transversal; by Proposition A.6 in the appendix, any injective homomorphism of differentiable groups must be a monomorphism. Proof. Let φ : Γ → ∆ be any such homomorphism. Let us temporarily assume that Γ and ∆ are groupoids over the same base manifold X and that φ covers the identity over X. Then for any given point x ∈ X the linear map T ⌢ x φ (0) will be injective if and only if the tangent base map T x φ = id T x X at x satisfies the condition Since φ is C ∞ -full, we can lift τ → h τ locally around zero to a C ∞ path τ → g τ ∈ Γ(x, −); it will not be restrictive to assume that g 0 = 1 x (for otherwise we can simply take τ → g τ g −1 0 ). Then . Now suppose φ : Γ → ∆ is completely general. As in the proof of Proposition 2.3, we can write φ as the composition of a homomorphism φ ′ covering the identity on the bases with a weak equivalence π. Using the faithfulness of π, it is straightforward to check that φ ′ must be itself C ∞ -full. Since weak equivalences are always faithfully transversal by Proposition 1.6, we are finally reduced to the special situation considered at the beginning.
Clearly, any C ∞ -full homomorphism φ : Γ → ∆ will induce an epimorphism of differentiable groups φ x x : Γ x x ։ ∆ φx φx at each base point x of Γ. Thus, by the remarks preceding the last lemma, when φ is also transversal the quotient homomorphism φ x x : Γ x x → ∆ φx φx will be an isomorphism (of differentiable groups); therefore, by the lemma, we will have an isomorphism between the effective infinitesimal model for Γ at x and that for ∆ at φx.

The category of reduced Lie groupoids
Throughout the rest of the article we shall let LGpd stand for the category of Lie groupoids and Lie groupoid homomorphisms. We shall use the notation 'h 1 ≡ h 2 (mod . ∆)' as an abbreviation for 'sh 1 = sh 2 = y, th 1 = th 2 and h −1 ∆ is a normal, 3 totally isotropic (abstract) subgroupoid of ∆, it follows that the binary relation ≡ (mod . ∆) thus defined on the arrows of ∆ is a (categorical, abstract) congruence (although in general not a regular congruence in the sense of Appendix A).
G G ∆ be any pair of homomorphisms between two given Lie groupoids Γ and ∆. By a natural congruence τ between φ and ψ, in symbols 'τ : φ . ⇒ ψ', we shall mean a map τ : X → ∆ of class C ∞ from the base manifold X of Γ into the manifold of arrows of ∆ such that τ(x) ∈ ∆(φx, ψx) for all x ∈ X and such that for all g ∈ Γ τ(tg)φ(g) ≡ ψ(g)τ(sg) (mod .

∆).
Obviously, any ordinary natural isomorphism τ : φ ⇒ ψ is a fortiori a natural congruence τ : φ . ⇒ ψ. The collection of all natural congruences between homomorphisms Γ → ∆ is closed under the obvious operations of composition τ ′ τ and inversion τ −1 . On each hom-set LGpd(Γ, ∆) the binary relation φ . ≡ ψ ⇔ there exists a natural congruence between φ and ψ is therefore an equivalence. If we now let LGpd · denote the subcategory of LGpd with the same objects (Lie groupoids) and with morphisms all those φ ∈ LGpd(Γ, ∆) such ∆ φx φx for every base point x of Γ, it is immediate to check that the above equivalence . ≡ gives rise to a categorical congruence on LGpd · . We shall let LGpd · / . ≡ denote the resulting quotient category whose morphisms are the .

≡-equivalence classes of morphisms in
LGpd · (compare [16, II.8]). The notation [φ] · will be used to indicate the . ≡-class of a homomorphism φ ∈ Mor(LGpd · ). Let E ⊂ Mor(LGpd · ) denote the collection of all the (C ∞ -)full and completely transversal homomorphisms of Lie groupoids (we know that any transversal homomorphism must lie within LGpd · , by the results of Section 1). By abuse of notation, we shall use the same letter to indicate the image of E under the quotient projection functor that passing from LGpd to LGpd · already makes a relevant difference from the point of view of what "reduced smooth stacks" turn out to be eventually in comparison with ordinary smooth stacks. Let the orthogonal group O(2) act on R 2 by matrix multiplication (canonical action). Similarly, let the special orthogonal group SO(3) act on R 3 by matrix multiplication.  The next examples demonstrate that the relation of natural congruence is quite coarser than the relation of natural isomorphism in most cases. We shall exhibit several pairs of homomorphisms φ, ψ ∈ Mor(LGpd · ) such that φ is naturally congruent but not isomorphic to ψ.
are naturally congruent, because all the isotropic arrows in SO(3)⋉ R 3 outside the origin are ineffective, but not naturally isomorphic (unless of course η = id), because all the isotropy groups in SO(3) ⋉ R 3 outside the origin are abelian. (b) Let ω : R → R be an arbitrary C ∞ real-valued function of one real variable. Let R = (R, +) [= the additive group of the real numbers] act on the product C × R by operating on the first factor by rotations with frequency ω: θ · (z, t) = (e iω(t)θ z, t). Any C ∞ real-valued function of one real variable ϕ : R → R such that |ϕ(t)| = |ω(t)| for all t gives rise to a Lie groupoid homomorphism which belongs to Mor(LGpd · ) and whose kernel contains the totally isotropic, normal subgroupoid K of R ⋉ [C × R] defined by the expression Since K is, in fact, a closed regular kernel of constant dimension in R⋉[C×R] (compare Appendix A and Example 3.8 below), this homomorphism factors through the quotient groupoid Γ = (R ⋉ [C × R])/K ⇒ C × R, thus giving rise to a Lie groupoid homomorphism φ : Γ → SO(3) ⋉ R 3 which still belongs to Mor(LGpd · ). Now, let ϕ 0 , ϕ 1 be any two functions as above, and let φ 0 , φ 1 denote the Lie groupoid homomorphisms Γ → SO(3) ⋉ R 3 they give rise to. Clearly, one has φ 0 .

Axiom II. Given any pair of morphisms*
(*for visual immediacy, we shall be using wavy arrows to represent morphisms belonging to E). Since φ belongs to E, by definition it must be transversal so that we may form the weak pullback (which in our case is clearly also a smooth groupoid) (the diagram here is supposed to commute up to natural isomorphism). By the lemmas 2.2 and 2.6, φ ∈ E implies pr ∆ ∈ E. Thus, we will be done if we set ∆ ′ = ∆ ψ ⊓ φ Γ ′ , φ ′ = pr ∆ and ψ ′ = pr Γ ′ , provided we show that pr Γ ′ belongs to Mor(LGpd · ). To this end, suppose h ′ is an ineffective isotropy arrow in ∆ ψ ⊓ φ Γ ′ . Since both ψ and pr ∆ belong to Mor(LGpd · ), so will do their composition ψ • pr ∆ and hence also the composite homomorphism φ • pr Γ ′ as this is naturally isomorphic to ψ • pr ∆ . 5 We will then have Γ. Now φ is faithfully transversal by Lemma 2.8. We can therefore invoke Corollary 1.4 to yield the desired conclusion that pr Γ ′ h ′ ∈ . Γ ′ .

Axiom III. Given any morphisms
Let X, Y and Y ′ respectively denote the base manifolds of the Lie groupoids Γ, ∆ and ∆ ′ . By assumption, we have φ•ψ 1 . ≡ φ•ψ 2 , in other words, there exists some natural congruence τ ′ : φ•ψ 1 We make use of the C ∞ -fullness of φ in the situation depicted below.
The map c : X ′ ։ X is an open cover, in particular, a surjective submersion. Hence it certainly makes sense to pull back the Lie groupoid Γ along it.
The resulting pullback groupoid, denoted Γ ′ = c * Γ ⇒ X ′ , will be a Lie groupoid as well, 6 and its canonical projection π onto Γ will be a weak equivalence of Lie groupoids and thus, a fortiori, an element of E and hence a morphism in the category LGpd · . We contend that the C ∞ map τ in (13) must be a natural congruence by the commutativity of (13) and (14). Moreover, for every arrow g ′ ∈ Γ ′ , 5 More in general, whenever φ . ≡ ψ ∈ LGpd(Γ, ∆) and φ ∈ LGpd · (Γ, ∆) then also ψ ∈ LGpd · (Γ, ∆). The proof is immediate. 6 It can always be assumed that X ′ is a smooth manifold. Indeed, in any case X ′ is a (non-empty) Hausdorff manifold of constant dimension, just because so is X. Since X is second countable, we can always find a countable open cover X ′′ ։ X subordinate to X ′ ։ X via some map X ′′ → X ′ . By Lemma 2.8, φ must be a faithfully transversal homomorphism. We immediately conclude from Corollary 1.4 that the following implication holds: We thus deduce from (15) that as desired.

3.6.
Our proof that the multiplicative system E in the category LGpd · ≡ for brevity. One starts by introducing the "category of right fractions" LE −1 : • The objects of LE −1 are simply those of L (namely Lie groupoids); • The morphisms in LE −1 (Γ, ∆) are equivalence classes of "spans" in L two such spans (α 1 , ε 1 ) and (α 2 , ε 2 ) being equivalent whenever there is a commutative diagram • The composition of morphisms in LE −1 is given by The reader might be wondering whether it is possible to give a more "geometric" characterization of the category which we have just defined (say, up to equivalence). For instance, one might consider the idea of focusing attention on some collection of smooth groupoids whose members are "geometrically reduced" in a suitable sense-for example, they do not contain non-trivial closed regular kernels of constant dimension (we account for the terminology used here in the appendix)-and then asking whether the morphisms in RedLGpd between groupoids belonging to this collection may be represented by morphisms of a more familiar type, like for instance the generalized homomorphisms of the usual Morita category. This sort of approach runs into a number of technical difficulties. (We shall hint at a couple of them presently.) Even if one is willing to believe that these difficulties may be overcome, the construction underlying Definition 3.7 has undeniable advantages, such as extreme simplicity, full generality, and a very convincing justification at both intuitive and technical level.
To begin with, we point out that all of the Lie groupoids occurring in the preceding examples 3.2 and 3.3 are "geometrically reduced" in the above sense. [Actually, this is true of Example 3.3(b) only when the set {ω 0} is dense within R. Ending up with only "geometrically reduced" groupoids was the main reason behind the apparently unnecessary complications in that example.] On the basis of those examples, it seems impossible to dismiss the notion of natural congruence altogether. Furthermore, even if we agree to talk only about "geometrically reduced" groupoids, it is still necessary to show that for each Lie groupoid a natural "geometrically reduced model" may be produced in a systematic way. It is not at all obvious how to do that in general. A sample of the pathologies which arise in practice and which make the task arduous is given in our next example.
is evidently normal, closed, and smooth since it can be parameterized by means of local C ∞ sections to the groupoid source projection (θ, l) → l. By the theory of regular kernels (reviewed at the beginning of Appendix A), the quotient groupoid is naturally equipped with the structure of a smooth groupoid. We contend that, provided the open set U = {ω 0} and the interior V of the vanishing locus of ω are both non-empty, the codomain Γ ′ of any Lie groupoid homomorphism φ : belonging to E must contain some non-trivial regular kernel which is also closed and of constant dimension. In particular, Γ is unlikely to admit a "geometrically reduced model" in the above sense. To begin with, since the existence of such non-trivial kernels is a Morita invariant property, by the argument already used in the proof of Proposition 2.3 it will not be restrictive to assume that φ is a full homomorphism covering the identity on L. Let us set K ′ = ker φ. By our hypothesis about φ, K ′ is a regular kernel in Γ and thus K ′ ⊂ . Γ. In particular for every l ∈ L such that ω([l]) 0 we must have By hypothesis, Γ ′ is a smooth groupoid, so K ′ must be closed and of constant dimension. Since the restriction of K ′ over L U coincides with the unit bisection of Γ, we have dim K ′ = dim L. Thus, any local C ∞ section to the projection K ′ ։ L is also a local parameterization of K ′ . This immediately implies that the subset is itself a closed regular kernel of constant dimension in Γ. Since K ′′ K ′ , its image φ(K ′′ ) will be a non-trivial closed regular kernel of constant dimension in Γ ′ .

Comparison with the category of reduced orbifolds
The elements of the class E are not the only morphisms of the category LGpd · which become invertible under the canonical functor LGpd By way of example, consider any Lie groupoid homomorphism ψ : Γ → ∆ for which there exists some element φ : ∆ ′ of E such that the composition φ • ψ also lies in E. Clearly, ψ must be a homomorphism in LGpd · , and its image under (16) must be invertible. We shall call any element of Mor(LGpd · ) with this property an effective equivalence. In order to maintain our intuitive interpretation of RedLGpd as a category of "effective transversal geometry types", we need to make absolutely sure-among other things-that the lemmas 2.7 and 2.8 continue to hold for effective equivalences.
There is a canonical functor LGpd · ⊂ LGpd → Top into the category of topological spaces and continuous maps, which to each Lie groupoid Γ ⇒ M assigns the corresponding orbit space M/Γ and to each Lie groupoid homomorphism ψ : Γ → ∆ the (continuous) map of M/Γ into N/∆ induced by ψ. Evidently, any two naturally congruent homomorphisms induce the same map between the orbit spaces. By the universal property of quotient categories [16, Section II.8], we obtain a well-defined functor LGpd · The generalization to effective equivalences of Lemma 2.8 and of its consequences, although conceptually not more involved than the generalization of Lemma 2.7 which we have just obtained, requires some extra preparation work. For every pair of Lie groupoids Γ ⇒ M, ∆ ⇒ N, let us define S Γ,∆ to be the set of all 4-tuples (x, y; θ, λ) consisting of a base point x ∈ M, a base point y ∈ N, a Lie group homomorphism θ : Γ x x → ∆ y y and a θ-equivariant linear map λ : . We have two obvious maps α Γ,∆ and β Γ,∆ of S Γ,∆ into respectively M and N and, for each triplet of Lie groupoids Γ, Γ ′ , Γ ′′ , an obvious composition operation Let S Γ,∆ denote the quotient of the set S Γ,∆ with respect to the equivalence relation x ′ and similarly c h mean "conjugation", and where ε(g) and ε(h) as usual mean "effect". Evidently, we have induced maps* [*the quotient composition operation being well defined essentially because of the formulas below, which hold for every isotropic arrow g ∈ Γ x x : We introduce an auxiliary category, Skel, which we call the category of "transversal skeletons" of Lie groupoids. Lie groupoids are the objects of Skel. LGpd · → Skel which to each Lie groupoid homomorphism ψ : Γ → ∆ assigns the global a Γ,∆ -section Reasoning as we did before, 7 we conclude that there must be a factorization of this functor through RedLGpd. LGpd For any effective equivalence of Lie groupoids φ : Γ → ∆ and for any base point Thus, we still have an isomorphism of the form (12) between the effective infinitesimal model for Γ at x and that for ∆ at φx.

Remark on terminology.
In [26], we called effective a Lie groupoid representation whose kernel consists of ineffective isotropic arrows. More in general, we may call effective a homomorphism of differentiable groupoids which enjoys the same property. Our terminology 'effective equivalence' is consistent with this use of the adjective 'effective'. Indeed, by 4.2, any effective equivalence is faithfully transversal and hence satisfies the hypotheses of Proposition 1.4(b), which then implies the desired property.
One of the claims we made in the introduction to the present article was that our notion of "reduced smooth stack" was going to generalize the notion of reduced orbifold. It is now time to substantiate that claim. In doing this, we shall make essential use of the existence of a calculus of fractions for the localized category LGpd · Without a calculus of fractions, the task of comparing the above-mentioned two notions would be very likely an insurmountable mess. We shall adopt the general point of view on orbifolds that is advocated for instance in [21,18]. In practice, for our purposes this means that the category of reduced orbifolds we want to compare with our category of reduced Lie groupoids is obtained in conformity with the following stepwise procedure. Start with the category of effective orbifold groupoids. 8 Then pass to the quotient category where any two homomorphisms are identified whenever there exists a natural isomorphism connecting them. Finally formally invert the morphisms corresponding to weak equivalences.
We shall place our discussion in a context which is slightly more general than strictly needed. Let effLGpd ⊂ LGpd denote the full subcategory consisting of all effective Lie groupoids. Since by definition the ineffective subbundle of an effective Lie groupoid is trivial, effLGpd is actually a full subcategory of LGpd · . Moreover, by the same token, the two equivalence relations ≡ (natural isomorphism) and . ≡ (natural congruence) turn out to coincide when restricted to the morphisms of this subcategory. We thus have a canonical imbedding of categories (i.e., a fully faithful functor which is "identical" on Let W ⊂ Mor(LGpd · ) denote the class of all weak equivalences of Lie groupoids, and set W eff := W ∩ Mor(effLGpd). It is standard routine to check that the localized category effLGpd /≡ [W −1 eff ] admits a calculus of right fractions. By the universal property of localization, the imbedding (18) induces a canonical functor

The above functor is fully faithful and hence an imbedding of categories.
Proof. (Fullness.) Let a span be given Γ representing a morphism in RedLGpd between two given effective Lie groupoids Γ and ∆. Since φ ∈ E is transversal, we must have φ( ∆ (= 1 because ∆ is effective) and therefore ker ψ ⊃ . Γ ′ = ker φ. Now, if as in the proof of Proposition 2.3 we decompose φ into an epimorphism identical on the bases φ ′ followed by a weak equivalence π, then from the first homomorphism theorem for Lie groupoids it follows that ψ admits a unique factorization ψ = ψ ′ • φ ′ through φ ′ . It is evident that the span (ψ ′ , π) [in which π ∈ W eff because Γ is effective and effectiveness is a Morita invariant property] represents the same morphism in RedLGpd as (ψ, φ) does.
To begin with, notice that since . Similarly, since . ∆ = 1, we must have ψ 1 • χ 1 ≡ ψ 2 • χ 2 . Now each φ i is a faithful functor, hence for every isotropic arrow g ′ in Γ ′ we must have Γ ′ , as noticed in the previous paragraph). Write φ as the composition of an epimorphism identical on the bases φ ′ : Γ ′ ։ Γ ′′ with a weak equivalence π : Γ ′′ → Γ. Necessarily Γ ′′ is effective, because so is Γ. By the first homomorphism theorem for Lie groupoids, we have a unique factorization χ i = χ ′ i • φ ′ of each χ i through φ ′ . Trivially, each χ ′ i must be a homomorphism in LGpd · . Now in general for any pair of homomorphisms α, β from Γ ′′ into any given other Lie groupoid. [Since φ ′ is identical on the bases, any natural isomorphism τ : α • φ ′ ⇒ β • φ ′ may also be interpreted as a natural isomorphism τ : By the same token, We thus obtain the following commutative diagram in effLGpd /≡ .
U U 7 w 7 w 7 w 7 w 7 w 7 w 7 w 7 w (φ 1 , φ 2 , π ∈ W eff ) (The notation [ ] is supposed to indicate the ≡-class of a homomorphism.) Let efforbGpd ⊂ effLGpd denote the full subcategory with objects all effective orbifold groupoids. (Recall that by an orbifold groupoid we mean a proper étale Lie groupoid; cf. Footnote 8.) By the universal property of localization, we have canonical functors ; here of course we have set W efforb := W ∩ Mor(efforbGpd). Each one of the two localized categories at the extremes of the sequence admits a calculus of right fractions. The right-hand canonical functor is trivially an imbedding, essentially because effectiveness is a Morita invariant property. The other canonical functor is also an imbedding. This is almost as trivial to see, by using the fact that any foliation groupoid is the codomain of a weak equivalence with domain an étale groupoid. 9 Combining 4.3 with the above remarks, we obtain:

Corollary 4.4. The canonical functor of localized categories
≡ [E −1 ] =: RedLGpd imbeds the category of reduced orbifolds into that of reduced Lie groupoids. 9 By a foliation groupoid, in general, we mean a differentiable groupoid of constant dimension which has only discrete (i.e., zero-dimensional) isotropy groups. For any foliation groupoid Γ ⇒ X there is some integer 0 ≦ r ≦ dim X such that dim T ⌢ x Γ (0) = r for all x ∈ X. Equivalently, the Γ-orbits all have When Γ ⇒ X is a smooth foliation groupoid, we can find a complete transversal in T : T → X with domain a smooth manifold T of dimension r. The pullback groupoid Π = in * T Γ ⇒ T will be a smooth groupoid with all orbits zero-dimensional and all isotropy groups discrete. Clearly, any such groupoid must be étale.
We define the kernel of an arbitrary homomorphism of differentiable groupoids φ : Γ → ∆ to be the (abstract, set-theoretic) subgroupoid of Γ In a broader sense, by a kernel in an arbitrary groupoid Γ we shall mean a totally isotropic subgroupoid which contains all the units of Γ and is normal (cf. Footnote 3 on page 15). Of course, the kernel of a homomorphism turns out to be a kernel in this sense. It is not hard to show that if φ : Γ ։ ∆ is an epimorphism of differentiable groupoids which covers the identity then ker φ is actually a regular kernel in Γ, that is to say, a kernel which is also a differentiable subgroupoid of Γ. 11 Moreover, under the same assumption, if the groupoid ∆ is Hausdorff then ker φ is a closed subset of the manifold Γ (1) . Conversely, let K be an arbitrary kernel in a given differentiable groupoid Γ ⇒ X. The equivalence relation R K on the arrows of Γ defined by is an (abstract, categorical) congruence on Γ. In this context, we shall write Γ/K for the quotient groupoid Γ/R K . The congruence R K is regular if, and only if, the kernel K is regular. Whenever K is regular and the base X is Hausdorff, the subset R K ⊂ Γ (1) × Γ (1) is closed if, and only if, so is the subset K ⊂ Γ (1) . Notice that if the groupoid Γ ⇒ X is of constant dimension and the kernel K is regular then, as a groupoid over X, K is of constant dimension if, and only if, the same is true of the quotient Γ/K, in which case It follows that for any regular kernel K in a smooth groupoid Γ the quotient differentiable groupoid Γ/K is smooth if, and only if, K is closed and of constant dimension. It is a well-known fact in the elementary theory of Lie groups that any bijective Liegroup homomorphism must be a diffeomorphism and hence an isomorphism; compare [2, Exercise 2.22 (2)]. The relevant property of Lie groups, here, is second countability. In fact, the statement in question is false for general (i.e., non-Lie) differentiable groups. By way of example, let G be the one-dimensional differentiable group obtained by endowing the additive group of euclidean 2-space (R 2 , +) with the one-dimensional differentiable structure resulting from the identification where the right-hand side denotes the disjoint union of uncountably many copies of R. The identifying map itself provides a bijective homomorphism of differentiable groups between G and (R 2 , +) [= standard, two-dimensional, euclidean Lie group] which is certainly not an isomorphism (not even a homeomorphism). 11 A differentiable subgroupoid of Γ is a subgroupoid Γ ′ (in the abstract, set-theoretic sense) such that Γ ′ (1) ⊂ Γ (1) is a differentiable submanifold and such that the source map of Γ restricts to a submersion of Γ ′ (1) onto a differentiable submanifold of Γ (0) . With the induced differentiable structure, a differentiable subgroupoid becomes a differentiable groupoid in its own right. f : G/K → H such that f =f • π. Evidentlyf must be bijective and thus, in view of Lemma A.1(b), a diffeomorphism.
We are now in a position to conclude that each one of the maps φ x : Γ x → ∆ x which φ induces between two corresponding source fibers is a submersion. In fact, our claim is a straightforward consequence of what we have already shown, Equation (2a), and the principality of the Lie-group bundles pr Γ x : Γ x ։ O Γ x and pr ∆ x : ∆ x ։ O ∆ x . Finally, for every arrow g ∈ Γ x the matrix expression id 0 * T g φ x associated with any choice of splittings to the surjective linear maps T g s Γ and T φ(g) s ∆ makes it evident that the linear map T g φ (1) must be surjective.
Recall that a differentiable groupoid Γ ⇒ X is locally transitive if the associated combined source-target map (s, t) : Γ → X × X is submersive, and transitive if in addition the same map is surjective. By a one-orbit groupoid we shall mean a groupoid whose combined source-target map is surjective.

Proposition A.3. Any one-orbit Lie groupoid is transitive.
Proof. A general differentiable groupoid Γ ⇒ X is locally transitive if, and only if, for every base point x ∈ X the restricted target map t x : Γ x → X is a submersion. If t x is surjective then the orbit immersion in Γ x : O Γ x X is a bijection. Moreover, if Γ is second countable and X is of constant dimension then the orbit O Γ x is a second countable manifold of constant dimension. We conclude from Lemma A.1(b) that whenever Γ ⇒ X is a one-orbit Lie groupoid the orbit immersion in Γ x is a diffeomorphism and consequently t x = in Γ x • pr Γ x is a submersion. A.4 (Counterexamples). (a) Let M = N ⊔ N denote the disjoint union of two copies of a smooth manifold N. Let G be any Lie group of positive dimension. Let Γ and ∆ respectively denote the trivial Lie-group bundles (G × N) ⊔ (G × N) = G × M → M and G × N → N. The map (1 G × id N ) ⊔ id G×N : Γ → ∆ (where 1 G indicates the constant endomorphism of G) is a surjective Lie-groupoid homomorphism for which the conclusion of Proposition A.2 fails. This shows that the hypothesis φ (0) = id in that proposition cannot be relaxed in any substantial way.
(b) Let G denote the one-dimensional differentiable group arising from the identification (19). Consider the two translation groupoids Γ = G ⋉ R 2 and ∆ = (R 2 , +) ⋉ R 2 , the group action in either case being given by the formula (s, t) · (a, b) = (s + a, t + b). Each of them is a one-orbit differentiable groupoid. However, Γ is not transitive. The isotropy groups of Γ and ∆ are all trivial. The "identical" homomorphism from Γ onto ∆ is bijective and covers the identity over Γ (0) = ∆ (0) = R 2 . However, none of the injective immersions induced between the orbits O Γ x O ∆ x (albeit bijective) can be a diffeomorphism (for reasons of dimension since dim O Γ x = 1 whereas dim O ∆ x = 2). For the sake of completeness, we record the following useful lemma, which shows that there is essentially no such thing as a theory of differentiable groups beyond the classical theory of Lie groups. The only way in which a general differentiable group may fail to be a Lie group is in having uncountably many connected components, each component being a perfectly nice, smooth manifold.

Lemma A.5. Every connected differentiable group is a Lie group.
Proof. Let G be any such group. Since translations in G are diffeomorphisms, G is of constant dimension. By a standard argument, any differentiable group is Hausdorff. Thus, the claim is essentially all about second countability.
The connectedness of G entails that if U is any neighborhood of the unit e then G = ∞ k=1 U k where U k = U · · · U (k-fold product in G). The product S T of any dense subsets S ⊂ V, T ⊂ W of two arbitrary subsets V, W of G must be dense within the product VW. Indeed, let ∈ V, ∈ W and let U ∋ be any open neighborhood. By the continuity of group multiplication, we may find open neighborhoods V ′ ∋ , W ′ ∋ so that V ′ W ′ ⊂ U. By density, there will be elements s ∈ S ∩ V ′ , t ∈ T ∩ W ′ . Then, st ∈ S T ∩ V ′ W ′ ⊂ S T ∩ U.
Let us fix an arbitrary local chart ϕ : U ≈ → R n for G with center at e = ϕ −1 (0). For every k ∈ N let us put U k = ϕ −1 B 1/k (0) , where B 1/k (0) denotes the open ball {x ∈ R n |x| < 1/k}. Since R ≈ ϕ Q n is a dense subset of U ≈ ϕ R n , we conclude from the above that R k must be dense within U k for all k and hence that ∞ k=1 R k must be dense within ∞ k=1 U k = G. Thus G has to contain some dense sequence {g l } l∈N . We contend that {V k g l (k, l) ∈ N × N} where V k = U k ∩ U −1 k must be a countable basis for the topology of G. Indeed, given any element of an open subset W of G we choose k so that V k V k ⊂ U k U k ⊂ W −1 and then l so that g l ∈ V k ; then clearly ∈ V k g l ⊂ W. Proposition A.6. Let φ : Γ → ∆ be a homomorphism between two arbitrary differentiable groupoids. Suppose that φ covers the identity and is faithful as an abstract functor. Then φ is a monomorphism.
Proof. In view of the hypothesis φ (0) = id, Equation (2b) implies that each one of the maps O φ x : O Γ x → O ∆ x induced by φ between two corresponding orbits is an injective immersion. As next step, we show that each isotropy homomorphism φ x x : Γ x x → ∆ x x is an immersion. Clearly, it will be enough to show that T 1 x φ x x is an injective linear map. By Lemma A.5, the identity component of Γ x x is a Lie group. The claim is then an immediate consequence of the naturality of the exponential map of a Lie group [2, (3.2), p. 23]. The proof proceeds by analogy with that of Proposition A.2.
Corollary A.7. Any fully faithful homomorphism of Lie groupoids which covers the identity is an isomorphism.