NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS

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1 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS OLEG CHTERENTAL April 30, 2009 These notes are an exposition of the beginning of Ehud Hrushovski s paper Groupoids, Imaginaries and Internal Covers. They assume a bakground onsisting of an introdutory level ourse in Model Theory. I would like to thank NSERC for the USRA that provided funding for the term and my supervisor Professor Rahim Moosa for his time and guidane and also Moshe Kamensky for some helpful disussions. 1. Definable Groupoids and Funtors A ategory C is a ertain 2-sorted struture with two sorts O and M denoting objets and arrows or maps respetively. There are two maps dom, od : M O representing domain and odomain of elements in M. If dom(f) = a and od(f) = b we write f : a b. There is a partial omposition : M dom,od M M : (f, g) f g, where M dom,od M = {(f, g) M M : dom(f) = od(g)} and an identity map Id : O M satisfying the assoiativity and identity axioms. We write ObC for the olletion of objets of C and MorC for the olletion of all maps in C or simply O and M if the ategory is lear. Also, Mor C (a, b) will be the olletion of maps with domain a and odomain b and C a = Mor C (a, a). We write Id a for the identity map on the objet a. Sometimes we omit the symbol when writing ompositions, so f g is written as fg. A groupoid G is a ategory with the property that for every f : a b there is g : b a suh that f g = Id b and g f = Id a. Suh a g is unique and we denote it f 1. A groupoid is onneted if eah set Mor G (a, b) is not empty. We will work with onneted groupoids. Let T be a theory with a universal domain U. Let Def(U) be the ategory whose objets are U-definable sets and U-definable maps between them. Note that definable by itself will mean -definable. A definable groupoid G (in U) is intuitively a groupoid whose sets of objets and morphisms are definable sets in U and the domain, odomain, omposition and identity maps are also definable. To be preise, we need two definable sets O, M and four definable maps (that 1

2 2 OLEG CHTERENTAL is, having definable graphs) dom, od : M O, : M dom,od M M and Id : O M. As well we need the groupoid axioms to hold for these maps. Note that in the above, the definability of M, dom and od means that M dom,od M is definable. By a definable family of definable subsets of some fixed sort, we mean a olletion of nonempty sets S a indexed by some definable set I of tuples a from U, and an L-formula φ(x, y) suh that φ(a, y) defines S a. Eah set S a is then {a}-definable and the set S = a I {a} S a is defined by (x I) φ(x, y). The projetion π : a I {a} S a I onto the first oordinate is a definable map and the S a s are anonially and definably isomorphi to the fibres, the {a} S a s, of this map. Conversely, if S and I are definable sets and π : S I is a surjetion and its graph (after swapping positions) {(b, a) : a I, b S, π(b) = a}, is defined by φ(x, y), then the fibre S a over a I is defined by φ(a, y). Thus definable families are just the fibres of a definable map. Note also that if F = {S a : a I} is a definable family of definable subsets then there is a definable equivalene relation on S suh that F is the set of -equivalene lasses. There are some definable families of definable sets at work in our definition of a definable groupoid G: (1) The family {G a : a O} is a definable family of definable subsets of M. Indeed if G M is defined by the formula (f M) (dom(f) = od(f)), then the G a s are the fibres of the definable map π : G O : f dom(f). (2) There is the definable map + : G π G G whih is the restrition of to G π G. This makes {G a : a O} into a definable family of definable groups. (3) More generally we have the definable map dom od : M O O. For eah (a, b) O O the fibre (dom od) 1 (a, b) = Mor G (a, b) is {a, b}-definable. Eah h Mor G (a, b) indues an {a, b, h}-definable isomorphism from G a to G b given by m hmh 1. Its graph is {(m, n) : m G a, n G b, n = hmh 1 }. This gives us a definable family of definable isomorphisms among the G a s.

3 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 3 In this definition the objets of a definable groupoid are elements of a definable set and the maps are also elements of a definable set. We will see that indeed any definable groupoid an be represented in suh a way that its objets are in fat definable sets and the maps are definable maps. First we need the notion of a definable funtor, from a definable groupoid G to Def(U). Assume that G is a definable groupoid in U. A definable funtor F : G Def(U) is a funtor where funx (1.1) X = a O{a} F (a) = {(a, d) : a O, d F (a)}, and funy (1.2)Y = {(a, b,, d, e) : a, b O, Mor G (a, b), (1.3) (1.4) d F (a), e F (b), F ()(d) = e} = {{(a, b, )} Graph(F ()) : a, b O, Mor G (a, b)}, are definable. Let s unpak this definition. For a O we have F (a) ObDef(U) so F (a) is a U-definable set. If the funtor is definable, the set X is definable and then the projetion π : X O, on the first oordinate is definable. Eah F (a) is the fibre π 1 (a) and is {a}- definable, so the F (a) s form a definable family of definable subsets. If F : G Def(U) is a definable funtor, then by identifying the elements of {F (G a ) : a O} (whih are morphisms in Def(U)) with their graphs, we get a definable family of definable groups suh that the natural ation of F (G a ) on F (a) is definable. This follows from the fat that Y in Equation 1.2 is definable. Here is an example, for any definable groupoid G. Consider the funtor F taking a to G a and f : a b to F (f) : G a G b : m f m f 1. One sees this is indeed a funtor. It is in fat a definable funtor. The set X is just and the set Y is {(a, d) O M : dom(d) = od(d) = a}, {(a, b,, d, e) O 2 M 3 : dom() = dom(d) = od(d) = a, od() = dom(e) = od(e) = b, d 1 = e}. Reall that a funtor is faithful if whenever f, g : a b are maps and F (f) = F (g) then f = g, that is, F indues an injetion Mor(a, b) Mor(F (a), F (b)) for all objets a, b Ob(G). The funtor in the above example is faithful if all the G a have trivial enter, sine if f, g : a b

4 4 OLEG CHTERENTAL and fmf 1 = gmg 1 for all m G a then (g 1 f)m = m(g 1 f) 1 for all m G a thus g 1 f = Id a. For any definable groupoid however, there is a faithful definable funtor into Def(U) as shown in [1]. Indeed one may take the funtor δ taking an objet a O to the {a}-definable set δ(a) = {g M : dom(f) = a}, and taking a morphism f : a b to the morphism δ(f) : δ(a) δ(b) : g gf 1. A onrete definable ategory is a pair (G, δ G ) where G is a definable ategory and δ G : G Def(U) is a faithful definable funtor. By the above paragraph, any definable groupoid an be made onrete with the appropriate funtor. 2. Two speial ases Let us now work out in detail [2, Example 1.1]. Let G be a definable groupoid. Note that if F : G Def(U) is a definable funtor, eah of the sets F (a) are definably isomorphi over parameters, sine G is onneted. However there is usually no anonial hoie of isomorphisms. The following examples onsider ases where there is a anonial hoie. eg1 Example 2.1. Suppose eah G a is trivial. Then for eah a, b O, Mor G (a, b) onsists of a unique morphism. In this ase if F : G Def(U) is a definable funtor, one an interpret without parameters a set S, definably isomorphi to eah F (a). Proof. If f, g Mor G (a, b) then g 1 f G a and sine it is trivial, g 1 f = Id a so f = g. Let F : G Def(U) be a definable funtor. The set S is onstruted as follows. Let E S = {(a, b, a, b ) X 2 : Mor G (a, a )F ()(b) = b }, and let S = X/E S where X is as in equation 1.1. One may hek that E S is in fat an equivalene relation. For example it is symmetri beause if there is a Mor G (a, a ) with F ()(b) = b then there is automatially 1 Mor G (a, a) with F ( 1 )(b ) = b by funtorality. The relation is definable sine it is given by (a, a,, b, b ) Y, where Y is the set desribed in equation 1.2. We are taking the quotient of a definable set by a definable equivalene relation and laiming it is definably isomorphi to eah F (a). The quotient is in U eq but using the disussion in the appendix we an pullbak to U to witness this isomorphism.

5 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 5 The definable set X = {(a, d) : a O, d F (a)} of equation 1.1 is a subset of some U k so extend E S to a definable equivalene relation on U k. This an be done by making all elements not in X to be equivalent. Then we wish to show that X/E S is definably isomorphi to F (q) for any fixed q O. Consider the map r : X/E S F (q) given by r([a, b]) = F ()(b) where Mor G (a, q) is the unique morphism from a to q. If (a, b) (a, b ) then F (d)(b) = b for the unique d Mor G (a, a ). Thus if Mor G (a, q) is the unique morphism from a to q, then d 1 is the unique morphism from a to q, so r([a, b ]) = F (d 1 )(b ) = F ()F (d 1 )(b ) = F ()(b) = r([a, b]), so r is well-defined as a funtion. It an be heked that r is indeed a bijetion. Now we hek that the funtion r f ES : X F (q) is definable, albeit with parameters, sine F (q) is only definable with parameters. This suffies by Proposition 6.2. Its graph is {(a, b, u) : (a, b) X, u F (q), (dom() = a od() = q F ()(b) = u)}. Now we work out [2, Example 1.2]. Example 2.2. If G is abelian, then the G a are all anonially isomorphi, and one an interpret without parameters a single group, isomorphi to all G a. Proof. Here an abelian groupoid is one in whih eah G a is abelian. Consider again the funtor F (a) = G a and F (f) : G a G b : m fmf 1. If g, h Mor G (a, b) then F (g) = F (h). Indeed we have h 1 g G a and by ommutativity we get (h 1 g)m = m(h 1 g) for all m G a whih expands to gmg 1 = hmh 1 for all m G a. This means that the groupoid H whih is the image of the funtor F has the property that any morphism set Mor H (G a, G b ) has a single element (the objets of H are the groups G a for a Ob G and the morphisms are the indued onjugation maps between these groups). Thus by Example 2.1 with the identity funtor Id on H, we get a set S = X/E S onstruted from H and we get an isomorphism r : S G q for any q ObG. Note that in this ase S onsists of pairs (G a, b) where b G a, under the equivalene (G a, b) (G a, b ) if F ()(b) = b for some element of Mor G (a, a ). We have r([g a, b]) = F ()(b) where is an element of Mor G (a, q). The map r = r q depends on q. The set S is anonial beause it has only one indued group struture from all the maps r q. That is, if you

6 6 OLEG CHTERENTAL pullbak the group struture of G q to S via r q and of G q to S via r q, then the two group multipliations on S agree. groupoid stable 3. Binding Groupoids We turn our attention to [2, Proposition 1.5]. A type definable set is the solution set of a partial type. In [2] this is also alled -definable. If the type is allowed to have potentially infinitely many but a small ardinality of variables we all the solution set -definable. By restriting to finitely many variables and inreasing the variables at eah stage, a -definable set an be seen as a projetive system of type definable sets. Let V be a olletion of sorts of U, losed under artesian produts. We work throughout in U eq and V eq when appropriate. We assume that V is stably embedded: If S is a sort of V eq and P is a U-definable subset of S n, then in fat P is definable with parameters from V. A definable set Q U is said to be internal to V if there is a tuple U suh that Q dl(v ). We then have the following proposition whih is [2, Proposition 1.5]. Proposition 3.1. Assume V is stably embedded in U and Q is a definable set of U that is internal to V. There exist onneted -definable groupoids G in U eq and G V in V eq, and relatively definable funtors F : G Def(U eq ) and F V : G V Def(V eq ) suh that ObG = ObG V { }, the groupoid G V is a full subgroupoid of G, the funtor F V is the restrition of F to G V, and F ( ) = Q. Moreover F (G ) with its ation on Q is type-definable and isomorphi to Aut(Q/V). First some remarks about the proposition. The group Aut(Q/V) is the group of permutations of Q that arise as restritions of automorphisms of U fixing V. Thus Aut(Q/V) = {f Q : f Aut(U/V)}. One may equivalently think of this as Aut(U/V)/Aut(U/(Q V)). Note that Aut(U/(Q V)) is indeed a normal subgroup of Aut(U/V). One onsequene of this theorem is that F (G ) = Aut(Q/V) is interpretable in U. Sine G is onneted, all the F (G a ) for a ObG V are definably isomorphi to F (G ) and hene Aut(Q/V), but not anonially. In partiular Aut(Q/V) is type definable without parameters in U eq but with parameters in V eq. Let us work out the proof of this proposition. The following lemmas will be needed. Lemma 3.2. If tp(a/v) = tp(b/v) then there exists σ Aut(U/V) suh that σ(a) = b.

7 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 7 ata Proof. This is essentially [3, Lemma 0.8] and is a onsequene of the stable embeddedness of V in U. Sine tp(a/v) = tp(b/v) we have tp(a) = tp(b). Thus by saturation there is γ Aut(U) with γ(a) = b. Note that extending the language by a onstant symbol a and interpreting it as a in U we get a new struture (U, a). Furthermore V is still stably embedded in (U, a), indeed any definable set D ontained in a sort from V is now definable possibly with the extra parameter a, and by the stable embeddedness of V in U it is still definable with parameters from V. Consider the automorphism γ V of V. By [4, Appendix, Lemma 1], we an extend this to an automorphism τ of (U, a), so τ(a) = a. Now let σ = γ τ 1. Then σ fixes V pointwise and σ(a) = b as required. Let be a tuple from U witnessing the internality of Q in V. Lemma 3.3. There is a tuple b V, a b-definable set Q b V eq and a b-definable bijetion f : Q Q b. Proof. First we show that there is a -definable set X V and a - definable surjetion g : X Q. Let a Q. Then there is a tuple d with elements from some sort S (depending on a) in V and φ a (x, y, ) an L-formula with parameter suh that {a} is defined by φ a (x, d, ). Consider the set X a = {d S : φ a (x, d, ) defines a singleton from Q}. This is nonempty as it ontains d. It is {}-definable by u(φ a (u, x, ) = u Q) u v((φ a (u, x, ) φ a (v, x, )) = u = v), where x is of the sort S. Consider the funtion g a : X a Q taking d to the singleton element in Q that is defined by φ a (x, d, ). One sees that it is also {}-definable and its image g a (X a ) is thus {}-definable. Now the set g a (X a ) ontains a, and so as a ranges over all elements in Q, we see that the olletion {g a (X a )} a Q is a over of Q. Consider the set {x Q x g a (X a )} a Q of formulas with parameter. This set is not realised by any tuple in U thus by saturation some finite subset is not realized. Hene there are a 1,..., a n Q suh that Q is overed by {g ai (X ai )} 1 i n. Eah X ai is ontained in a sort from V, we an take the produt of these sorts to get another sort from V (sine it is losed under artesian produts) and find a set X, in this new sort, that is definably isomorphi to i X a i. Note that to onstrut the produt and embed our sets into it, we need -definable elements whih may have to be named if

8 8 OLEG CHTERENTAL not already available. This set X is -definable and we get a -definable surjetion g : X Q by letting g Xai = g ai. From this g : X Q we onstrut the desired f. For a Q let X a X be the fibre of g over a. Then X a is a-definable. Let φ(x, d) with d U eq be a ode for X a. Sine X a V, by stable embeddability it is b-definable by some formula ψ(x, b) for b V. Now then d dl(b), as it is defined in the variable y by x (φ(x, y) ψ(x, b)), thus we have d V eq. Let S be the sort of V eq ontaining d. Sine d is also in dl(a), let ϕ(x, a, ) define {d}. Consider the following -definable subset of Q S: {(a, d ) : ϕ(x, a, ) defines {d } and φ(x, d ) is a ode for X a }. Note that for any a Q there is at most one possible d, sine it would have to be a ode with φ(x, d ) for X a. Thus the relation is atually a -definable funtion f a : U a C a for -definable sets U a Q and C a S ontaining a and d respetively. We an safely assume that it is surjetive and note that it is also injetive, for if f a (a ) = d = f a (a ) then φ(x, d ) is a ode for both X a and X a. The fibres are then equal and so a = a. Thus f a is a bijetion. Now the U a s over Q so as before we may assume that some finite olletion U a1,..., U an overs Q. Furthermore by taking appropriate boolean operations we may assume the U ai s are disjoint. We onsider the sorts that ontain the C ai s and take a produt of them and find a new sort S of V eq with C = i C a i S. Then by gluing the funtions f ai together we have a single -definable bijetion f : Q C. Sine C V eq is -definable, by stable embeddablility it is b-definable for some finite tuple b V and we are done. Now let f : Q Q b be as in the above lemma. Sine V is stably embedded, by [4, Appendix, Lemma 1] we have tp(/(dl() dl(v))) tp(/v). We have dl(v) = V eq. Thus by extending b, we may assume that b = dl() V eq so that imply (3.1) tp(/b) tp(/v) and b is an infinite tuple of elements of V eq. Note that the ardinality of b depends on the size of the language and so an be taken to be small. The original finite tuple b we will refer to as b. auto Remark 3.4. Note that if b V and U are tuples suh that tp(b) = tp(b ) then by saturation there is an automorphism σ of U suh that σ(b) = b and σ() =. Then the bijetion f : Q Q b

9 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 9 disussed earlier gives rise to a bijetion f : Q Q b through σ. If φ(x, y, b, ) defines f (x) = y then φ(x, y, b, ) defines f (x) = y. Thus for all x Q we have U = φ(x, f (x), b, ) and so U = φ(σ(x), σ(f (x)), b, ). This gives the identity σ Qb f = f σ Q = f σ() σ Q. Sometimes we refer to f without mentioning the impliit b if it is lear from the ontext. Let ObG V be the set of solutions of tp(b) and let ObG = ObG V { } where is any -definable element in U. Then ObG V is -definable sine it is just the set defined by the type in infinitely many variables determined by b and ObG is defined by {φ (x = ) : φ tp(b)}. We will first define the morphisms from to all b and use this to define the set of morphisms between arbitrary elements. For b ObG with b one should think of the set Mor G (, b ) as onsisting of those funtions f : Q Q b where tp(b) = tp(b ). To show that this is -definable first let M(, b ) = {(, b, ) : tp(b) = tp(b )}. Now this is a -definable set sine it is essentially the realizations of tp(b). We put an equivalene relation on M(, b ) by saying (, b, ) (, b, d ) if they indue the same funtions, that is if f = f d. Sine f is definable by a formula in b and, the equivalene is definable. Then we think of Mor G (, b ) as M(, b )/, a -definable set in U eq. The morphisms Mor G (b, ) are treated in muh the same way: as the set M(b, ) = {(b,, ) : tp(b) = tp(b )}, modulo the appropriate equivalene. We think of Mor G (b, ) as onsisting of the maps f 1 : Q b Q for tp(b) = tp(b ). Similarly we define Mor G (b, b ) for b, b. One should think of this as the set of maps of the form f f 1 where tp(b) = tp(b ) = tp(b ). We define it by first letting M(b, b ) = M(b, ) M(, b ). This, as essentially seen before, is a -definable set. Then we take the equivalene relation that says (b,,,, b, ) (b,, d,, b, d ) if f f 1 = f d f 1 d, whih is definable. In this way we again see Mor G (b, b ) as a -definable set in V eq. The omposition in G should be nothing more than a refletion of omposing the indued f s, f 1 f s or f f 1 s as regular funtions. For example onsider an element [(b,, )] of Mor G (b, ) and

10 10 OLEG CHTERENTAL [(, b, )] of Mor G (, b ). Then define [(, b, )] [(b,, )] = [(b,,,, b, )], whih is an element of Mor G (b, b ). That this is well-defined follows from the identities imposed by the equivalene relations. Indeed assume that [(b,, )] = [(b,, d )] and [(, b, )] = [(, b, d )], whih implies f = f d and f = f d. Then we expet that [(b,,,, b, )] = [(b,, d,, b, d )], whih is the ase as f f 1 = f d f 1 d. Let us now define Mor G (, ) and its omposition. One should essentially think of Mor G (, ) as the set of funtions of the form f 1 f. The definition of omposition in Mor G (, ) also serves as a template for defining omposition in the remaining ases in G. To define Mor G (, ) let M(, ) = b M(, b ) M(b, ), where b ranges over elements satisfying tp(b ) = tp(b). This is the set of tuples (, b,, b,, ) with tp(b) = tp(b ) = tp(b ). We mod out by the equivalene (, ) (d, d ) if f 1 f = f 1 d f d. Note that M(, ) is -definable as it onsists of pairs (b ), (b ) of realizations of tp(b). The equivalene, as before, is definable. Thus we get that Mor G (, ) is a -definable set in U eq. Let us onsider omposition in the ase of Mor G (, ). Define a ternary relation R on M(, ) as follows. Let if (, b, d, b,, d ), (, b,, b,, ), (, b, e, b,, e ) R, f 1 d f d f 1 f = f 1 e f e. This is not a binary operation on M(, ) sine there an be many andidates (e, e ) for the right hand side. However one we mod out by the equivalene relation it beomes a binary operation on Mor G (, ). To be speifi, modding out by the equivalene means to reate a ternary relation R/ on Mor G (, ) where a triple ([u], [v], [w]) is in R/ if (u, v, w) is in R. One an see that this is a well-defined ondition. Let us see how eah element of Mor G (, ) indues an element of Aut(Q/V). An element of Mor G (, ) is an equivalene lass [(, b,,, b, )],

11 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 11 representing a funtion of the form f 1 f where tp(b) = tp(b ) = tp(b ). Claim 3.5. The funtion h : Mor G (, ) Aut(Q/V) : [(, b,,, b, )] f 1 f, is a omposition preserving bijetion from Mor G (, ) to Aut(Q/V). Note that through this bijetion Mor G (, ) inherits the obvious ation on Q. The bijetion is essentially the restrition to Mor G of the as yet to be defined funtor F : G Def(U). Proof. First let us show that f 1 f Aut(Q/V). Equation 3.1 says that tp(/b) tp(/v) and sine tp(b) = tp(b ) = tp(b ) we get that tp( /V) = tp( /V). Then by Lemma 3.2 there is σ Aut(U/V) suh that σ( ) =. The last part of Remark 3.4 essentially gives us that σ Qb f = f σ Q. Note that σ extends to V eq in a unique way and sine it fixes V pointwise it fixes V eq and in partiular Q b pointwise. Thus we get f = f σ Q, from whih it follows that f 1 f Aut(Q/V). By the nature of the equivalene on M(, ), the map h is indeed a funtion and does not depend on the representative of the equivalene lass. Thus the map is well-defined. One an hek also that it preserves omposition. Also from the equivalene we see it is injetive. It remains only to show it is surjetive. If we have σ Aut(Q/V), then σ fixes V and V eq, thus we have σ Qb = Id so by Remark 3.4 we have f σ() σ Q = f. Hene, σ Q = f 1 σ() f, proving surjetivity of h. Now via h, we get an ation of Mor G (, ) on Q. Indeed for q Q and we have [(, b,,, b, )] q = f 1 f (q). This ation is definable. Indeed onsider the type definable subset R of M(, ) Q 2 ontaining tuples (, b,,, b, ), q, f 1 (f (q)), where tp(b) = tp(b ) = tp(b ) and q Q. Consider the equivalene relation on this set by taking from before on the first oordinate, equality on the seond and third oordinates. Then R/ gives the graph of the ation of Mor G (, ) on Q. The rest of the ases for defining omposition in G are handled in similar ways as presented above.

12 12 OLEG CHTERENTAL equiv nostru The funtor F : G Def(U) is defined by F ( ) = Q and F (b ) = Q b on objets and it takes any morphism to the orresponding funtion f, f 1 f or f f Generalized Imaginaries Let T be an extension of the omplete theory T in a language ontaining the language of T and one additional sort S. Every model M of T is of the form (M, S M ) where M is a model of T and S M is the domain of the sort S. Note that if M is a universal domain for T then is M one for T. Proposition 4.1. Assume that M = (M, S M ) is a model of T. The following are equivalent: (1) S M dl(m) (2) The restrition map Aut(M ) Aut(M) is an isomorphism. (3) S is a sort of M eq. First we need a lemma: Lemma 4.2. The set M is stably embedded in M iff M indues no new struture on M. That is, if X M n is definable in M (with parameters in M ), then it is definable in M (with parameters in M). Proof. Clearly if M indues no new struture then M is stably embedded. Assume that M is stably embedded. First we show that for any small A M and a M we have tp M (a/a) tp M (a/a). Indeed if b M is a realization of tp M (a/a) then there is an automorphism σ of M that fixes A pointwise and σ(a) = b. By [4, Appendix, Lemma 1] σ extends to an automorphism σ of M. Clearly then σ witnesses the fat that tp M (b/a) = tp M (a/a) as desired. Now we show that M indues no new struture on M. Assume X M n is M -definable via the formula φ(x, m). By stable embeddedness of M we may assume m M. Let Γ = {ψ(x, m) L(M) : M = x(φ(x, m) ψ(x, m))}, where ψ L(M) means ψ is a formula in the language of the sorts of M. Consider the set Γ(x, m) { φ(x, m)}. Assume it is realized by some tuple b M. We show that tp M (b/m) {φ(x, m)} has a realization. Assume not, then some finite subset has no realization. By taking onjuntions we may assume that ϕ(x, m) φ(x, m) has no realization for some ϕ tp M (b/m). Then M = x(φ(x, m) ϕ(x, m)). Thus ϕ Γ tp M (b/m). But this is a ontradition sine ϕ tp M (b/m).

13 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 13 Thus there must be a M suh that a realizes tp M (b/m) {φ(x, m)}. By the disussion at the beginning of the proof, we must have that a realizes tp M (b/m), but this inludes φ(x, m) whih ontradits the fat that a realizes φ(x, m). Thus Γ(x, m) { φ(x, m)} is not realized in M and by saturation some finite subset is not realized in M. Let Σ Γ be finite suh that Σ(x, m) { φ(x, m)} is not realized in M. Let ϕ(x, m) be the onjuntion of all formulas in Σ. Then M = x(ϕ(x, m) φ(x, m)), and based on the definition of Γ we see that in fat M = x(ϕ(x, m) φ(x, m)). Thus X is defined by ϕ and so M indues no new struture on M. Now we an proeed with the proof of Proposition 4.1. Proof. (1 = 2) Note that the restrition map is a group homomorphism. First we show it is one to one. Assume that σ M = Id M. Let a S M. By assumption there is a formula φ(x, m) with m M defining a. Then σ(a) is defined by φ(x, σ(m)) = φ(x, m) so σ(a) = a and σ = Id M. To show that the map is surjetive note that by [4, Appendix, Lemma 1] it is enough to show that M is stably embedded in M. Assume X M is defined by a formula φ(x, m, a) for m M and a S M. Then by assumption eah o-ordinate of a is defined by a formula with parameters from M. From this one sees that we may define X using only parameters from M and so M is stably embedded. (2 = 1) Sine the restrition map is surjetive, every automorphism of M extends to one of M, so by [4, Appendix, Lemma 1] M is stably embeddeded in M. Now let a S M. Again by [4, Appendix, Lemma 1] there is a small set M 0 M suh that tp(a/m 0 ) tp(a/m). If b is a realization of tp(a/m 0 ) then tp(b/m 0 ) = tp(a/m 0 ) and so tp(b/m) = tp(a/m). Thus by Lemma 3.2 there is an automorphism σ of M fixing M pointwise and σ(a) = b. But sine the restrition map is injetive, we must have that σ is the identity on M so a = b. Thus by saturation a dl(m 0 ) dl(m). (3 = 1) An element of S M is of the form a/e for some a M and E(x, y) a -definable equivalene relation on M. Then a/e is defined by the a-formula in y that says there is x M suh that E(x, a) and f E (x) = y where f E is the natural projetion M S M = M/E. (1 = 3) Using a similar argument to that in Lemma 3.3, we get a -definable surjetion f : X S M where X M n is a -definable set. The surjetion is in the language of M. The indued equivalene

14 14 OLEG CHTERENTAL fingen relation E on X is -definable and it is a subset of M 2n. Sine 1 = 2, the restrition map Aut(M ) Aut(M) is surjetive so M is stably embedded in M. By Lemma 4.2 E an be defined by a formula in the language of M. Thus the natural projetion f E : X X/E whih is defined entirely in M, has the same fibres as f and through this we may identify S M with X/E. The above result gives us some alternate haraterizations of a sort in M eq. As a generalization, say that the sort S is a finite generalized imaginary sort, if the restrition map Aut(M ) Aut(M) has finite kernel and is surjetive. Note that the kernel is the binding group of S. The following extends Proposition 4.1: Proposition 4.3. Assume that M is stably embedded. Then the following are equivalent: (1) S dl(am) where a al(m). (2) S M is a finite generalized imaginary sort. Proof. (1 = 2) We must show that the kernel of the restrition map is finite. Assume σ Aut M (M ). Note that sine S dl(am), the map σ depends entirely on its ation on a, that is, if τ Aut M (M ) and σ(a) = τ(a) then σ = τ. Sine a al(m), there only finitely hoies for σ(a) and thus only finitely many suh maps σ. (2 = 1) It suffies to find a tuple a M suh that S dl(am). Indeed by stable embeddability, tp(a/m) M is the orbit of a under Aut(S/M) whih is finite by (2). So tp(a/m) M is finite and by a stable embeddedness argument, a al(m). Assume there is a finite tuple a S suh that for every σ Aut(S/M) if σ(a) = a then σ = Id S. We show that then S dl(am). Indeed let b S. By [4, Appendix, Lemma 1] there is a small M 0 M suh that tp(a/m 0 ) tp(a/m) and tp(b/m 0 ) tp(b/m). We show that b dl(am 0 ). Indeed let f Aut am0 (M ). Sine f fixes am 0 we have tp(f(b)/am 0 ) = tp(b/am 0 ) and thus tp(f(b)/am) = tp(b/am). By a slight variant of the proof of Lemma 3.2 there is σ Aut am (M ) suh that σ(b) = f(b). But by our initial assumption sine σ fixes both a and M pointwise we must have σ(b) = b and so f(b) = b. To onstrut the tuple a, onsider an enumeration f 0,..., f n of Aut(S/M) with f 0 = Id S. For 1 i n let a i S be suh that f i (a i ) a i. Then a = a 1 a 2... a n satisfies our ondition sine if f Aut(S/M) is suh that f(a) = a then f an only be the identity. A finite generalized imaginary sort is a speial ase of the more general notion of an internal generalized imaginary sort. An internal generalized imaginary sort S is a sort where we require the kernel Aut(S/M)

15 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 15 of the restrition map Aut(M ) Aut(M) to have small ardinality. The following is an extension of Proposition 4.3 to internal generalized imaginaries based on [5, Proposition 15]. Proposition 4.4. Suppose M is stably embedded in M. Then the following are equivalent: (1) S is internal to M. (2) S is an internal generalized imaginary sort. (3) The binding group Aut(S/M) with its ation on S is definable in M. Proof. (1 = 3) This is the ontent of Proposition 3.1. (3 = 2) If the binding group is definable then it an be assumed small. (2 = 1) We proeed essentially as in the proof of Proposition 4.3 by finding a small set A suh that for every f Aut(S/M), if f fixes A pointwise then f S = Id S. Then we show that S dl(am) and we finish by showing that one an then find a finite tuple a A suh that S dl(am). Let {f α } α<κ be an enumeration of Aut(S/M) with f 0 = Id S. Choose as before a α S suh that f α (a α ) a α for 0 < α < κ. Let A be the set of all a α s. First we show that S dl(am). Let s S. Then by stable embeddedness there is a small M 0 M suh that tp(s/m 0 ) tp(s/m). We show that s dl(am 0 ). Assume f Aut AM0 (M ). Then tp(s/m 0 ) = tp(f(s)/m 0 ) so tp(s/m) = tp(f(s)/m). Now add all of A as onstant symbols to M. Then M is still stably embedded in this new struture. Sine f fixes A pointwise one sees that it is an automorphism of this new struture as well. Thus tp(s/m) = tp(f(s)/m) holds in the new struture. By stable embeddedness there is an automorphism σ of M fixing M and A pointwise suh that σ(s) = f(s). But by onstrution of A we must have that σ is the identity and so f(s) = σ(s) = s as required. Now using an argument similar to that in Lemma 3.3 we an find an A-definable surjetion from a power of M to S. Sine this formula will use only finitely many elements from A, we an find a finite tuple a A suh that S dl(am) as required. 5. Finite Internal Covers We are now in a position to define a finite internal over. Let N be a struture and M a union of some of the sorts in N. We say N is a finite internal over of M if M is stably embedded in N and Aut(N/M) is finite.

16 16 OLEG CHTERENTAL Assume G 1 and G 2 are onrete definable ategories in U. We have previously defined a definable funtor from a definable ategory to Def(U). A funtor F : G 1 G 2 between onrete definable ategories will be alled definable if the funtor δ 2 F : G 1 Def(U) is a definable funtor. redue 6. Appendix The following is used in Example 2.1. We need some fats about M eq given an L-struture M. Note that if φ( x) is an L-formula and ā M then M = φ(ā) iff M eq = φ(ā). We need also the following fat: Proposition 6.1. Let φ(x 1,..., x k ) be an L eq -formula where x i is of the sort S Ei. Then there is an L-formula ψ(y 1,..., y k ) suh that M eq = ψ(y 1,..., y k ) φ(f E1 (y 1 ),..., f Ek (y k )). Proof. The proof is by indution on the omplexity of φ. First we need to show the following laim about terms. If t(x 1,..., x k ) is an L eq -term with x i of sort S Ei and the term maps to the sort S E, then there is an L- term t (y 1,..., y k ) suh that t(f E1 (y 1 ),... f Ek (y k )) = f E (t (y 1,..., y k )). Clearly this holds if t is a onstant, sine then it is an L-term already. If t(x 1,..., x k ) = x i then take t (y 1,..., y k ) = y i. By assumption E = E i and so the result holds in this ase. We also must hek the funtion ase so assume g is an n-ary funtion symbol in L. Then its input variables eah have sort S = so onsider for 1 j n terms t j (x 1,..., x k ) whih map to the sort S = and x i is of sort S Ei. Then by indution there are L-terms t j(y 1,..., y k ) satisfying t j (f Ei (y 1 ),..., f Ek (y k )) = t j(y 1,..., y k ). Thus g(..., t j (f Ei (y 1 ),..., f Ek (y k )),...) = g(..., t j(y 1,..., y k ),...) = f = (g(..., t j(y 1,..., y k ),...)), so g(..., t j(y 1,..., y k ),...) is the desired L-term. Now we an proeed with the proof. We illustrate just the ase where φ is the equality of two L eq -terms: t 1 (x 1,..., x k ) = t 2 (x 1,..., x k ), where the terms are of sort S E and x i is of sort S Ei. Let t j(y 1,..., y k ) for j = 1, 2 be as desribed above. Then M eq = φ(f E1 (y 1 ),..., f Ek (y k )) iff f E (t 1(y 1,..., y k )) = f E (t 2(y 1,..., y k )), that is, iff and iff M = E(t 1(y 1,..., y k ), t 2(y 1,..., y k )), M eq = E(t 1(y 1,..., y k ), t 2(y 1,..., y k )),

17 NOTES ON GROUPOIDS, IMAGINARIES AND INTERNAL COVERS 17 pull where E(x, y) is the L-formula defining the equivalene relation E. Thus we an take ψ to be E(t 1(y 1,..., y k ), t 2(y 1,..., y k )). Suppose A M k and A M l are definable sets defined by φ( x) and φ (ū) respetively. Also let E( x, ȳ) and E (ū, v) be definable equivalene relations on M k and M l respetively. We have that A/E M k /E and A /E M l /E. Indeed A/E and A /E are definable sets in M eq defined by x(ȳ = f E ( x) φ( x)), and ū( v = f E (ū) φ (ū)), respetively in the variables ȳ and v. In these formulas x and ū are of the sort S =, that is to say eah of the k and l variables are of this sort. Note that the expression f E ( x) means that f E is applied to eah omponent of x. If we wish to show that some map between A/E and A /E is definable, we may do so in M eq but there is a ondition that one may hek stritly in M. Proposition 6.2. Let A, A be definable sets and E, E definable equivalene relations on them as above and assume g : A/E A /E is any map. Then g is a definable map in M eq iff there is a definable set S A A in M suh that (r, s) S iff g(r/e) = s/e. Proof. First assume that g is definable in M eq. Let ϕ( x 1, x 2 ) define g where x 1 has sort S E and x 2 has sort S E. Then by Proposition 6.1 there is an L-formula ψ(ȳ 1, ȳ 2 ) suh that M eq = ψ(ȳ 1, ȳ 2 ) ϕ(f E (ȳ 1 ), f E (ȳ 2 )). Then ψ(ȳ 1, ȳ 2 ) defines the appropriate set S A A. Conversely assume that S A A is definable and satisfies the ondition. Then we an define g by g(x) = y for x A/E and y A /E iff there is (r, s) S suh that x = r/e and y = s/e. As a speial ase, if E is equality and we wish to hek that g : A/E A is a definable isomorphism in M eq, we need only hek that the map g f E is definable in M and that g is bijetive. Referenes GK [1] J. Goodrik and A. Kolesnikov, Groupoids, Covers, and 3-Uniqueness in Stable Theories, goodrik/researh.html 31 Ot H [2] E. Hrushovski, Groupoids, Imaginaries and Internal Covers, arxiv: math/ v1 16 Mar H2 [3] E. Hrushovski, Notes for Model Theory 2, ehud/mt2/ CH [4] Z. Chatzidakis, E. Hrushovski, Model Theory of Differene Fields, Trans. Am. Math. So. 351 No. 8, , 8 April 1999.

18 18 OLEG CHTERENTAL K [5] M. Kamensky, Definable groups of partial automorphisms, arxiv: math/ v3 15 Mar M [6] R. Moosa, Some Elementary Fats About M eq.