Extending Hrushovski's groupoid-cover correspondence using simplicial groupoids
aa r X i v : . [ m a t h . L O ] S e p Extending Hrushovski’s groupoid-cover correspondenceusing simplicial groupoids
Paul Wang
ABSTRACT. Hrushovski’s suggestion, given in [”Groupoids, imaginaries and inter-nal covers,”
Turkish Journal of Mathematics , 2012], to capture the structure of the1-analysable covers of a theory T using simplicial groupoids definable in T is realizedhere. The ideas of Haykazyan and Moosa, found in [”Functoriality and uniformity inHrushovski’s groupoid-cover correspondence,” Annals of Pure and Applied Logic , 2018]are used, and extended, to define an equivalence of categories. Finally, a couple ofexamples are studied with these new tools.
Contents
Acknowledgements
Most of the work presented here was done under the supervision of Martin Bays. Hishelp was invaluable in correcting and improving ideas, as well as for suggesting directionsof research.I would also like to thank Martin Hils, for pointing out a mistake which led to abetter understanding of one of the examples presented here.1ther improvements, such as the section on type-definable groupoids, were suggestedby Rahim Moosa, who is also one of the authors of a paper which inspired the presentwork.Finally, thanks to the Ecole Normale Superieure for the great teachers, and class-mates, I have, as well as for the housing and funding provided, which represent a much-appreciated help in my studies.
When studying many-sorted first-order structures, one may pick a structure U , and adda new sort S . If this does not create new definable sets in U , the extended structureis called a cover of U . Under an additional hypothesis of internality, it was shown byHrushovski in [Hru12] that covers came with definable binding groupoids , which are ageneralized version of automorphism groups. Up to a suitable notion of equivalence, theinternal covers of a fixed structure U are characterized by their binding groupoids.This correspondence between internal covers and definable groupoids was then ex-tended in [HM18], in two ways : First, it was shown to come from an equivalence ofcategories. Then, it was extended to the case of fibrations of internal covers, or 1-analysable covers, under a (rather strong) hypothesis of independence.In this paper, we pursue the work done in [HM18]. The aim is to extend the corre-spondence by dropping the hypothesis of independence found in [HM18].First, we recall some results regarding the internal and independent cases. We thenwrite down the details of a proof left to the reader at the end of [HM18]. The aim is tofind ideas that can later on be extended to the more general case at hand.Then, sections 4 and 5 are aimed at realizing Hrushovski’s suggestion to deal withnon-independent 1-analysable covers by using simplicial groupoids instead of groupoids.Several new ideas appear, the most crucial one being that of coherence of morphisms ina simplicial groupoid.Then, two examples of non-internal and non-independent 1-analysable covers arestudied, using the new tools given by simplicial groupoids. The aim is to find explicitdescriptions of the binding simplicial groupoids of these covers, and deduce a few results.Finally, in section 7, we drop yet another hypothesis, of finite generatedness of thelanguage, and aim to generalize the correspondence again, using type-definable simplicialgroupoids. In this section, our goal is to recall some results in the internal and independent cases.We also use this opportunity to give more detailed proofs that the structures defined in[HM18](Definition 3.9 and Remark 3.10) are indeed covers. The methods and ideas weuse here will be generalized in the later sections.2 .1 General definitions and context
From now on, U denotes a saturated model of a theory T that admits elimination ofimaginaries.We will assume that all the theories we study have saturated models of the samecardinal κ . Definition 2.1.
Let T ′ be an extension of T , possibly with additional sorts. Let U ′ bea saturated model of T ′ , and U := U ′ | T . We say that U is stably embedded in U ′ whenthe map Aut( U ′ ) → Aut( U ) is surjective.In this case, we also say that T ′ is a cover of T , or that U ′ is a cover of U .If there exist definable bijections between the new sorts and definable sets of U , thecover is internal . Proposition 2.2.
Let T ′ be an extension of T , possibly with additional sorts. Let U ′ bea saturated model of T ′ , and U := U ′ | T . The following are equivalent :1. U is stably embedded in U ′ .2. Any subset of a product of sorts of U that is definable with parameters from U ′ isdefinable with parameters from U .3. For all tuples b ⊆ U ′ of small length, there exists a small subset A of U such that tp ( b/A ) | = tp ( b/ U ) . See [CH99] (Appendix, Lemma 1) for a proof of the equivalences.
Definition 2.3. A category is a two-sorted structure, with a set of objects, a set ofmaps with source and target, an associative composition, and an identity map for eachobject.The following definitions are the same as those in [HM18]. For the full formal defi-nitions, we refer the reader to this paper. Definition 2.4.
1. A groupoid is a category where each morphism has an inverse.It is connected if two arbitrary objects are isomorphic. It is canonical if twoisomorphic objects are equal.2. A definable (concrete) groupoid is given by a definable set of definable objects( O i ) i ∈ I , and a definable set M of definable bijections between these objects. See[HM18](Definition and Remark 3.4).We require that the bijections define a category which is a groupoid. We alsoassume the objects to be pairwise disjoint.3. A concrete groupoid G is finitely faithful if, for each object O of the groupoid G ,there exists a finite tuple x of elements of O whose pointwise stabilizer in the groupAut G ′ ( O ) is trivial. 3. If A is a definable set, a definable groupoid over A is a uniformly definablefamily ( G a ) a ∈ A of definable connected groupoids that are pairwise disjoint. See[HM18](Definition 5.1) Definition 2.5.
1. Let ( U , S ) be a cover of U , and A ⊆ U be a 0-definable set. Thecover is if there exists a 0-definable surjective map f : S → A such that each fiber S a is internal to U . In other words, f and S define a familyof internal covers ( U , S a ).2. If, additionally, we have Aut( S/ U ) = Q a Aut( S a / U ), we say that the fibers areindependent.3. Let ( U , S ) be a cover of U , with a 0-definable relation R ⊆ B × S , where B is0-definable in U . If b ∈ B , we define R b ⊆ S as the set of elements s ∈ S such that | = R ( b, s ). We also define the structure ( U , R b ) as that induced by the b -definablesets.If there exist finitely many relations R ( x , y ) , ..., R k ( x k , y ) that are 0-definable in( U , S ) and such that, for each element b in B , the relations R i ( x i , b ) generate thelanguage of ( U , R b ) over that of U , then we say that the language of ( U , R b ) is finitely generated over that of U , uniformly for b ∈ B .A special case of this definition is that where B = A = n , and there exists a 0-definable map f : S ։ A such that R ( b, s ) is defined as ” f ( s ) is some coordinateof b ”. Remark 2.6.
Note that the condition ”the language of ( U , S a ) is finitely generated overthat of U , uniformly in a ∈ A ” is not the same as finite generatedness of the languageof the whole cover ( U , S ) over that of U .The second example in section 6 can be rearranged to build a cover satisfying thelatter condition, but not the former. Example 2.7. If A, K are groups that are 0-definable in U , and S is a group, with a0-definable exact sequence 1 → K → S → A →
1, then ( U , S ) is 1-analysable over A ,with non-independent fibers in general. We first recall the existence of the binding groupoid of an internal cover :
Theorem 2.8 ([Hru12], [HM18]) . Let U ′ = ( U , S ) be an internal cover of U whoselanguage is finitely generated over that of U . Then there exist 0-definable connectedgroupoids G in U and G ′ in U ′ eq , satisfying the following conditions :1. The groupoid G ′ is an extension of G , with S as the only added object.2. The automorphism group of S over U satisfies Aut U ′ ( S/ U ) = Aut G ′ ( S ) .
4e give the same proof as in [HM18](Theorem 3.5), because one idea that appearsin said proof will be used later on.
Proof.
Let us first pick a definable bijection f b : O a → S , where a ∈ U and b ∈ U ′ .We wish to use f and O to define a groupoid. However, we need to restrict the set ofparameters that are allowed, in order to satisfy condition 2.Let q := tp ( b/ U ) . By stable embeddedness, there exists a small A ⊆ U such that p := tp ( b/A ) | = q . If c, d | = p , then c ≡ U d . In particular, f c , f d are bijections from O a to S , thus the map ( f d ◦ f − c ) ∪ id U is a well-defined automorphism of U ′ over U .We have thus proved : p ( x ) ∪ p ( y ) | =”( f y ◦ f − x ) ∪ id U ∈ Aut( U ′ / U )”.By compactness, we find a formula ψ ( z, α ) ∈ p such that ψ ( x, α ) ∧ ψ ( y, α ) | = ”( f y ◦ f − x ) ∪ id U ∈ Aut( U ′ / U )”. We may assume α = a . The following formula enables usto restrict the set of objects : θ ( t ) := ∀ x, y [ ψ ( x, t ) ∧ ψ ( y, t ) → ”( f y ◦ f − x ) ∪ id U ∈ Aut( U ′ / U )”]The objects of G ′ are the O a ′ , where | = θ ( a ′ ), along with S . The morphisms from O a ′ to S are the f b ′ , where | = ψ ( b ′ , a ′ ). The other morphisms are defined using compositionsand inverses. O a O a S SS O af − b ◦ f b f b f b ′ ◦ f − b f − b f − b f b ′ Let’s check that
Hom ( O a ′ , S ) ◦ Hom ( S, O a ′ ) = Aut( S/ U ), for any a ′ satisfying θ ( t ).The properties of θ and ψ imply that Hom ( O a ′ , S ) ◦ Hom ( S, O a ′ ) ⊆ Aut( S/ U ).Conversely, if σ ∈ Aut( S/ U ) , then, for any f b ′ ∈ Hom ( O a ′ , S ), we have σ = f σ ( b ′ ) ◦ f − b ′ ∈ Hom ( O a ′ , S ) ◦ Hom ( S, O a ′ ).Finally, by stable embeddedness, the groupoid G := G ′ | U is 0-definable in U . Remark 2.9.
One of the main ideas in this theorem is to possibly add many objects that classify the various ”kinds of bijections” between S and the O a , to ensure that twobijections f b and f b ′ of the ”same kind” yield a model-theoretic automorphism f b ′ ◦ f − b .Let G be a 0-definable connected groupoid in U . For the remainder of this subsec-tion, we let U ′ = ( U , O ∗ , M ∗ , R ) denote the extension of U built from G as defined in[HM18](3.9). Proposition 2.10.
The structure U ′ is interpretable in U and thus saturated.Proof. Using constants to create duplicates, we are able to follow the construction in[HM18](3.9) and interpret the whole structure of U ′ in U . For instance, an element x inthe sorts of U is interpreted by x × { } , whereas an element in O ∗ is interpreted by anelement of U × { } , the first coordinate itself being given by the construction in [HM18].Similarly, a morphism in M ∗ is interpreted by an element of U × { } . The new relation R is interpreted the obvious way, again following the construction of U ′ .5 roposition 2.11. The structure U ′ yields an internal cover of U .Proof. From the construction, it follows that the new sorts are definably isomorphic todefinable sets in U . In fact, all the morphisms in the extended groupoid G ′ with source O ∗ yield definable bijections with O ∗ . As for the set of morphisms M ∗ , it is almostin bijection with the set of morphisms in G . Using elimination of imaginaries, one canadd the missing copies and get a definable subset X of a product of sorts in U , with adefinable bijection X ≃ M ∗ . Thus, since U ′ is saturated, it remains to show that no newstructure is induced on the sorts of U . To do this, we shall prove that automorphismsof U can be extended to automorphisms of U ′ .Let σ ∈ Aut( U ). Keeping the notations in [HM18], let O i be the object whose copyis O ∗ . Let f c : O i → O σ ( i ) be a morphism in the original groupoid. We shall define anaction on the elements of O ∗ and on the morphisms in M ∗ .If x ′ is an element of O ∗ , let x be the element of O i whose copy is x ′ . Then, we shalldefine σ ( x ′ ) := ( f − c σ ( x )) ′ , i.e., it is the copy in O ∗ of the image of x under f − c ◦ σ .Now, if m is a morphism in M ∗ , we shall distinguish different cases, depending onits domain and codomain : • If neither the domain nor the codomain of m is O ∗ , then we let σ act on m as itdid in U . • If m : O ∗ → O j for any j in U , let n : O i → O j be the copy of m . Then σ ( n ) ◦ f c : O i → O σ ( j ) . Therefore, let σ ( m ) : O ∗ → O σ ( j ) be the copy of σ ( n ) ◦ f c . • Similarly, if m : O j → O ∗ for any j in U , let σ ( m ) : O σ ( j ) → O ∗ be the copy of f − c ◦ σ ( n ), using the same notations as before. • Finally, if m : O ∗ → O ∗ , let σ ( m ) : O ∗ → O ∗ be the copy of f − c ◦ σ ( n ) ◦ f c .Now, studying the various cases, we can show that this extension of σ preserves therelation R that defines the behaviour of the bijections in M ∗ . For instance, using thesame notations as before, let m : O ∗ → O j , x ′ ∈ O ∗ and y = m ( x ′ ) ∈ O j . We compute σ ( m )( σ ( x ′ )) = σ ( n ) ◦ f c ( σ ( x ′ )) = σ ( n ) ◦ f c ( f − c ◦ σ ( x )) = σ ( n )( σ ( x )) = σ ( n ( x )) , where x ∈ O i is the copy of x ′ . Since we had assumed that m ( x ′ ) = n ( x ) = y , we havethe equality σ ( m )( σ ( x ′ )) = σ ( m ( x ′ )).Finally, it is not hard to see what the inverse of σ should be. For the action on O ∗ ,instead of applying σ and then f − c , we should apply f c and then σ − . The action onthe morphisms is defined in an analogous way, using f − c and σ − . Definition 2.12.
1. Let A be a 0-definable set. We define the notion of A just as in [HM18] (Definition 5.1). We shallalso refer to these groupoids as groupoids over A .2. If G is a groupoid over A , and a is an element of A , we let G a denote the connectedcomponent of G over a . It is an a -definable connected groupoid.6 onstruction 2.13. Let
G → A be a 0-definable family of connected groupoids in U .Let U ′ be the structure defined in [HM18](5.6). As in the case of a connected groupoid, itis defined by choosing one object in each isomorphism class of G , and creating appropriatecopies to yield a one-object extension of each isomorphism class. Proposition 2.14.
The theory of U ′ is a cover of the theory of U .Proof. Let V ′ = ( V , S, M ∗ , M ∗ → A, S → A, R ) be a saturated model of the theory of U ′ ,such that V is elementarily equivalent to U . We shall prove that V is stably embeddedin V ′ using automorphisms.Let σ be an automorphism of V . For each a in A , we pick an index i a ∈ I a and amorphism f a : S a → O i a .We also pick a morphism m a : O i σ ( a ) → O σ ( i a ) . Such an m a exists because σ ( i a ) isin I σ ( a ) , for σ is an automorphism of V , and because the connected components of G aregiven by the map I → A .Now we shall define the action of σ on S and M ∗ .On S a , the function σ shall act as the function f − σ ( a ) m − a σf a : S a → S σ ( a ) .For the action on M ∗ , the point is to use the f a to get a morphism in G , then apply σ to it, then use morphisms again to get the appropriate domain and codomain. • If m : O j → S a , we define σ ( m ) = f − σ ( a ) m − a σ ( f a m ) : O σ ( j ) → S σ ( a ) . • If m : S a → O j , we define σ ( m ) = σ ( mf − a ) m a f σ ( a ) : S σ ( a ) → O σ ( j ) . • Finally, if m : S a → S a , we define σ ( m ) = f − σ ( a ) m − a σ ( f a mf − a ) m a f σ ( a ) : S σ ( a ) → S σ ( a ) .We can then check that σ preserves the relation R , and commutes with the maps M ∗ → A , S → A . Remark 2.15.
In the previous case, with only one isomorphism class, the morphism m a was given by the function defining O ∗ as a copy of O i . The morphism n a was givenby the function f − c . In this section, we write down the details of the proof of the equivalence of categoriesthat were left to the reader at the end of section 5 of [HM18].We find that all the ideas of the internal case extend naturally to the independentcase. In fact, most of the time, dealing with independent fibers amounts to dealing witheach fiber individually, without having to worry about coherence.7 .1 Morphisms of groupoids
Here, we define what morphisms of groupoids are. Our definition is almost the same asthat of [HM18](4.2), with only a small change to the condition (B) found there.
Definition 3.1.
Let G , G be 0-definable groupoids in U . A definable morphism H isa non-empty 0-definable family of definable functions between objects of G and objectsof G , as defined in [HM18](4.2), that satisfies the following conditions :(A) : For each pair of maps h p : O ,i → O ,j , h m : O ,k → O ,l in H , the sets of maps h p ◦ Hom G ( O ,k , O ,i ) and Hom G ( O ,l , O ,j ) ◦ h m are equal.(B’) : The set of maps H is stable under precomposition with morphisms of G , andunder postcomposition with morphisms of G .We recall the condition (B) given in [HM18](4.2) : (B) : The set of maps H is stableunder postcomposition with morphisms of G .Note that the remark found at the end of the definition in [HM18](4.2) is incorrect: If we are given a groupoid morphism H : G → G satisfying conditions (A) and (B’),and if we pick a nonempty proper 0-definable subset X of the set of objects of G , wecan restrict H to the set of functions whose domain is one of the objects in X . We geta nonempty set of functions satisfying (A) and (B), but it wouldn’t satisfy (B’).We now prove that isomorphic connected groupoids are equivalent in the sense of[Hru12] (Section 3), which was not proved in full generality in [HM18]. Proposition 3.2.
Let H : G → G be an isomorphism between 0-definable connectedgroupoids. Then there exists a 0-definable connected groupoid G , whose set of objects isthe disjoint union of the objects of the G i , and 0-definable groupoid embeddings f i : G i →G .Proof. The set of objects of G is defined as the disjoint union of Ob ( G ) and Ob ( G ).The morphisms are defined as follows : • Between objects of G , the morphisms are copies of morphisms of G . • Between objects of G , the morphisms are copies of morphisms of G . • If O is an object of G and O is an object of G , the set of morphisms from O to O is the set of maps in the morphism H that are bijections from O to O . • Conversely, the set of maps from O to O is the set of inverses of maps in H thatare bijections from O to O .Composition is defined as expected, using composition of maps.Using condition (A), we check that if h : O → O and g : O ′ → O are morphisms in G , then h ◦ g is a (copy of a) morphism in G , and thus is a morphism of G . Symmetrically,if h : O → O and g : O → O ′ are morphisms in G , then g ◦ h is a (copy of a) morphismin G . 8ow, condition (B) shows that we may postcompose morphisms between objects of G and objects of G with morphisms in G .Using conditions (A) and (B’), we show that we can also precompose with morphismsin G : Let h : O → O be a function in H , let α : O ′ → O be a morphism in G . Let h ′ ∈ H be a map from O ′ to O . Such a map exists by condition (B’). Using condition(A), there exists a morphism β : O → O in the groupoid G such that β ◦ h ′ = h ◦ α .By condition (B’), the function β ◦ h ′ is still in H , and thus so is h ◦ α .We can also precompose inverses of maps in H with maps in G and postcomposeinverses of maps in H with maps in G .Finally, this groupoid G is 0-definable, since both the G i and H are. The embeddings f i : G i → G are the obvious ones.Now, if two groupoids are not connected, we want to specify which connected com-ponents are ”linked together” by the morphisms of groupoids between them. Definition 3.3.
Let
A, B be 0-definable sets. Let R ⊆ A × B be a definable relation. Let G , G be groupoids over A and B respectively. A morphism of groupoids H : G → G is said to be compatible with the relation R if, for all a in A , for all b in B , there existsa map in H from some object of G ,a to some object of G ,b if and only if | = R ( a, b ).If A = B , unless otherwise stated, we shall consider only morphisms of groupoidsthat are compatible with the equality relation.In the rest of this section, we let FCG A denote the category of finitely faithful 0-definable connected groupoids over A , whose morphisms are defined as above. It isindeed a category.On the other hand, let ACIF A be the category of 1-analysable covers over A withindependent fibers and finitely generated languages, defined just as in [HM18](betweenProposition 5.7 and Example 5.8). The morphisms are morphisms of covers compatiblewith the maps S → A . The same arguments as in [HM18](4.1) show that compositionof morphisms is well-defined and yields a category. In fact, the proof there deals withthe more general case of covers with one extra sort, so it can be applied to our specificcase of 1-analysable ones. C : FCG A → ACIF A Definition 3.4.
Let h : G → G be a morphism of groupoids. Let G ′ and G ′ be theextensions of the G i defined in [HM18](5.6). For each a in A , let h a : O ,i a → O ,j a beone of the maps in the morphism h , and let f a : S ,a → O ,i a , g a : O ,j a → S ,a bemorphisms in G ′ and G ′ respectively.The morphism C( h ) is defined as the theory of (C( G ) ∪ C( G ) , S a g a ◦ h a ◦ f a ). Proposition 3.5.
The theory of C( h ) depends only on the morphism h , and not onthe choices made. More precisely, different choices of maps lead to structures that areisomorphic over U . roof. Let f ′ a , g ′ a , h ′ a , i ′ a , j ′ a be other possible choices of maps for the index a . Using theconnectedness of the groupoid, and preservation of the set of morphisms in h under preand post composition by groupoid morphisms, we may assume that only h ′ a is differentfrom h a , all else being equal. This is the same as assuming that the following diagramcommutes, and replacing h ′ a with h ′′ a .S ,a O ,i a O ,j a S ,a O ,i ′ a O ,j ′ a f a f ′ a h ′′ a g a h ′ a g ′ a Now, using property ( A ) of the morphism h , we get : h ′′ a ◦ σ a = h a , for some σ a ∈ Aut G ( O ,i a ).Thus, we have f − a ◦ σ a ◦ f a ∈ Aut G ( S ,a ) = Aut( S ,a / U ) .By independence of the fibers, we can then construct an automorphism ρ of thestructure C( G ) ∪ C( G ) that fixes U pointwise, given by the union of the f − a ◦ σ a ◦ f a and id C( G ) . Finally, we check that g a h ′′ a f a ρ = g a h ′′ a f a f − a σ a f a = g a h ′′ a σ a f a = g a h a f a = ρg a h a f a . Thus, the structures given by h a , f a , g a and h ′′ a , f a , g a are isomorphic over U . Lemma 3.6.
The theory of C( h ) is a cover of the theory of U .Proof. We assume C( h ) to be saturated. We shall prove that any automorphism of U canbe lifted into an automorphism of C( h ). Let σ ∈ Aut( U ) . Let σ ′ ∈ Aut(C( G ) ∪ C( G ))be an extension of σ .Now, we shall use the proof of the previous proposition : we notice that the maps σ ′ ( f a ) , σ ′ ( g a ) , σ ( h a ) represent another possible choice for the construction of C( h ). In-deed, since σ ′ is an element of Aut(C( G ) ∪ C( G )), it preserves the structure of the0-definable groupoids G ′ and G ′ , and the 0-definable set of maps given by h . Therefore,there exists some automorphism τ of C( G ) ∪ C( G ) fixing U pointwise such that, for all a in A , we have τ σ ′ ( g a ) σ ( h a ) σ ′ ( f a ) = g σ ( a ) h σ ( a ) f σ ( a ) τ . Let ρ = τ σ ′ . We check that this ρ commutes with the map S a ∈ A g a h a f a . Let a be an element of A . We compute : ρg a h a f a = τ σ ′ g a h a f a = τ ◦ [ σ ′ ( g a h a f a )] ◦ σ ′ = g σ ( a ) h σ ( a ) f σ ( a ) τ σ ′ = g σ ( a ) h σ ( a ) f σ ( a ) ρ. Finally, since τ fixes U pointwise, the function ρ extends σ . Proposition 3.7.
The theory of C ( h ) is a cover of both C( G ) and C( G ) .Proof. Let σ be an automorphism of C( G ). By the previous lemma, we may assumethat σ fixes U pointwise. We need to define the action of σ on each of the S ,a . Notethat since σ fixes U pointwise, an extension of σ should fix each fiber S ,a setwise.Now, let a be an element of A . We notice that the map f a σ | S ,a f − a is a morphismin G . Thus, by condition (A) of the morphism h , there exists some morphism γ in G such that h a f a σ | S ,a f − a = γh a . Now, since G ′ , is a groupoid, there exists a morphism σ ,a ∈ Aut G ′ ( S ,a ) such that g a γ = σ ,a g a . 10 ,a O ,i O ,j S ,a S ,a O ,i O ,j S ,af a h a g a σ f − a h a ∃ γ g a ∃ σ ,a Finally, we have g a h a f a σ | S ,a = g a γh a f a = σ ,a g a h a f a . We then use the family ofthe σ ,a and the independence of the fibers in C( G ) to define an automorphism σ ofC( G ) that fixes U pointwise and which agrees with the σ ,a . Thus, the map σ ∪ σ isan automorphism of (C( G ) ∪ C( G ) , S a g a ◦ h a ◦ f a ) that extends σ .Similarly, C( G ) is stably embedded in (C( G ) ∪ C( G ) , S a g a ◦ h a ◦ f a ). Proposition 3.8.
The action of C on the morphisms defines a functor C : FCG A → ACIF A .Proof. We need to show that C sends identity morphisms of groupoids to identity mor-phisms of covers, and that it preserves composition.For the first part, using proposition 3.5, we may assume that each h a is the identitymorphism of some object O a in the groupoid G . We may also assume that the maps g a and f a are inverses of one another. Thus, the map S a g a ◦ h a ◦ f a is the identity of thenew sort S . Therefore, the cover C ( id G ) is indeed the identity morphism of C ( G ).Regarding composition, we use proposition 3.5 again. Let h : G → G , h : G → G be morphisms. Let h : G → G be the morphism h ◦ h . Then we mayassume that the morphisms picked to define C ( h ) and C ( h ) are consistent with theones picked to define C ( h ). More precisely, we may assume that, for each a ∈ A , themorphisms g a , h a and f a were picked to get the equality : g a h a f a g a h a f a = g a h a f a Indeed, the map f a g a is a morphism in the groupoid G . Therefore, by property(B) of the groupoid morphism h , the map f a g a h a is again a map in the morphism h . Thus, the map h a f a g a h a is a map of the groupoid morphism h ◦ h = h . S ,a O ,i a O ,j a S ,a O ,k a O ,l a S ,a O ,i a O ,l a f a f a h a g a f a h a g a id h a id g a By proposition 3.5, we may assume that h a = h a f a g a h a and f a = f a and g a = g a . In such a case, the desired equality holds, i.e. the above diagram commutes.Thus, from the definition of the composition of morphisms of covers, we get C ( h ◦ h ) = C ( h ) ◦ C ( h ) . .3 The functor G : ACIF A → FCG A Definition 3.9.
Let ( U ∪ U , h : S → S ) be a morphism in the category ACIF A .We assume the structures to be saturated. Let G , G be the liaison groupoids G ( U ), G ( U ). Let G ′ , G ′ be the extensions of these groupoids that are 0-definable in U eq and U eq respectively. The groupoid morphism G ( h ) is defined as the set of mor-phisms H := { g a h a f a : O ,i a → O ,j a | a ∈ A, i a ∈ Ob ( G ,a ) , j a ∈ Ob ( G ,a ) , f a ∈ Hom G ′ ( O ,i a , S ,a ) , g a ∈ Hom G ′ ( S ,a , O ,j a ) } .This set H of morphisms is a 0-definable morphism of groupoids over A in ( U ∪ U , h : S → S ) eq . We wish to get a set of morphisms 0-definable in U with a 0-definable mapto A . We shall use a compactness argument similar to the one in [HM18](5.6, § Lemma 3.10.
Let χ be a formula defining in ( U ∪ U , h : S → S ) eq the set of (codesof ) maps described above, and let φ ( x, y, t ) define the action of those maps. Then thereexists a 0-definable set N ⊆ U and a formula ψ ( x, y, u ) in the language of U such thatthe set of maps defined by the ψ ( x, y, n ) , n ∈ N is equal to the set of maps defined by the φ ( x, y, t ) , t ∈ χ ( U ∪ U ) . Remark 3.11.
We only need the sets of maps to be equal. There may be redundantparameters, but we shall use elimination of imaginaries in U to deal with them afterwards. Proof.
By stable embeddedness, for each parameter t satisfying χ , there exists a for-mula ψ t ( x, y, z ) in the language of U and a parameter z t in the sorts of U such that | = ψ t ( x, y, z t ) ↔ φ ( x, y, t ). Thus, the following set of formulas is inconsistent with thetheory of ( U ∪ U , h ) eq : { χ ( s ) } ∪ {¬ ( ∃ z ∀ x ∀ y ψ t ( x, y, z ) ↔ φ ( x, y, s )) | t ∈ χ ( U ∪ U ) } .By compactness, some finite fragment Σ is inconsistent. Let ψ ( x, y, z ) , ..., ψ d ( x, y, z d )be the formulas appearing as subformulas in Σ . We first reduce to the case where all the z i are in the same product of sorts of U . Let’sassume z i lives in the product of sorts P i , for i = 1 , ..., d. By elimination of imaginaries,there exists a sort Y in U with 0-definable constants 1 , , ..., d . We then work in theproduct of sorts Y × P × ... × P d , and modify the formulas ψ i to accept the parametersonly when the first coordinate is equal to i , in which case the parameter is used byprojecting onto P i . Let ψ ( x, y, u ) be the disjunction of the new formulas ψ i .Finally, the set N is the set of parameters n ∈ Y × P × ... × P d such that the formula ψ ( x, y, n ) defines one of the maps in H . Thus, N is 0-definable in ( U ∪ U , h ) eq . Bystable embeddedness and saturation, it is 0-definable in U .Now, to get a 0-definable map to A , we only need to replace N with { ( a, n ) | n ∈ N ∧ ∃ i ∈ Ob ( G ,a ) ∃ j ∈ Ob ( G ,a ) n : O i ≃ O j } . Proposition 3.12.
The set of maps G ( h ) depends only on the theory of ( U ∪ U , h ) ,and is a morphism in FCG A .Proof. Let h ′ : S → S such that ( U ∪ U , h ) ≡ ( U ∪ U , h ′ ). By saturation, thereexists an isomorphism σ : ( U ∪ U , h ) ≃ ( U ∪ U , h ′ ). By stable embeddedness of12 in ( U ∪ U , h ), we may assume σ U = id . Thus, if τ = σ | U ∈ Aut( U / U ) = Q a Aut G ′ , a (S , a ), we get τ h = h ′ . Thus the families of maps induced by h and h ′ areequal.We now need to check conditions (A) and (B’). The latter is easier to see, sincepostcomposing a map in Hom G ′ ( S ,a , O ,j a ) ◦ h a ◦ Hom G ′ ( O ,i a , S ,a ) by a morphism in Hom G ( O ,j a , O ,k a ) yields a map in Hom G ′ ( S ,a , O ,k a ) ◦ h a ◦ Hom G ′ ( O ,i a , S ,a ) . Similararguments work for precomposition.Let us now check condition (A). Let α , β be morphisms in G ′ with target S ,a , let α , β be morphisms in G ′ with source S ,a . We shall prove that, for each morphism α in G such that the composition α h a α α is well-defined and has the same source as β h a β , there exists a morphism β in G such that α h a α α = ββ h a β . A symmetricalargument will then give the converse. Let such a morphism α be. Then, the morphism α αβ − is in Aut G ′ ( S ,a ) = Aut( S ,a / U ). By stable embeddedness and independence offibers, the morphism α αβ − can be extended to an automorphism σ ∈ Aut( U / U ). Bystable embdeddness again, this automorphism can be extended to an automorphism τ of U ∪ U , h . We then have α h a α αβ − = α h a τ = α τ h a . Here, α τ is a morphismof the connected groupoid G ′ ,a , and so is β . Moreover, these morphisms have the samesource, which is S ,a . Thus, there exists a morphism β in G ′ ,a such that ββ = α τ .Thus α h a α α = ββ h a β . Since β is defined between objects of G ,a , it is in fact amorphism in G ,a . Proposition 3.13.
The action of G on the morphisms of ACIF A defines a functor.Proof. The action on the identity morphisms is pretty clear : since we may pick anymap h : S → S , as long as it yields the desired theory, we may assume h = id S . Inthat case, the set of maps in G ( h ) is just the set of morphisms in G , which is indeedthe identity of the groupoid.For composition, we use the same argument as in [HM18](4.11). Let ( U ∪ U , h ),( U ∪ U , h ) be morphisms in ACIF A . The maps in G ( h ) ◦ G ( h ) are of the form α h ,a α β h ,a α , whereas those in G ( h ◦ h ) are of the form β h ,a h ,a β . Us-ing the identity of S ,a , which is indeed a morphism in G ′ , we deduce that G ( h ◦ h ) ⊆ G ( h ) ◦ G ( h ). For the other inclusion, let α h ,a α β h ,a α be a map in G ( h ) ◦ G ( h ). Then, the map α β is an automorphism of S ,a in G ′ . Thus, nystable embeddedness and independence of fibers, it can be extended by identity to anautomorphism of U fixing U pointwise. By stable embeddedness again, this automor-phism can be extended to an automorphism σ of ( U ∪ U , h ). We then compute α h ,a α β h ,a α = α h ,a σh ,a α = α h ,a h ,a σα . Here, σα is a morphism in G ′ . This proves the equality G ( h ◦ h ) = G ( h ) ◦ G ( h ). η : id ACIF A ≃ CG The following proposition is a relativization of proposition 4.2 in [HM18], under an extraassumption. 13 roposition 3.14.
Let U , U be objects in the category ACIF A . Let h : S → S be amap compatible with the maps S → A , S → A . Assume that U is stably embedded in ( U ∪ U , h ) . Then ( U ∪ U , h ) is a morphism in ACIF A if and only if, for all a ∈ A ,the map h a ∪ id U : ( U , S ,a ) → ( U , S ,a ) yields a morphism of covers from ( U , S ,a ) to ( U , S ,a ) .Proof. We shall prove that U is stably embedded in ( U ∪ U , h ). Let σ be an automor-phism of U . We want to extend it to an automorphism of ( U ∪ U , h ). We may assumethat σ fixes U pointwise. Thus, for each a in A , σ restricts into an automorphism σ a of ( U , S ,a ). Since ( U , S ,a , S ,a , h a ) is a cover of ( U , S ,a ), there exists an automorphism τ a ∈ Aut(( U , S ,a , S ,a , h a ) / U ) extending σ a . Using independence of fibers and stableembeddedness of the ( U , S ,a ) in ( U , S ), the union of maps τ := S a τ a is an automor-phism of U ∪ U extending σ . By construction, it commutes with h . Thus U is stablyembedded in ( U ∪ U , h ), as desired. Corollary 3.15.
Let U , U be objects in the category ACIF A . Let h : S → S be amap compatible with the maps S → A , S → A . Assume that U is stably embedded in ( U ∪ U , h ) . Then ( U ∪ U , h ) is an isomorphism of covers if and only if, for each a in A , ( U , S ,a , S ,a , h a ) is an isomorphism of covers. Corollary 3.16.
Let U , U be objects in the category ACIF A . Let h : S → S be amap compatible with the maps S → A , S → A . Assume that U is stably embeddedin ( U ∪ U , h ) . Then ( U ∪ U , h ) is an isomorphism in ACIF A if and only if, for all a ∈ A , the map h a ∪ id U : ( U , S ,a ) → ( U , S ,a ) is a bijective bi-interpretation. Definition 3.17.
Let V = ( U , S, S → A ) be an object of ACIF A . Let G ′ ( V ) be theextension of the liaison groupoid G ( V ) with one extra object S a in each isomorphismclass, such that Aut G ′ ( V ) ( S a ) = Aut( S a / U ). Let ( U , O ∗ ) be the cover C ( G ( V )). Let G ′′ ( V ) be the extension of G ( V ) defined in C ( G ( V )) eq by formally creating copies ofobjects and morphisms of G ( V ).Then η V is defined as ( V ∪ C G ( V ) , S a h a g a ), where, for each a in A , i a is some objectin G ( V ) a , and the maps h a and g a are arbitrary morphisms in Hom G ′′ ( V ) ( O ,i a , O ∗ ,a ) and Hom G ′ ( V ) ( S a , O ,i a ) respectively. S a O ,i a O ∗ ,ag a h a Lemma 3.18.
The theory of η V is a cover of U .Proof. Let σ be an automorphism of U . Let σ ′ be an automorphism of ( V ∪ C G ( V )) eq that extends σ . We shall proceed as in lemma 3.5, by exhibiting another possible choiceof family of morphisms, and finding an automorphism connecting the two choices. Foreach a in A , let h ′ a = σ ′ ( h σ − ( a ) ) and g ′ a = σ ′ ( g σ − ( a ) ). Since σ ′ is an automorphism, h ′ a is a morphism in the groupoid G ′′ ( V ) with target O ∗ ,a and source in the objects of G ( V ). Similarly, g ′ a is a morphism in the groupoid G ′ ( V ) with source S ∗ ,a and target14n the objects of G ( V ), and the composition h ′ a g ′ a is well-defined. Now, there existmorphisms α a and β a in G ( V ) such that h ′ a = h a α a and g ′ a = β a g a . Again, there existsa morphism τ a in Aut G ′ ( V ) ( S a ) = Aut( S a / U ) such that g a τ a = α a β a g a . So we have h a g a τ a = h a α a β a g a = h ′ a g ′ a . S a O i O i O ∗ ,a S a O i O j O j S a O ig a g ′ a β a h a g a α a h ′ a τ a g a α a β a Now, using independence of fibers, the morphisms τ a define an automorphism of V fix-ing U pointwise, which we extend by identity on O ∗ .This way, we define an automorphism τ of ( V ∪ C G ( V )) such that, for all a in A , we have h a g a τ σ ′ = τ σ ′ ( h σ − ( a ) ) σ ′ ( g σ − ( a ) ) σ ′ = τ σ ′ h σ − ( a ) g σ − ( a ) . The automorphism τ σ ′ has the desired properties. Proposition 3.19.
The theory of η V is an isomorphism between V and C ( G ( V )) .Proof. We will prove that V is stably embedded in η V = ( V ∪ C ( G ( V )) , S a h a g a ) . Similararguments show that C ( G ( V )) is stably embedded in η V = ( V ∪ C ( G ( V )). Then, η V willbe a morphism of covers. The map S a h a g a being bijective, the morphism will thus bean isomorphism.Let σ be an automorphism of V . By the previous lemma, we may assume σ to fix U pointwise. We only have to extend σ to the sort O ∗ in a way that make it commute withthe map S a h a g a and be an automorphism of C ( G ( V )). Luckily, since the map S a h a g a is abijection, there is only one way to do so. Since σ fixes U pointwise, we may consider σ a = σ | S a , for a in A . The map σ a is an element of Aut V ( S a / U ) = Aut G ′ ( V ) ( S a ). Therefore,the map g a σ a g − a is a morphism in G ( V ), so the map h a g a σ a g − a h − a is an automorphismof O ∗ ,a in G ′′ ( V ). It can therefore be extended into an automorphism of ( U , O ∗ ,a ) fixing U pointwise. By stable embeddedness, this can be extended into an automorphism of C ( G ( V )). Finally, by independence of fibers, the map S a h a g a σ a g − a h − a ∪ σ S ∪ id U is anautomorphism of V ∪ C ( G ( V )) that commutes with S a h a g a . Proposition 3.20.
The family of maps η defines a natural isomorphism between theidentity functor of ACIF A and the functor CG .Proof. We have to prove that the naturality squares commute. U CG ( U ) U CG ( U ) η U h CG ( h ) η U
15n this situation, we recall that, by lemmas 3.5 and 3.18 and lemma 2.4 of [HM18],we may freely pick the morphisms defining the η U i and CG ( h ) among all the possiblechoices, as long as they have appropriate domains and codomains. With this in mind,we may in fact assume that the diagram with the concrete maps defining the morphismsof covers commutes, not just the diagram with the theories. ε : id F CG A ≃ GC Definition 3.21.
Let
G → A be an object of F CG A . Let G ′ be the extension of G constructed with the cover C ( G ) by creating adequate copies of objects and morphismsin G . Let S denote the new sort in the cover C ( G ). Let G ′′ ( C ( G )) be the extension of G ( C ( G )) with one extra object S a in each isomorphism class.The set of maps of ε G is defined as ε G := { g ◦ f : O ,i → O ,i | i ∈ Ob ( G ) , i ∈ Ob ( GC ( G )) , ∃ a ∈ A, g ∈ Hom G ′′ ( C ( G )) ( S a , O ,i ) ∧ f ∈ Hom G ′ ( O ,i , S a ) } . Proposition 3.22.
The set of maps ε G is an isomorphism in F CG A .Proof. As each of the maps is a bijection, it suffices to check that ε G is a morphism.First, regarding 0-definability in U , the same arguments as in 3.10 yield a 0-definableset in U .Then, as we have already seen a few times, condition (B) of being a morphism iseasy to check, since each map is of the form g ◦ f , where g is a morphism in G ′′ ( C ( G )).Now, let’s check condition (A). Let a be an element of A . Let g, g ′ be morphismsin G ′′ ( C ( G )) with source S a and targets in G ( C ( G )). Let f, f ′ be morphisms in G ′ withtarget S a and sources in G . Let γ be a morphism in G such that gf γ has the same domainas g ′ f ′ . We seek some morphism β in G ( C ( G )) such that βg ′ f ′ = gf γ . We notice that h := f γf ′− is an element of Aut G ′ ( S a ). Using the properties of G ′ and G ′′ ( C ( G )), wededuce that f γf ′− ∈ Aut G ′ ( S a ) = Aut C ( G ) ( S a / U ) = Aut G ′′ ( C ( G )) ( S a ). Thus, the map gf γf ′− is a morphism in G ′′ ( C ( G )). Since it is in the same connected component as themorphism g ′ , there exists a morphism β in G ′′ ( C ( G )) such that βg ′ = gf γf ′− . S a O ,j O ,j ′ O ,i O ,i ′ S ag ∃ ? βf γ f ′ h gh g ′ Since the source and target of β are objects of G ( C ( G )), it belongs to the full sub-groupoid G ( C ( G )). The other inclusion in condition (A) is proved similarly. Proposition 3.23.
The family of maps ε defines a natural isomorphism between theidentity functor of F CG A and the functor GC .Proof. We have to prove that the naturality squares commute. Given a morphism h : G → G in F CG A , and an element a of A , this amount to checking that the followingsquare commutes : 16 ,a GC ( G ) a G ,a GC ( G ) aε G a h a GC ( h ) a ε G a Checking this consists in writing down the maps in GC ( G ) a ◦ ε G a and in ε G a ◦ h a as compositions of maps in all the groupoids involved, finding subterms τ in thecompositions that define automorphisms of the object S a , using the binding groupoidstatements to extend the τ to automorphisms of one of the internal covers above a ,and using stable embeddedness to extend them again to automorphisms of C ( h ) =( C ( G ) ∪ C ( G ) , S a g a h a f a ). We shall now try and generalise the constructions to the case where the fibers are notassumed to be independent. We follow the suggestion made in [Hru12], in the remarkfollowing Lemma 3.1, to use simplicial groupoids.
Definition 4.1.
1. If A is a set, and n is an integer, we let A = n denote the set oftuples ( a , ..., a n ), where the a i are elements of A that are pairwise distinct. Wealso let A <ω denote the collection of finite subsets of A .2. If c is a finite tuple, we use the notation ” a ∈ c ” to denote ” a is some coordinateof the tuple c ”. We also use the notation ” c ⊆ d ”, where c, d are tuples, to denote”for all elements a , if a ∈ c , then a ∈ d ”.3. If f : S → A is a map, if n is an integer and c is an element of A = n , we define theset S c as S c := S a ∈ c S a ⊆ S . Definition 4.2.
Let A be a definable set. A definable simplicial groupoid G over A is given by a sequence of 0-definable groupoids G n , for n ≥
1, along with injectivemorphisms of groupoids ι n,m : G n ֒ → G m , for n < m , such that :1. For each n , G n is a groupoid over A = n /E n , where E n is the equivalence relationassociated to the action of the symmetric group S n on the set A = n . Informally, G n is over the finite subsets of A of size n .2. The operation ( n < m ) ι n,m behaves well with compositions.3. For each n < m , the morphism of groupoids ι n,m is compatible with the relation R ( c, d ) ↔ c ⊆ d . See definition 3.3 Remark 4.3.
The morphisms of groupoids ι n,m defining the simplicial structure onlylink the connected components one would expect them to.17 efinition 4.4. A morphism of simplicial groupoids is defined as a family of morphismsof groupoids that is compatible with the inclusion morphisms defining the simplicialstructures.More explicitly, if ( G , ( ι n,m )) , ( G , ( ι n,m )) are simplicial groupoids over A , then amorphism H : G → G is given by a family of morphisms H n : G ,n → G ,n that arecompatible with the equality relations R n ⊆ ( A = n /E n ) × ( A = n /E n ), and such that, forall integers n, m such that 1 ≤ n < m , the following diagram of morphisms of groupoidscommutes : G ,m G ,m G ,n G ,nH m ι n,m H n ι n,m Remark 4.5.
The reader might be wondering why we didn’t use a definition fromcategory theory to define simplicial groupoids as simplicial objects in the category ofgroupoids.The main reason is to avoid degeneracies, which are painful to deal with. In fact,most of the constructions that appear below are much easier within our framework.However, an algebraic topologist more technically gifted than the writer might disagree,and find that having to deal with the degeneracies is worth it.We now prove a general lemma that will be useful when studying 1-analysable covers.
Lemma 4.6.
Let ( U , T, f : T → A ) be a cover of U .Let τ : T → T be a map compatible with f : T → A .Then the map τ ∪ id U is an automorphism of ( U , T ) if and only if, for all finitesubsets c ⊆ A , its restriction is an automorphism of ( U , S a ∈ c T a ) .Proof. One direction is clear, as the structure on ( U , S a ∈ c T a ) is the structure induced bythe c -definable sets.Conversely, let’s assume τ ∪ id U satisfies the local condition. Let b be a finite tupleof elements of ( U , T ), and R be a 0-definable relation in ( U , T ) such that ( U , T ) | = R ( b ).Let c be a tuple of elements of A such that b ⊆ U ∪ S a ∈ c T a . Let R ′ be the restriction ofthe relation R to U ∪ S a ∈ c T a . By definition of the structure of U ∪ S a ∈ c T a , the relation R ′ is 0-definable in ( U , S a ∈ c T a ).Therefore, ( U , S a ∈ c T a ) | = R ′ (( τ ∪ id U )( b )). So ( U , T ) | = R (( τ ∪ id U )( b )) In this subsection, we let U ′ = ( U , S, f : S → A ) denote a 1-analysable cover of U withthe following properties : 18 For each integer n ≤
1, for each finite tuple c ∈ A = n , the language of ( U , S a ∈ c S a )is finitely generated over that of U , and this happens uniformly for c in A = n . Seedefinition 2.5 for a formal description of ”uniformly finitely generated languages”. • For each finite subset C ⊆ A , the structure ( U , S a ∈ C S a ), with the structure inducedby the C -definable sets of ( U , S ), is stably embedded in ( U , S ). Theorem 4.7.
There exist simplicial groupoids G and G ′ , over the set A , that are 0-definable in U and U ′ eq respectively, with the following properties: • The simplicial groupoid G ′ is an extension of G with the extra object S c in theconnected component over c . Moreover, the set-theoretic inclusions between the S c belong to the simplicial groupoid.More explicitly : for each degree n ≥ , for each element c ∈ A = n /E n , the connectedgroupoid G ′ n,c is an extension of the connected groupoid G ′ n,c , with only one extraobject, whose underlying set is the set S c = S a ∈ c S a . Moreover, if we have n < m , c ∈ A = n /E n , d ∈ A = m /E m and c ⊆ d , then the morphism of groupoids ι ′ n,m : G ′ n →G ′ m contains the set-theoretic inclusion map S c ֒ → S d . • The automorphism groups
Aut G ′ ( S c ) , along with their action on S c , are canonicallyisomorphic to the groups Aut( S c / U ) .Proof. We shall define a simplicial groupoid which captures the whole structure of thecover U ′ and generalizes the case of independent fibers.For each positive integer n , and each tuple c ∈ A = n , we notice that the structure( S c , U ) is an internal cover of U . Using the hypothesis of uniformly finitely generatedlanguage, we can find a groupoid G n over A = n which is 0-definable in U , with an extension G ′ n that is 0-definable in ( U ′ ) eq . This extension has exactly one extra object S a ∪ ... ∪ S a n in the connected component over ( a , ..., a n ) and satisfies the binding groupoid statement: Aut G ′ n ( S a ∪ ... ∪ S a n ) = Aut( S a ∪ ... ∪ S a n / U ).Now, we want groupoids that are over A = n /E n , and not over A = n . To do this, itsuffices to glue together the groupoids that correspond to a common orbit under theaction of S n . This is made possible by the existence of the common object S c , whoseautomorphism group is the same in all the connected components where it appears.We define the simplicial structure using set-theoretic inclusions. For each inclusionof finite subsets of A , c ⊆ d , we find the definable map S c ֒ → S d . We then createsets of maps between the groupoids G ′ c by precomposing and postcomposing these mapswith morphisms in the groupoids G ′ n . To see that these sets of maps define groupoidmorphisms, we need to use the second hypothesis on ( U , S ), that enables us to liftautomorphisms and get surjective group homomorphisms Aut( S d / U ) ։ Aut( S c / U ), foreach inclusion c ⊆ d : Proposition 4.8.
Let ι be the sets of inclusion maps defined above. Then these sets ofmaps induce groupoid morphisms between the G ′ n , that define a simplicial groupoid. roof. The main point is to check condition (A). We shall first prove the followingpartial result : If ι : S c → S d is a set-theoretic inclusion, and σ, τ are elements ofAut( S c ) and Aut( S d ) respectively, then there exist elements σ ′ and τ ′ in Aut( S c ) andAut( S d ) respectively making the following diagrams commute : S c S c S c S c S d S d S d S dσι ι ∃ σ ′ ι ι ∃ τ ′ τ We first use the binding groupoid statements. We define binary relations R i on S c : R i ( x, ) ↔ f ( x ) = c i . These relations are 0-definable in ( U , S c ), and thus preserved by σ . Therefore, we see that σ is a union σ c ∪ ... ∪ σ c n of automorphisms of the S c i , for c i acoordinate of c . We can also check that the restriction S c i ∈ d σ c i is indeed an automorphismof S d . This restriction is the automorphism τ ′ we were looking for.Now, to find σ ′ , we use the second hypothesis on the cover ( U , S ) . We have assumedthat τ induces an automorphism of the structure ( U , S a ∈ d S a ). Since ( U , S a ∈ c S a ) is a coverof ( U , S a ∈ d S a ) by hypothesis, such an automorphism exists.Since the maps in ι are of the form αι β , where α and β are morphisms in the G ′ n ,the result proved above yields the full condition (A). Remark 4.9.
It is somewhat surprising that condition (A), which looks a bit formal,should translate as the model-theoretic stable embeddedness condition above.
Proposition 4.10.
The simplicial groupoid built here has a specific property, in additionto being finitely faithful : Let d be a finite subset of A , and c , ..., c k define a partition of d . Let O i d , O i c , ..., O i ck be objects in the corresponding groupoids. Let ι : O i c → O i d ,..., ι k : O i ck → O i d be maps belonging to the inclusion morphisms of groupoids. Thenthe map ι ⊔ ... ⊔ ι k : O i c ⊔ ... ⊔ O i ck → O i d is a bijection that identifies O i d with thedisjoint union O i c ⊔ ... ⊔ O i ck .Proof. The statement is true for the objects S d , S c j and the set-theoretic inclusions.Now, condition (A) enables us to compare this setting to any other setting. Indeed,if m ab : O i ab → S ab is any morphism in G ′ , condition (A) implies that there existmorphisms m a : O i a → S a and m b : O i b → S b in the degree 1 groupoid such that thefollowing diagram commutes : 20 a O i a S ab O i ab S b O i b ι a m a m ab ι b m b In a nutshell : O i a ⊔ O i b ≃ S a ⊔ S b = S ab ≃ O i ab We will call this property the disjoint union property . Remark 4.11.
These properties of the binding simplicial groupoids we build are pre-served under isomorphisms of simplicial groupoids. Thus, our future assumptions on thesimplicial groupoids we work with will be justified.
Proposition 4.12.
Let (Aut( S d / U ) ։ Aut( S c / U )) c ⊆ d be the projective system of groupsmentioned at the end of definition 4.7. Then its projective limit in the category of groupsis isomorphic to the automorphism group Aut( S/ U ) . Proof.
Using the second hypothesis on the cover ( U , S ), and lemma 4.6, we see thateach element of the projective limit yields an automorphism of S over U . Conversely, anautomorphism of S over U induces a family of automorphisms in the Aut( S c / U ). Sucha family is in the projective limit, since the projections act as restrictions.The next proposition is a generalization of proposition 3.6 in [HM18]. It merelyconsists in making the compactness argument found there uniform over A . Proposition 4.13.
Let V = ( U , S, S → A ) be a 1-analysable cover of U over A . As-sume that the automorphism groups Aut( S a / U ) along with their actions on the S a areuniformly interpretable in V . Then the language of ( U , S a ) is finitely generated over thatof U , uniformly in a .Proof. By compactness and 1-analysability, there exists a 0-definable set B in V with a0-definable surjective map g : B → A , a 0-definable set X in U and a 0-definable map F : S × A B → X such that for all a in A , for all b in B a , the map F b is an injection F b : S a ֒ → X .Let G ( x, y, b, b ′ ) be the formula ”There exists some a in A such that the map F − b ◦ F b ′ : S a → S a is a well-defined bijection sending x to y ”. Now, we shall use the uniforminterpretability of Aut( S a / U ). Let p ( v, z ) be the partial type { z ∈ A } ∪ { ” G v is apermutation of S z ” } ∪ {∀ x ( R (( G v ∪ id U )( x ); z ) ↔ R ( x ; z )) | R : 0-definable relation in U } .We know that p ( v, z ) | = G v ∈ Aut( S z / U ). Thus, by compactness, there existsformulas R ( x, y ) , ..., R n ( x, y ) in the language of V such that, for all a in A , for all v such that G v is a permutation of S a , the map G v ∪ id U is an automorphism of ( U , S a ) ifand only if it preserves the relations R ( x, a ) , ..., R n ( x, a ) . a in A , let Σ a be the language generated by the relations R i ( x, a ) alongwith the relation R ( x, y, b ), the latter being defined as F b ( x ) = y . We thus get a familyof finitely generated languages that vary uniformly over A . It remains to show that theselanguages generate the languages L a of ( U , S a ). To simplify notations, let V a and V ,a be the structure ( U , S a ) in the languages L a and L ∪ Σ a respectively. By saturation, itsuffices to prove that Aut( V a ) = Aut( V ,a ). By stable embeddedness, it suffices to showthat Aut( V a / U ) = Aut( V ,a / U ). Let σ be an automorphism of V ,a fixing U pointwise.Let b be a parameter in B such that F b : S a ֒ → X is an injection. Since the formuladefining F is in the language Σ a , we know that F σ ( b ) is also an injection from S a to X .In addition, its image is the same as that of F b , for X is a subset of U and σ fixes U pointwise. We now compute, for x in S a : F b ( x ) = σ ( F b ( x )) = F σ ( b ) ( σ ( x )). Thus themap F − σ ( b ) ◦ F b is equal to the map σ | S a . Finally, if v = ( σ ( b ) , b ), applying the propertyof the map G v described in the third paragraph of our proof, we deduce that σ is anautomorphism of V a , as desired. Corollary 4.14.
Let V = ( U , S, S → A ) be a 1-analysable cover of U over A . Let n bean integer. Assume that the automorphism groups Aut( S c / U ) along with their actionson the S c are uniformly interpretable in V , for c ∈ A = n . Then the language of ( U , S c ) is finitely generated over that of U , uniformly in c .Proof. We first note that the automorphism groups Aut( S c / U ) along with their actionson the S c are uniformly interpretable in V , for c ∈ A = n , if and only if their counterpartsAut( Q a ∈ c S a / U ) are.Moreover, the languages of the covers ( U , S c ) are uniformly finitely generated over U , for c ∈ A = n , if and only if the same holds true for the covers ( U , Q a ∈ c S a ).Thus, if g : S n → A n is the map f × ... × f , and S ′ = g − ( A = n ), we can applyproposition 4.13 to the 1-analysable cover ( U , S ′ , g : S ′ → A = n ). This yields the result. Remark 4.15.
As in [HM18] (Proposition 3.6), this result shows that the condition ofuniformly finitely generated language is necessary in order to get 0-definable groupoidsand not just type-definable ones.
In this subsection, we let G be a finitely faithful 0-definable simplicial groupoid over A satisfying the properties of proposition 4.10. Definition 4.16. A commuting system of inclusions is a family of objects ( O i c ) c ⊆ A, | c | <ω with one object in each connected component, along with a family of maps ( ι c,d : O i c ֒ → O i d ) c ⊂ d , such that, if c ⊂ d ⊂ e , then ι d,e ◦ ι c,d = ι c,e . Proposition 4.17.
Let G be a definable simplicial groupoid over A . Then there existsa commuting system of inclusions in G .
22e build such a system inductively. We will first need to prove weaker results :
Lemma 4.18.
Let a , b , c be finite subsets of A . Let O a , O ab , O ac , O abc be objects in G , inthe appropriate connected components. Let ι : O a ֒ → O abc , ι : O ac ֒ → O abc be inclusionsin the morphisms of groupoids defining the simplicial structure of G . Then there existsa unique inclusion ι : O a ֒ → O ac making the following diagram commute : O abc O ac O a ι ι ∃ ! ι Proof.
Uniqueness comes from injectivity of ι . Let us now prove existence.First, we recall the equality of sets of maps : Hom ( O a , O abc ) = Hom ( O ac , O abc ) ◦ Hom ( O a , O ac )So there exist inclusion maps j , j such that j ◦ j = ι . Then, by condition(A), there exists σ ∈ Aut( O ac ) such that ι σ = j . Then, the map ι := σj satsifies ι ι = ι σj = j j = ι , as desired. O abc O abc O ac O ac O aid ∃ j ∃ σι ι σj ∃ j Lemma 4.19.
Let k be an integer. Let X ⊂ P ( A ) be a finite collection of finite subsets of A . Let’s assume that | X | = 2 k − , and that the relation of inclusion on X is isomorphicto that of the nonempty subsets of some B ⊆ A , where | B | = k . Let ( O x ) x ∈ X be a familyof objects of G in the appropriate connected components. Then there exists a commutingsystem of inclusions between the O x .Proof. We prove the result by induction on k . For k = 1, there is nothing to prove.Now, let k ≥
2, let X ⊂ P ( A ) and ( O x ) x ∈ X be as in the lemma. Let x , ..., x k bethe minimal elements of X . Using the induction hypothesis, there exists a commutingsystem of inclusions for the family ( O x ) x ∈ X,x k x . There also exists a commuting systemof inclusions for the family ( O x ) x ∈ X,x k ⊂ x . It remains to add x k to the latter system, andconnect the two systems together. 23rom now on, to simplify notations, we shall assume that the x i are in fact elementsof A , and that X = P ( B ) for B = { x , ..., x k } .To add x k to the system of subsets of B that strictly contain x k , pick one inclusion O x k ֒ → O B . Then, if i, j = k , since there are already inclusions as in the followingdiagram, we can apply lemma 4.18 : O B O x j x k ; ; ①①①①①①①①① O x i x k c c ❋❋❋❋❋❋❋❋❋ O x k O O ∃ ! ι i < < ①①①①①①①① ∃ ! ι j c c ❋❋❋❋❋❋❋❋ Now, using injectivity of all the inclusions, commutativity of these squares, and thefact that the system between the ( O x ) x ∈ X,x k ⊂ x is commutative, we can prove that theall new squares created by our new maps ι i commute. For instance, if i = j , we studythe following diagram : O B O x i x j x k g O O O x j x k f j : : ✉✉✉✉✉✉✉✉✉ g j F F ✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍✍ O x i x k f i d d ■■■■■■■■■ g i X X ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ O x k ι i : : ✉✉✉✉✉✉✉✉✉ ι j d d ■■■■■■■■■ Here, all the arrows except ι i and ι j belong to the commuting system between the( O x ) x ∈ X,x k ⊂ x . So the two upper triangles commute. Moreover, by definition of ι i and ι j , the outer square commutes, i.e. g j ι j = g i ι i . Thus, we find gf j ι j = g j ι j = g i ι i = gf i ι i .Since g is injective, we deduce f j ι j = f i ι i .Now, it remains to connect the objects that contain x k with those that do not containit. Again, pick an arbitrary inclusion f : O x ...x k − ֒ → O B . Then, if B ′ = B \ { x i , x k } ,apply lemma 4.18 in the following setting : O B O B ′ x i f ; ; ✇✇✇✇✇✇✇✇✇ O B ′ x k c c ●●●●●●●●● O B ′ O O ∃ ! j B ′ ; ; ✇✇✇✇✇✇✇✇ c c ●●●●●●●● B ′ x i = { x , ..., x k − } . The map in the middle is defined preciselyas the composition of the maps on the left. The map O B ′ x k → O B in the upper rightcorner belongs to the system ( O x ) x ∈ X, x k ⊂ x . We find a uniquely determined inclusion j B ′ : O B ′ ֒ → O B ′ x k that makes the left triangle commute. By definition, this extendedsystem commutes.Now, deal with subsets B ′ = B \ { x i , x j , x k } . There is already a unique inclusionmap O B ′ ֒ → O B , given by any path ending with f : O B O B ′ x i x j f O O O B ′ x i : : ✉✉✉✉✉✉✉✉✉ O B ′ x j O O O B ′ x k X X ✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶ O B ′ O O d d ❏❏❏❏❏❏❏❏❏ ∃ ! j B ′ : : ttttttttt The map O B ′ x k → O B in the upper right corner belongs to the system ( O x ) x ∈ X, x k ⊂ x .Again, using injectivity and commutativity of the previous smaller systems, we findthat these new maps define a commuting system.Then deal with the cases B ′ = B \ { x t , x i , x j , x k } in a similar way, applying lemma4.18 to find uniquely determined inclusions j B ′ : O B ′ ֒ → O B ′ x k .By decreasing induction, we finally find uniquely defined maps j { x i } : O x i ֒ → O x i x k to complete our system. Note that once f was chosen, all the maps yet to be definedwere uniquely determined.Using the same ideas as in the third paragraph of this proof, namely, injectivity andcommutativity of the ”older” smaller systems, we check that the whole system commutes.We are now ready to prove proposition 4.17 : Proof of proposition 4.17.
Let κ := | A | . Let us pick an enumeration of A : A = { a α | α <κ } . For α < κ , let A α denote the subset { a β | β < α } . We intend to define an increasingfamily of commuting systems of inclusions on the A α , and then take the increasing unionof these partial systems.This inductive construction uses both saturation and the ideas in the proof of lemma4.19.Let γ < κ . Assume that there exists an increasing family of commuting systems ofinclusions on the A β , for β < γ . We want to find a commuting system of inclusionson A γ . We may assume that the ordinal γ is a successor ordinal, otherwise just takethe union of the previously defined systems. Let us write γ = α + 1. Pick any object25 i aα in the component of G over a α . We wish to add this object to the system. Thisconsists in finding objects O i Baα , for B a finite subset of A α , along with inclusions ι B ′ ,B : O i B ′ ֒ → O i B , for B ′ ⊂ B finite subsets of A α , when a α ∈ B .We shall view the codes of the inclusion maps we are looking for as variables in apartial type .In fact, commutativity of the finite diagrams we are considering is expressible in thelanguage of U . Moreover, the number of variables is the number of new inclusions wehave to define, so its cardinal is bounded by α . Similarly, the set of parameters of thepartial type we define is given by the codes of the inclusions that are already defined.This set of parameters is thus also bounded in cardinality by α .Thus, if we can show that the partial type expressing commutativity of all the diagramsinvolved is consistent, we can use saturation of U to extend our system by ”adding thepoint a α ”. Let us now prove the consistency of this partial type.Let A , ..., A n be finite subsets of A α . We wish to find commuting systems of inclu-sions for A ∪{ a α } , ..., A n ∪{ a α } that extend the preexisting systems defined on A , ..., A n respectively.It suffices to consider the finite set A ∪ ... ∪ A n ⊆ A α , for which we have a commutingsystem of inclusions that contains the ones on the A i . Indeed, managing to extend thissystem to A ∪ ...A n ∪ { a α } is enough.Now, we claim that the proof of the induction step for lemma 4.19 shows that, given acommuting system of inclusions on some finite set A ′ ⊂ A , one can ”add a point a α toit” and get an extended commuting system on A ′ ∪ { a α } . Definition 4.20.
Let G , G be 0-definable simplicial groupoids over A with the disjointunion property. Let H : G → G be a morphism of simplicial groupoids. Let ι =( O i c → O i d ) c ⊂ d ∈ A <ω and ι = ( O j c → O j d ) c ⊂ d ∈ A <ω be commuting systems of inclusionsin G and G respectively.Let B ⊆ A . Let ( h a : O i a → O j a ) a ∈ B be a family of maps belonging to the morphismof groupoids H . This family is called coherent , with respect to ι and ι , if, for eachfinite subset c ⊂ B , there exists a higher degree map h c in H such that, for each element a ∈ c , the following diagram commutes : O i c O j c O i a O j a h c h a ι ι We shall now prove that coherent families of maps always exist, for B = A . Lemma 4.21.
Let G , G be 0-definable simplicial groupoids over A with the disjointunion property. Let H : G → G be a morphism of simplicial groupoids. Let ι =( O i c → O i d ) c ⊂ d ∈ A <ω and ι = ( O j c → O j d ) c ⊂ d ∈ A <ω be commuting systems of inclusionsin G and G respectively. Let m a , ..., m b be maps in H , and f be a map in H in higherdimension, such that the following diagram commutes : i c O i a O i b O j a O j b O j c fι ι m a m b ι ι Let d be an element of A . Then there exists a degree 1 map m d : O i σ ( d ) → O σ ( i d ) belonging to H , and a higher dimension map g : O i σ ( cd ) → O σ ( i cd ) in H making thefollowing diagram commute : O i cd O i a O i b O i d O j a O j b O j d O j cd gι ι ι m a m b m d ι ι ι Proof.
We recall that the inclusions ι and ι belong to commuting systems. Thus,commutativity of the diagram above is equivalent to that of this one : O i cd O i c O i a O i b O i d O j a O j b O j d O j c O j cd ∃ ? gι ι fι ι m a m b ∃ ? m d ι ι ι ι H is a morphism of simplicial groupoids, by condition (A) (see the proof oflemma 4.18), there exists a higher dimension morphism g making the following squarecommute : O i cd O i c O j c O j cd ∃ gι f ι Applying condition (A) again, this time to the rightmost part of the big diagram, wefind an appropriate morphism m d . Proposition 4.22.
Let G , G be 0-definable simplicial groupoids over A with the disjointunion property. Let H : G → G be a morphism of simplicial groupoids. Let ι = ( O i c → O i d ) c ⊂ d ∈ A <ω and ι = ( O j c → O j d ) c ⊂ d ∈ A <ω be commuting systems of inclusions in G and G respectively.Then there exists a coherent family of maps ( m a : O i a → O j a ) a ∈ A belonging to themorphism of simplicial groupoids H .More generally, for all finite B ⊆ A , any coherent family of maps ( m a : O i a → O j a ) a ∈ B can be extended into a coherent family of maps ( m a : O i a → O j a ) a ∈ A .Proof. We shall build the additional morphisms inductively, using the previous lemma.Let us take an enumeration of A : A = { a α | α < κ } that begins with the elementsof B . We will build inductively a family of degree 1 morphisms m α : O i aα → O j aα satisfying the following coherence property :For all β < κ , for all finite tuples c = ( a, ..., b ) ⊆ { a α | α ≤ β } , there exists a map f belonging to H , in degree | c | , making the following diagram commute : O i c O i a O i b O j a O j b O j c ∃ fι ι m a m b ι ι
28e start with the construction up to the finite ordinal | B | .Let β be an ordinal smaller than κ . Given the construction for all ordinals α < β , weshall extend it by realizing a partial type whose parameters are the morphisms alreadybuilt. Let p β ( m ) be the set of formulas containing the formula ” m ∈ Hom G ( O i aβ , O j aβ )”and, for each tuple c = ( a, ..., b ) ⊆ { a α | α ≤ β } , the formula expressing :”There exists a higher dimension morphism g : O i c → O j c in H making the followingdiagram commute : O i c O i a O i b O j a O j b O j c gι ι m a m b ι ι where m a β is a notation for m .”Using the previous lemma and the induction hypothesis, we see that this set offormulas is consistent : if c , ..., c m are finite tuples, by the commutativity of the systemsof inclusions ι and ι , and condition (A), it suffices to find a coherence morphism for thelarger tuple c ...c n . The previous lemma shows that such a coherence morphism exists.Moreover, the set of parameters of p β ( m ) is small enough in cardinality. Thus, bysaturation, this partial type is realized in U .Having introduced these two notions of coherence, we can now define the coverassociated to a simplicial groupoid : Theorem 4.23.
Let G be a 0-definable simplicial groupoid in U over A , with the disjointunion property. Then there exists a cover ( U , O ∗ ) of U , along with an extension G ′ of G that is 0-definable in ( U , O ∗ ) , such that : • The cover ( U , O ∗ ) is 1-analysable over A , and satisfies the two technical hypothesesgiven at the beginning of subsection 4.1. • The simplicial groupoid G ′ is an extension of G with the extra object O ∗ ,c in theconnected component over c . • The automorphism groups
Aut G ′ ( O ∗ ,c ) , along with their action on O ∗ ,c , are iso-morphic to the groups Aut( O ∗ ,c / U ) .Proof. Pick a commuting system of inclusions ι : O i c ֒ → O i d , in the groupoid morphismsdefining the simplicial groupoid, where c ⊆ d are finite tuples of elements of A .Create a copy O ∗ ,a of O i a , with a map f a : O ∗ ,a → O i a identifying the two sets. Thenew objects in degree n are the O ∗ ,a ∪ ... ∪ O ∗ ,a n . Use the inclusions to build bijections29 a ...a n : O ∗ ,a ∪ ... ∪ O ∗ ,a n ≃ O i a ...an . These bijections are morphisms in the extendedgroupoid. They make the following diagrams commute : O ∗ ,a ∪ ... ∪ O ∗ ,a n O ∗ ,a O ∗ ,a n ... ...O i a O i an O i a ...an f a ...an ι ι f a f an ι ι Here, the ι are the set-theoretic inclusions, and the ι are in the commuting systempicked earlier. Note that we only consider the case of pairwise distinct a i .Extend the morphisms of groupoids, i.e. inclusions, in the only way that makes themaps f a ...a n , n ≥ , a , ..., a n ∈ A isomorphisms of these groupoids.Define a multi-sorted extension of U , using the whole simplicial groupoid structure: There are new sorts O ∗ = S a ∈ A O ∗ ,a and M ( n ) ∗ for the groupoid in degree n , and newsorts N ( i,j ) ∗ for the morphisms from the degree i groupoid to the degree j groupoid. Thenew relations are the ones interpreting the morphisms in the new sorts as bijections.We call O ∗ the object sort.We define the structure U ′ as the induced structure on ( U , O ∗ ). Proposition 4.24.
The extension of G in the structure ( U , O ∗ , M ( n ) ∗ , N ( i,j ) ∗ ) is a sim-plicial groupoid over A .Proof. First, in each degree, the extra objects O ∗ ,c and the bijections f c : O ∗ ,c → O i c define groupoids, just as in the internal or independent cases.Then, we need to check that the extended inclusions satisfy condition (A). We knowthat the maps f c connect the inclusions belonging to ι to the set-theoretic ones in ι .Then, since the inclusion morphisms of groupoids in G satisfy condition (A), the extendedones also do.Finally, let us show that all this data defines a simplicial groupoid, i.e. all the requiredrelations between the inclusions hold. First, they hold for the set-theoretic maps. Thus,by condition (A), they hold globally for the sets of maps that define the morphisms ofgroupoids. Proposition 4.25.
The structure ( U , O ∗ ) is stably embedded in the simplicial groupoidstructure ( U , O ∗ , M ( n ) ∗ , N ( i,j ) ∗ ) . roof. Using finite faithfulness, we see that the morphism sorts are included in thedefinable closure, in ( U , O ( n ) ∗ , M ( n ) ∗ , N ( i,j ) ∗ ), of the object sorts.Now, to prove stable embeddedness of U in this new structure, we shall proceedmore carefully than in the case of independent fibers. Just as in proposition 2.14, if σ isan automorphism of U , in order to define its action on the morphisms in the extendedgroupoid, we need to find maps m a : O i σ ( a ) → O σ ( i a ) in the degree 1 groupoid. Then,we will be able to extend σ as follows: if, for instance, m : O ∗ ,a → O j is a degree 1morphism in the extended groupoid, we define σ ( m ) as the morphism σ ( mf − a ) m a f σ ( a ) : O ∗ ,σ ( a ) → O σ ( j ) .However, this time we need to make sure our choices of maps come from morphismsin the higher degree groupoids. This is necessary in order to define the action of σ onhigher degree morphisms in a way that is compatible with the inclusions. Proposition 4.26.
The structure ( U , O ∗ ) is a 1-analysable cover of U .Proof. We wish to extend automorphisms of U using the isomorphisms f − a : O i a ≃ O ∗ ,a .Using proposition 4.22, we find morphisms m a : O i σ ( a ) → O σ ( i a ) in the originalgroupoid, with the following coherence property : For all finite tuples c = ( a, ..., b ) ⊆ A ,there exists a morphism f in the groupoid of degree | c | making the following diagramcommute : O i σ ( c ) O i σ ( a ) O i σ ( b ) O σ ( i a ) O σ ( i b ) O σ ( i c ) ∃ fι ιm a m b σ ( ι ) σ ( ι ) We can thus extend the action of σ to the degree 1 groupoid : if, for instance, m : O ∗ ,a → O j is a degree 1 morphism in the extended groupoid, we define σ ( m ) as themorphism σ ( mf − a ) m a f σ ( a ) : O ∗ ,σ ( a ) → O σ ( j ) .Then, using the coherence property of the family of maps we built, we find uniquelydetermined maps m c : O i σ ( c ) → O σ ( i c ) , c ∈ A <ω lifting the degree 1 morphisms m a .In fact, uniqueness comes from the disjoint union property proved in 4.10, and forcescommutativity of the coherence diagrams between these maps. Using these higher degreebijections, we define σ on the higher degree groupoids in a similar way. See proposition2.14 for the full definition.It remains to define the action of σ on the inclusions. The distinguished ”set-theoretic”inclusions O ∗ ,a ∪ ... ∪ O ∗ ,a n → O ∗ ,a ∪ ... ∪ O ∗ ,a n +1 are sent to their counterparts O ∗ ,σ ( a ) ∪ ... ∪ O ∗ ,σ ( a n ) → O ∗ ,σ ( a ) ∪ ... ∪ O ∗ ,σ ( a n +1 ) . 31o check that this defines an automorphism of the extended simplicial groupoid,viewed as a first-order multi-sorted structure, we use the 0-definability in U of the smallersimplicial groupoid structure, the fact σ is an automorphism of U , and the coherenceproperty described above :As a first step, let’s check that, if the left square of the following diagram commutes,then so does the right square : O O ∗ ,ab σ ( O ) O ∗ ,σ ( ab ) O O ∗ ,a σ ( O ) O ∗ ,σ ( a ) nι ′ ι σ ( n ) σ ( ι ′ ) σ ( ι ) n σ ( n ) Here, ι is the distinguished set-theoretic inclusion, whereas ι ′ is some inclusion inthe original simplicial groupoid. We recall that σ ( n ) = f − σ ( ab ) m − ab σ ( f ab n ) and σ ( n ) = f − σ ( a ) m − a σ ( f a n ). Thus we can compute : σ ( ι ) σ ( n ) = σ ( ι ) f − σ ( ab ) m − ab σ ( f ab n )= σ ( ι ) f − σ ( ab ) m − ab σ ( f ab n )= f − σ ( a ) ιm − ab σ ( f ab n )= f − σ ( a ) m − a σ ( ι ) σ ( f ab n )= f − σ ( a ) m − a σ ( ιf ab n )= f − σ ( a ) m − a σ ( f a ι n )= f − σ ( a ) m − a σ ( f a n ι ′ )= f − σ ( a ) m − a σ ( f a n ) σ ( ι ′ )= σ ( n ) σ ( ι ′ ) . In other words, the following diagram commutes :32 i σ ( a ) O ∗ ,σ ( a ) O i σ ( ab ) O ∗ ,σ ( ab ) O σ ( i a ) O σ ( i ab ) σ ( O ) σ ( O ) f − σ ( a ) f − σ ( ab ) ι σ ( ι ) m − a m − ab σ ( ι ) σ ( f a n ) σ ( f ab ι n ) σ ( ι ′ ) σ ( f ab n ) The bottom part of the diagram commutes because σ is both an automorphism of thedisjoint union of the extended groupoids ⊔ n G ′ n and an automorphism of the simplicialgroupoid G , and because the following diagram commutes : O i a O i ab O ∗ ,a O ∗ ,ab O O ιf a ι f ab n ι ′ n Similar computations deal with the cases where the objects O , O may be equal tothe new objects O ∗ ,a or O ∗ ,ab . In these cases, we use the description of the automorphismsof the O ∗ ,c as the conjugates of the automorphisms of O i c by the maps f c .Therefore, defining σ on the new inclusions with σ ( αι ′ β ) := σ ( α ) σ ( ι ′ ) σ ( β ) yields a well-defined automorphism of the structure composed of the family of the extended groupoids G ′ n along with the extended inclusion morphisms between them. Remark 4.27.
A cautious reader might object that we need to take saturated models ofthe theories we define, before we can use automorphism arguments to prove that we havedefined a cover. This issue can be addressed by noticing that the inductive constructionsthat constitute the core of the proofs only use condition (A) for the various morphismsof groupoids involved, which is preserved under elementary equivalence.
Proposition 4.28.
The cover built from the simplicial groupoid G does not depend onthe choices made.Proof. Let us assume we had picked another commuting system of inclusions ι ′ , withobjects O j c , and copying maps f ′ c : O j c → O ′∗ ,c , for c in A <ω . Using proposition 4.22,we find bijections g c : O i c → O j c in the simplicial groupoid G that make the followingdiagrams commute : 33 i c O j c O i a O j a ι g c ι ′ g a Now, we define an isomorphism h between the two extended simplicial groupoids,using the only maps h c that make the following diagrams commute : O i c O j c O ∗ ,c O ′∗ ,cf c g c f ′ c h c This isomorphism of simplicial groupoids defines an isomorphism between the struc-tures ( U , O ∗ ) and ( U , O ′∗ ), that fixes U pointwise. Proposition 4.29.
Let c ∈ A = n be a finite tuple. Then the group Aut( O ∗ ,c / U ) alongwith its action on O ∗ ,c , is isomorphic to the group Aut G ′ ( O ∗ ,c ) . Proof.
Let σ be an automorphism in Aut( O ∗ ,c / U ). If m ∈ Hom G ′ ( O ∗ ,c , O i c ) and x ∈ O ∗ ,c ,we compute : m ( x ) = σ ( m ( x )) = σ ( m )( σ ( x )). The first equality comes from the fact m ( x ) ∈ U . The second one comes from the 0-definability of the action of the morphisms,which is therefore preserved by σ . Thus σ | O ∗ ,c = σ ( m ) − ◦ m ∈ Aut G ′ ( O ∗ ,c ) . For the converse, we use proposition 4.22. We are given an element σ c of Aut G ′ ( O ∗ ,c ),and wish to extend it into an automorphism of the whole structure ( U , O ( n ) ∗ , M ( n ) ∗ , N ( i,j ) ∗ )fixing U pointwise. We shall find morphisms σ d ∈ Aut G ′ ( O ∗ ,d ) for each finite tuple d in A , with the following coherence condition : if a is a subtuple of b , then the followingdiagram commutes : O ∗ ,b O ∗ ,b O ∗ ,a O ∗ ,aσ b ι ισ a For instance, let’s assume that we start with an automorphism σ abc of the object O ∗ ,abc .The degree 1 morphisms σ a , σ b , σ c are the ones uniquely determined by the commuta-tivity of the following diagrams : O ∗ ,abc O ∗ ,abc O ∗ ,abc O ∗ ,abc O ∗ ,abc O ∗ ,abc O ∗ ,a O ∗ ,a O ∗ ,b O ∗ ,b O ∗ ,c O ∗ ,cσ abc ι ι σ abc ι ι σ abc ι ισ a σ b σ c
34y proposition 4.22, this finite coherent family can be extended into a full coherentfamily ( σ a ) a ∈ A of degree 1 automorphisms of the O ∗ ,a . Because of the disjoint unionproperty, the higher degree automorphisms σ c are uniquely determined by the σ a .Finally, the action on morphisms is defined either as precomposition with the appro-priate σ c , postcomposition with the inverse of such a map, conjugation by it, or identity,depending on the domain and codomain of the morphism. Corollary 4.30.
Let n be a positive integer. Then the language of ( U , S a ∈ c O ∗ ,a ) is finitelygenerated over that of U , uniformly for c in A = n .Proof. We use proposition 4.13. The automorphism groups Aut( S a ∈ c O ∗ ,a / U ) are uni-formly interpretable in U ′ , since they are defined in the groupoid of degree n . Proposition 4.31.
Let C ⊆ A be a finite subset. Then the structure ( U , S a ∈ C O ∗ ,a ) isstably embedded in ( U , O ∗ ) .Proof. The proof is similar to that of propositions 4.26 and 4.29. The automorphisms σ we begin with are already defined on some of the new objects and morphisms. Forinstance, we are already given morphisms m a = σ | O ∗ ,a : O ∗ ,a → O ∗ ,a for a in C . Infact, by finite faithfulness, we can extend σ in a unique way to the simplicial groupoidrestricted to C . It suffices to check that these maps define a coherent family, with respectto the set-theoretic inclusions, and to apply proposition 4.22. From now on, we let V = ( U , S, p : S → A ) be a 1-analysable cover satisfying theconditions given at the beginning of subsection 4.1. The simplicial groupoids G ( V ) and G ′ ( V ) are the binding simplicial groupoids definable in U and V eq respectively. As in thecase of independent fibers, we shall also work with the cover C G ( V ) = ( U , O ∗ ) and thesimplicial groupoids G ( C G ( V )) and G ′′ ( C G ( V )). Theorem 4.32.
There exists an isomorphism of covers η V : V ≃ CG ( V ) . Proof.
Pick a coherent systems of inclusions ι = ( O i c → O i d ) in G ( V ). Then pickfamilies of morphisms ( g a : S a → O i a ) a and ( h a : O i a → O ∗ ,a ) a that belong to G ′ ( V ) and G ′′ ( C G ( V )) respectively, and which are coherent , with respect to ι and the set-theoreticinclusions. By proposition 4.22, such families exist. We shall prove that the structure( V ∪ CG ( V ) , S a ∈ A h a g a ) defines an isomorphism of covers.First, let us show that U is stably embedded in this structure. Let σ be an auto-morphism of U . We extend it arbitrarily into an automorphism of V ∪ CG ( V ). Then,the maps σ ( h a ) and σ ( g a ) are morphisms in G ′′ ( CG ( V )) and G ′ ( V ) respectively. Sincethe set theoretic inclusions are 0-definable, they are preserved by σ. Thus, the family ofmaps ( σ ( h a ) σ ( g a )) is coherent with respect to the set-theoretic inclusions in S and O ∗ ,just as ( h a g a ) is. As a consequence, the family of maps ( σ ( g − a ) σ ( h − a ) h σ ( a ) g σ ( a ) ) is also35oherent. To sum up, for all finite tuples c = ( a, ..., b ) ⊆ A made of pairwise distinctelements, there exist higher degree morphisms g, g ′ in G ′ ( V ) and h, h ′ in G ′′ ( CG ( V ))making the following diagram commute : S σ ( a ) O i σ ( a ) O ∗ ,σ ( a ) O σ ( i a ) S σ ( a ) S σ ( c ) O i σ ( c ) O ∗ ,σ ( c ) O σ ( i c ) S σ ( c ) S σ ( b ) O i σ ( b ) O ∗ ,σ ( b ) O σ ( i b ) S σ ( b ) g σ ( a ) h σ ( a ) σ ( h − a ) σ ( g − a ) ∃ g ιι ∃ h ∃ h ′− σ ( ι ) σ ( ι ) ∃ g ′− g σ ( b ) h σ ( b ) σ ( h − b ) σ ( g − b ) In fact, the maps g and h come from the coherence of the families ( g a ) and ( h a ). Besides,the families ( h − a ) and ( g − a ) are also coherent, with respect to the set-theoretic inclusionsand ι . Applying the model-theoretic automorphism σ to the commutative diagramsshows that the families ( σ ( h − a )) and ( σ ( g − a )) are coherent with respect to the set-theoretic inclusions and σ ( ι ). This implies the existence of the maps g ′ and h ′ .Now, let τ be the map S a ∈ A σ ( g − a ) σ ( h − a ) h σ ( a ) g σ ( a ) . Using lemma 4.6, we see that τ ∪ id U is an automorphism of V that fixes U pointwise, and sends the family of maps ( h σ ( a ) g σ ( a ) )to the family of maps ( σ ( h a ) σ ( g a )). Finally, the map ( τ ∪ id U ) − ◦ σ is an automorphismof ( V ∪ CG ( V ) , S a ∈ A h a g a ) extending σ .Thus, U is stably embedded in the structure ( V ∪ CG ( V ) , S a ∈ A h a g a ).Finally, we prove that V is stably embedded in ( V ∪ CG ( V ) , S a ∈ A h a g a ), the otherstable embeddedness statement being similar. Let σ be an automorphism of V , whichwe may assume to fix U pointwise. We define the action of σ on O ∗ by conjugating σ with the map S a ∈ A h a g a . Let τ be the map ( S a ∈ A h a g a ) ◦ σ | S ◦ ( S a ∈ A h a g a ) − . From the binding groupoid statements in V and CG ( V ), and lemma 4.6, this map τ ,when extended by the identity on U , is an automorphism of the structure CG ( V ) . Thus,the map τ ∪ σ is an automorphism of ( V ∪ CG ( V ) , S a ∈ A h a g a ) extending σ .So V is stably embedded in ( V ∪ CG ( V ) , S a ∈ A h a g a ). Remark 4.33.
This theorem implies that any interpretation of the extended simplicialgroupoid G ′′ ( CG ( V )) yields an interpretation of the cover V itself.A more precise study of this problem could hypothetically yield criteria for the in-terpretability of the cover V in the original structure U .For instance, if one could, inside the groupoid G , find canonical groupoids in eachdegree, and a 0-definable commuting system of inclusions between the unique objects ofthese canonical groupoids, one would be able to reconstruct the cover CG ( V ) inside U ,and thus interpret the cover V in the structure U itself.36 Functoriality
From now on, we let AC A denote the category of 1-analysable covers over A that satisfythe conditions given at the beginning of subsection 4.1, and SF CG A denote the categoryof the simplicial groupoids over A that satisfy the properties in proposition 4.10.We recall that morphisms of simplicial groupoids are sequences of morphisms ofgroupoids that are compatible with the inclusion morphisms defining the simplicial struc-tures. They are required to satisfy condition ( C ) in each connected component. G : AC A → SF CG A Definition 5.1.
Let h : U → U be a morphism of 1-analysable covers over A . Wedefine the simplicial groupoid morphism G ( h ) as follows : In degree n , the set of mapsdefining the groupoid morphism between the degree n groupoids G ( U ) n and G ( U ) n is given by G ( h ) n := { α ◦ h c ◦ α : O → O | c ∈ A = n , O ∈ Ob ( G ( U ) n ) , O ∈ Ob ( G ( U ) n ) , α ∈ Hom G ′ ( U ) n ( O , S ,c ) , α ∈ Hom G ′ ( U ) n ( S ,c , O ) } . Proposition 5.2.
For each integer n , the set of maps G ( h ) n is a 0-definable morphismof groupoids from G ( U ) n to G ( U ) n , that only depends on the theory of ( U ∪ U , h ) .Proof. As in the independent case, if we pick another map h ′ : U → U such that ( U ∪ U , h ) ≡ ( U ∪ U , h ′ ), then, taking saturated models, the structures are isomorphic. Sothere exists an automorphism σ of U ∪ U such that σh = h ′ σ . By stable embeddednessof U in ( U ∪ U , h ), we may assume that σ is the identity on U . So we get σh = h ′ .The binding groupoid statements in U enable us to conclude that, in each connectedcomponent, the sets of maps are equal.Now, we need to check that the sets of maps satisfy the morphism conditions. Weshall use the results proved in the simpler cases. We notice that, for each tuple c in A = n , the map h c : S ,c → S ,c defines a morphism of covers. Moreover, the connectedcomponents over c of the binding groupoids G ′ ( U ) and G ′ ( U ) are the binding groupoidsof the covers ( U , S ,c ) and ( U , S ,c ). Now, these covers are internal, so we already knowthat the set of maps { α ◦ h c ◦ α : O → O | O ∈ Ob ( G ( U ) n ) , O ∈ Ob ( G ( U ) n ) , α ∈ Hom G ′ ( U ) n ( O , S ,c ) , α ∈ Hom G ′ ( U ) n ( S ,c , O ) } defines a morphism of groupoids. Proposition 5.3.
The family of morphisms of groupoids G ( h ) n defines a morphism ofsimplicial groupoids.Proof. We need to check that the following diagrams commute, for each choice of inclu-sion ι : G ( U ) n +1 G ( U ) n +1 G ( U ) n G ( U ) nG ( h ) n +1 ι ιG ( h ) n
37t is easier to prove that the diagrams involving the extended groupoids commute : G ′ ( U ) n +1 G ′ ( U ) n +1 G ′ ( U ) n G ′ ( U ) nG ( h ) n +1 ι ιG ( h ) n Since all the sets of maps involved are morphisms of groupoids, we only need to checkthat, in each connected component, there exists a common map between the compositemorphisms of groupoids. To simplify notations, we assume that the inclusion maps thatdefine the morphism of groupoids ι are the set-theoretic inclusions. Let c = ( a , ..., a n )be a tuple in A n and d = ( a , ..., a n +1 ) be a tuple in A n +1 containing c . We claim thatthe map x ∈ S ,c h ( x ) ∈ S ,d belongs to both ι ◦ G ( h ) n and G ( h ) n +1 ◦ ι . Proposition 5.4.
The action of G on morphisms defines a functor G : AC A → SF CG A .Proof. The fact that G sends identities of AC A to identities of SF CG A is proved in thesame way as in the independent case.Now, it remains to check that finite composition is preserved by G . Working fiber-wise, this follows from the case of internal covers. C : Iso ( SF CG A ) → AC A Let H : G → G be an isomorphism in SF CG A . We wish to define a morphism of covers C ( H ) : C G → C G . As in the independent case, we shall use the extended groupoids G ′ and G ′ defined in the covers C G and C G respectively. Definition 5.5.
Let ι , ι be commuting systems of inclusions in G and G respectively.Let ( f c ), ( h c ), ( g c ) be families of maps that are coherent with respect to ι , ι and theset-theoretic inclusions. In other words, the following diagrams commute : S ,a O ,a O ,a S ,a S ,c O ,c O ,c S ,c S ,b O ,b O ,b S ,bf a h a g a f c ι ι h c ι ι g c f b h b g b We define C ( H ) as the theory of ( C G ∪ C G , S a ∈ A g a h a f a ) . First, as in the independent case, we need to make sure the choices made do notchange the theory of C ( H ). 38 emma 5.6. Different choices for the families of maps ( g c ) , ( h c ) , ( f c ) , ι , ι yield struc-tures ( C G ∪ C G , S a ∈ A g a h a f a ) that are isomorphic over U .Proof. We shall combine the ideas of the proof of proposition 3.5 and the coherenceconditions found in this section. Let ι ′ and ι ′ be different choices of commuting systemsof inclusions, and ( f ′ c ) , ( g ′ c ) and ( h ′ c ) be different choices of coherent families. We want toreduce to the case where only the f ′ c differ. To do this, we need to look at the followingcommutative diagrams : S ,c O ,c O ,c S ,c O ′ ,c O ′ ,c S ,c S ,a O ,a O ,a S ,a O ′ ,a O ′ ,a S ,af c h c g c f ′ c h ′ c g ′ c f a h a g a f ′ a h ′ a g ′ a Here, the vertical lines are inclusions, the non-dotted ones being the set-theoretic inclu-sions, the others being the ι i or ι ′ i , i = 1 , n and each tuple c in A = n , condition (A) of the morphism H n ,and bijectivity of the maps in H , implies the existence of a unique morphism σ c ∈ Aut G ′ ( S ,c ) = Aut( S ,c / U ) making the following diagram commute S ,c O ,c O ,c S ,c O ′ ,c O ′ ,c S ,c S ,a O ,a O ,a S ,a O ′ ,a O ′ ,a S ,af c ∃ ! σ c ∃ ! h c g c f ′ c h ′ c g ′ c f a ∃ ! h a g a f ′ a h ′ a g ′ a Now, by uniqueness, the family ( σ c ) n ∈ N , c ∈ A n makes the following diagram commute,for all finite tuples c and for all a in c : 39 ,c O ,c O ,c S ,c O ′ ,c O ′ ,c S ,c S ,a O ,a O ,a S ,a O ′ ,a O ′ ,a S ,af c σ c h c g c f ′ c h ′ c g ′ c f a σ a h a g a f ′ a h ′ a g ′ a Finally, using the criterion given in lemma 4.6, we find that the map τ := S a σ a ∪ id U isan automorphism of C G . Thus the map ρ := τ ∪ id C G is an automorphism of C G ∪ C G that fixes U pointwise and satisfies ρ ◦ ( S a ∈ A g a h a f a ) = ( S a ∈ A g ′ a h ′ a f ′ a ) ◦ ρ .We shall now prove that this C ( H ) defines a morphism of covers. We will proceedas in lemma 3.6 and proposition 3.7. Lemma 5.7.
The theory of C ( H ) is a cover of the theory of U .Proof. The same proof as in lemma 3.6 works.
Proposition 5.8.
The theory of C ( H ) is a cover of both C G and C G .Proof. Adapting the proof of proposition 3.7 to the need for higher degree coherencemorphisms, we find a way of extending automorphisms of C G into automorphisms of( C G ∪ C G , S a ∈ A g a h a f a ) :Let σ be an automorphism of C G . We may assume that it fixes U pointwise.Using the binding groupoid statement for G ′ in C G and condition (A) of the morphismof groupoids H , there exists, for each tuple c , a unique morphism σ ,c ∈ Aut G ′ ( S ,c )such that σ ,c g c h c f c = g c h c f c σ | S c . By uniqueness again, the family of maps ( σ ,a ) a ∈ A is coherent. Thus, it defines an automorphism σ of C G over U , with the equality : σ g a h a f a = g a h a f a σ . Proposition 5.9.
The definition of C yields a functor C : Iso ( SF CG A ) → AC A .Proof. As in the independent case, the action on the identities is easy enough to under-stand, and composition can be dealt with by picking literal compositions of morphismswhen choosing the h a , f a and g a . See proposition 3.8 for the details. Remark 5.10.
Under additional hypotheses, such as finitariness of all the automor-phism groups in the groupoids, or some stable embeddedness statements, we can definethe functor C on morphisms of groupoids, and not just isomorphisms of groupoids.40 .3 Comparing simplicial groupoids Definition 5.11.
Let G be an element of SF CG A . Let G ′ be the extension of G definingthe extra structure in the cover C ( G ) = ( U , S ). Let G ′′ ( C ( G )) be the binding groupoidof the cover C ( G ) . Let n be an integer. We define the set of maps ε G ,n := { g ◦ f : O ,i → O ,i | i ∈ Ob ( G n ) , i ∈ Ob ( GC ( G ) n ) , ∃ c ∈ A = n , g ∈ Hom G ′′ ( C ( G )) ( S c , O ,i ) ∧ f ∈ Hom G ′ ( O ,i , S c ) } . Proposition 5.12.
The sets of maps ε G ,n define an isomorphism ε : G → G ( C ( G )) in SF CG A .Proof. The same proof as in proposition 3.22 shows that, in each degree, the set of maps ε G ,n is an isomorphism of groupoids. It remains to show that this family of maps iscompatible with the inclusions that define the simplicial structures of G and G ( C ( G )).This can be checked using the same ideas as in proposition 5.3, namely, extending thegroupoid morphisms to G ′ and G ′ ( C ( G )), and checking the conditions there using thedistinguished objects S c . Theorem 5.13.
The functors G : Iso ( AC A ) → Iso ( SF CG A ) and C : Iso ( SF CG A ) → Iso ( AC A ) are equivalences of categories.Proof. The only statement not already proved is the fact that ε : id SF CG A → GC and η : id AC A → CG are natural isomorphisms. Working fiberwise, this is implied by thecase of internal covers. Conjecture 5.14.
The categories AC A and SF CG A are equivalent. Moreover, thisequivalence can be proved by extending the functor C : Iso ( SF CG A ) → Iso ( AC A ) tonon-isomorphisms using a similar definition. In this section, we intend to understand the general constructions in two examples. ( Z / Z ) ω In this subsection, we will try and apply our general constructions to the example ofthe two-sorted structure V := (( Z / Z ) ω , ( Z / Z ) ω , ι, π ). Here, the maps ι and π are thegroup morphisms that appear in the following exact sequence :0 ( Z / Z ) ω ( Z / Z ) ω ( Z / Z ) ω ι π Note that the F -vector space ( Z / Z ) ω does not eliminate imaginaries, so we haveto work in (( Z / Z ) ω ) eq roposition 6.1. Let ( σ c ) c ∈ A ≤ be an element of the projective limit of the restrictedsystem : (Aut( S c )) c ∈ A ≤ .Then this family induces an automorphism of the structure ( U , S ) over U .Proof. We define the map σ := S a ∈ A σ a .We first compute that, on each fiber S a , the map σ a is of the form x x + x a ,where x a is some element of S . Indeed, if x, y ∈ S a , then there exists some z ∈ ( Z / Z ) ω such that x − y = ι ( z ). Since σ a is an automorphism of S a over U , we compute : σ a ( x ) − σ a ( y ) = σ a ( ι ( z )) = ι ( σ a ( z )) = ι ( z ) = x − y. Thus σ a ( x ) − x = σ a ( y ) − y =: x a . Then, since each σ a ∪ σ b ∪ σ c is an automorphism of ( U , S a ∪ S b ∪ S c ), the followingcondition appears : x a + b = x a + x b for all a, b ∈ A . In particular, x = 0. Therefore,the map σ ∪ id U is a group automorphism commuting with ι and π . It is thus anautomorphism of ( U , S ). Remark 6.2.
In this case, the degenerate simplicial groupoid defined with G ′ n := G ′ if n ≥
3, and identities in higher degrees instead of the inclusion morphisms, captures thewhole structure of the cover ( U , S ) over U . Corollary 6.3.
The sort ( Z / Z ) ω is not internal to ( Z / Z ) ω .Proof. The automorphism group Aut( S/ U ) is isomorphic to the group End(( Z / Z ) ω ).The latter has cardinality 2 ℵ . Thus, it cannot be interpreted in ( Z / Z ) ω . Remark 6.4.
In this example, the groupoids in degree 1 can be chosen to be canonical ones, i.e. with one object in each isomorphism class. Namely, the object is the kernel K that appears in the short exact sequence, and the bijections K ≃ S a are the translations.However, in higher degrees, the non-existence of a section of the short exact sequencemakes canonical groupoids harder, or even impossible to find. For instance, if a + b = c in ( Z / Z ) ω , then you have to create one object in G a,b,c for each possible value of( x a + x b − x c ) ∈ K , if x a ∈ S a , x b ∈ S b and x c ∈ S c . These values classify the ”kinds ofbijections” K ⊔ K ⊔ K ≃ S a ⊔ S b ⊔ S c mentioned in remark 2.9. And, since sections donot exist, the 0-definable value 0 ∈ K is usually invalid, since no bijection realizes it. d ( dxx ) = 0 Let M | = DCF . Let C be the field of constants. Let’s consider the following definablesubgroup of M × : S := { x ∈ M × : d ( dxx ) = 0 } .We notice that the 2-sorted structure ( C, S ) comes with a definable short exactsequence 1 → C × → S → C → Proposition 6.5.
Let ( σ c ) c ∈ C ≤ be an element of the projective limit of the restrictedsystem : (Aut( S c )) c ∈ C ≤ . In other words, we are given a family of automorphisms σ a ∈ Aut( S a / U ) , such that, for each triple ( a, b, c ) , the map σ a ∪ σ b ∪ σ c is an automorphismof S a ∪ S b ∪ S c over U .Then this family induces an automorphism of the structure ( C, S ) over C . roof. We define the map σ := S a ∈ C σ a . We know that σ a : x x × λ a , where λ a issome element of S = C × .When considering σ a,b,a + b , the following condition appears : λ a + b = λ a × λ b for all a, b ∈ C . In particular, λ = 1.We now have to check that this condition is enough to get an automorphism of thestructure induced by the differential field M .Using quantifier elimination in DCF , we reduce to the case of formulas of the form Q ( x , ..., x n ) = P ( x , d ( x ) , ..., x n , d ( x n ) , ..., d m ( x n )) = 0, where the coefficients of P arein C . We already know that σ preserves multiplication and differentiation on C, S . Claim 6.6.
The fibers S a are linearly independent over C .Proof. The differential is C -linear, and each fiber S a is made of eigenvectors associated tothe eigenvalue a . An argument from linear algebra tells us that eigenvectors associatedto distinct eigenvalues have to be linearly independent.Let us assume that x , ..., x n ∈ S are roots of the differential polynomial Q above.We write Q ( x , ..., x n ) as a sum of products of elements of S . By the claim, all theseterms are in the same fiber. Let t , ..., t k ∈ S a be these terms. Each of them is a productof some of the d j ( x i ), with possibly a coefficient in C . By hypothesis, on S a , the map σ acts as multiplication by λ a . Thus, we have σ ( t ) + ... + σ ( t k ) = λ a ( t + ... + t k ) = 0.Moreover, we recall that σ preserves multiplication in S , differentiation and product withelements of C . So each σ ( t r ), for r = 1 , ..., k , is a product of some of the d j ( σ ( x i )), withthe same coefficient in C as the term t r . Thus, P ( σ ( x ) , d ( σ ( x )) , ..., d m ( σ ( x n ))) = 0, asdesired. Proposition 6.7.
Let G ≤ C be a finitely generated subgroup. Let s : G → S be a groupmorphism such that f ◦ s = id G , where f : S → C is the surjection in the short exactsequence mentioned above.Then there exists a section of f which is a group morphism that extends s .Proof. We will need the following lemma :
Lemma 6.8.
Let x = x be an element of S . Then, there exists a coherent system ofroots of x . More precisely, there exists a family ( x n ) n ≥ of elements of S such that, forall k, l ≥ , we have ( x kl ) k = x l .Proof. We use ω -saturation of the ambient structure. In fact, the conditions can beexpressed straightforwardly in a type, with ω variables (one for each x n ) and 1 parameter(which is x ). To prove consistency, we only need to pick a root of x of high enoughdegree, and take its powers.Finally, to prove that the x n are in S , we use the formula : n dx n x n = d (( x n ) n )( x n ) n = dxx ∈ C .Thus dx n x n ∈ C . 43ow, since G is a finitely generated torsion-free abelian group, we can find a Z -basis g , ..., g k ∈ G . This basis is also a Q -basis of the vector space V generated by G . Let W ≤ C be a Q -vector space such that V ⊕ W = C . Note that this direct sum is also adirect sum of abelian groups. Thus, it suffices to define a group morphism on W that issection of f : S → C , and to extend s : G → S into a group morphism s : V → S suchthat f ◦ s = id V .We will detail the construction for V , the one for W being slightly easier. For i = 1 , ..., k ,let ( h i, n ) n be a coherent system of roots of s ( g i ). If x = λ g + ... + λ k g k ∈ V , wherethe λ i = a i b i are in Q , we define s ( x ) := k Q i =1 ( h i, bi ) a i . Note that, if ab = cd , then h i, d =( h i, ad ) a = ( h i, bc ) a . Thus, ( h i, d ) c = ( h i, bc ) ac = ( h i, b ) a . So s is well-defined on V .Let us now check that s is a group morphism. Let x = a b g + ... + a k b k g k and y = c d g + ... + c k d k g k . So x + y = a d + c b b d g + ... + a k d k + c k b k b k d k g k .By definition, we have s ( x + y ) = k Q i =1 ( h i, bidi ) a i d i + b i c i .= k Q i =1 ( h i, bidi ) a i d i × k Q i =1 ( h i, bidi ) b i c i = k Q i =1 ( h i, bi ) a i × k Q i =1 ( h i, di ) c i = s ( x ) s ( y ).Thus, s is a group morphism. We notice that it extends the morphism that was definedon G . It remains to check that it is a section of f . Let 1 ≤ i ≤ k , let b ≥
1. Wewant to prove that f s ( b g i ) = b g i . Note that this is enough, since id V and f s are groupmorphisms, and the b g i generate the abelian group V .We compute : bf ( s ( b g i )) = f (( s ( b g i )) b ) = f s ( g i ) = g i , since s | G was assumed to be asection of f . Thus, since this equality holds in C , which is torsion-free and divisible, wehave b g i = f ( s ( b g i )), as desired. Proposition 6.9.
Let B = { b , ..., b k } be a finite subset of C . Let f : B → S be suchthat, whenever there is a relation of the form P i n i b i = 0 , then the relation Q i f ( b i ) n i = 1 holds.Then, f extends uniquely into a group morphism f : < B > → S .Proof. Uniqueness is clear, for the extension has to be defined by the following formula: P i n i b i Q i f ( b i ) n i . For existence, it suffices to check that, if P i n i b i = P i m i b i ,then Q i f ( b i ) n i = Q i f ( b i ) m i , for all tuples of integers ( n , ..., n k ) , ( m , ..., m k ). This isprecisely given by the hypothesis on f . Remark 6.10.
Even though this last result shows that automorphisms of finite unionsof fibers can be extended, there is an issue with uniform finite generatedness of thelanguages in degree 2 and above. Namely, if a ∈ C , and if n is an integer, the relation x n = y is definable in ( U , S a ∪ S n.a ). Since n can be arbitrarily large, uniform finitegeneratedness of the languages does not hold.44 roposition 6.11. The binding simplicial groupoid of ( U , S ) is type-definable, and canbe chosen to be canonical.Proof. Let n be an integer. We shall now define the degree n binding groupoid. Foreach element c ∈ A = n , we define the only object O c as the set S a ∈ c C × × { a } .Let X c be the set of tuples of couples of the form (( x , a ) , ( x , a ) , ..., ( x n , a n )), wherethe tuple ( a , ..., a n ) defines the same set as c , and x , ..., x n are elements of C × .Let E c be the quotient of X c under the relation ”being equal up to permutation”.We now define the set M c as the set of elements m of E c such that, for any repre-sentative (( x , a ) , ( x , a ) , ..., ( x n , a n )) of m , for any tuple of integers ( k , ..., k n ), if therelation P i k i a i = 0 holds in the group C , then the relation Q i x k i i = 1 holds in thegroup C × .Note that the set M c always contains the class of ((1 , a ) , ..., (1 , a n )). Moreover, itis type-definable, uniformly over A = n . The action of M c on O c is given by fiberwisetranslation.We thus get a type-definable groupoid in U . Using propositions 6.7 and 6.9, it isnot hard to extend this groupoid by adding the object S c in the component over c , for c ∈ A = n .Indeed, let us define the morphisms O c → S c as the fiberwise translations by elementsof the S a , for a ∈ c , that satisfy the condition of proposition 6.9. Then, the group ofgroupoid automorphisms of S c is the group of fiberwise translations by elements of C × that also satisfy the condition of proposition 6.9. Thus, by propositions 6.7 and then6.5, these bijections are in fact elements of Aut( S c / U ). Remark 6.12.
Contrary to the previous example, the groupoids can be chosen to be canonical in all degrees, for there exist enough sections C → S . In other words, thedistinguished value 1 (the identity element in S ) can be picked. However, these groupoidsare type-definable without parameters, and not 0-definable, as we shall prove next. Proposition 6.13.
Let n ≥ . Then, the languages of the covers ( S c , U ) are not uni-formly finitely generated over the language of U , for c ∈ A = n , even though each languageis finitely generated.Proof. We shall prove the fact for n = 2, the other cases being similar. By contradiction,assume that the languages of the internal covers ( S a ∪ S b , U ) are uniformly finitelygenerated, for a, b ∈ A , a = b . Then, the automorphism groups Aut( S a ∪ S b / U ), alongwith their actions on the S a ∪ S b , are uniformly definable in ( U , S ). Let φ ( a, b, m ) be aformula defining ” m is the code of an automorphism of S a ∪ S b over U .” Let ψ ( a, b, m, s, s ′ )be a formula defining the graphs of the maps defined by such codes m . We may assumethat φ and ψ are given as in proposition 6.11 above.Moreover, propositions 6.5, 6.7 and 6.9 give us a set of definable relations that gen-erate the languages at stake. Indeed, if k, l ∈ Z , we define the relation R k,l ( x, y ) as x k = y l . Then, by the results proved in the propositions mentioned above, we have :45 ” m defines, via ψ ( a, b, m, s, s ′ ), a map on S a ∪ S b which preserves the relationinduced by R k,l on ( S a ∪ S b ) ” — k, l ∈ Z } | = φ ( a, b, m ). Here, the variables are both a, b and m .Thus, by ω -saturation and compactness, a finite fragment of the set of relations R k,l suffices to ensure that the map defined by some code m is an automorphism. The crucialpoint is that this happens uniformly in a, b . Now, take an integer N that bounds the sizeof the integers k, l that appear in such a finite fragment. Pick some non-zero element a ∈ A . The relation R N, ( x, y ) has to be preserved by any automorphism of S a ∪ S N.a .However, there exist tuples m that code permutations of S a ∪ S N.a that preserve therelations R k,l , for | k | , | l | ≤ N , and which do not preserve the relation R N, . Simply takeany ”generic” pair of elements λ, µ of S - in particular, with λ N = µ - and considerthe map m λ,µ which acts as translation by λ and µ on S a and S N.a respectively.Finally, the relations induced by the R k,l , for | k | , | l | ≤ N , on ( S a ∪ S N.a ) are trivial,and thus automatically preserved by the map m λ,µ . However, the relation R N, is notpreserved by this map, which is a contradiction. Theorem 6.14.
The group S is isomorphic, but not definably so, to the direct product C × C × . The cover ( C, S ) is interpretable in the field C .Proof. The existence of group-theoretic sections of the short exact sequence 1 → C × → S → C → S is isomorphic to a semidirect product C × ⋊ C .Since the group S is abelian, this semidirect product is in fact a direct product.Note that the cover is not internal, for there are at least 2 | C | elements in the au-tomorphism group Aut(( C, S ) /C ). Thus, there are no definable group isomorphisms S ≃ C × × C .Interpretability in the field C follows, for the proof of proposition 6.5 shows thatthe structure induced on S by the differential field is only the group-theoretic structureinduced by the short exact sequence mentioned above. Remark 6.15.
Here, group isomorphisms S ≃ C × × C are built using families ofmorphisms f a : C × → S a that are coherent relative to the set-theoretic inclusions onboth sides. Since the inductive construction of such coherent families involves manychoices, it is not surprising to find that they are not definable.Moreover, the theorems proved in the next section should help understand why in-terpretability of S in C , in this example, is not a coincidence. In this section, we present the type-definable groupoids that appear when the technicalhypotheses of uniformly finitely generated languages are not satisfied, just as in theexample above. We shall call them ”*-definable” or ”type-definable” indiscriminately,for these are the only kinds of type-definable groupoids we will consider here.46rom now on, we study 1-analysable covers that may not satisfy the hypothesis ofuniform finite generatedness of the languages, and keep only the following hypothesis :For all finite subsets
C, D ⊆ A , the structure ( U , S a ∈ C S a ), with the structure induced bythe C -definable sets of ( U , S ), is stably embedded in ( U , S a ∈ C ∪ D S a ). Definition 7.1.
1. A concrete groupoid is ∗ -definable if its set of objects and itsset of morphisms are type-definable without parameters, with the types havingpossibly infinitely many variables.We still require the morphisms and objects to be definable maps and definablesets, whose definitions only use finite fragments of the tuples involved. We shallrefer to these finite fragments as the concrete parts of the infinite tuples.We weaken the assumption that two distinct objects (resp. morphisms) definedistinct sets (resp. bijections). Instead, we only require that two distinct objects(resp. morphisms) that define the same set (resp. bijection) have the same concretepart.So, each object (resp. each morphism) in the abstract groupoid is defined byan infinite tuple, and such an infinite tuple contains a finite tuple which is thecode, in the model-theoretic sense, of a definable set (resp. a definable bijection).Type-definability means that finite tuples (that code definable bijections) codemorphisms between objects if and only if they realize a specific type over theinfinite tuples that code the objects. Note that the infinite tuples defining themorphisms are merely made of a concrete part, and the infinite tuples defining thesource and target of the morphism.We suggest the reader have a look at the proof of theorem 7.3 to understand whywe picked this notion for type-definable groupoids.2. A concrete simplicial groupoid is ∗ -definable if the groupoids in each degree andthe inclusion morphisms are ∗ -definable.We will restrict to the case where the *-definable (simplicial) groupoids are natu-rally projective limits of 0-definable (simplicial) groupoids. More precisely, we askthat there exist coverings of the index sets of variables by finite subsets containingthe concrete parts, with the following properties : • The objects defined by restriction to these finite sets of coordinates are 0-definable (simplicial) groupoids. • The union of two finite subsets belonging to the covering is also an elementof the covering.3. A morphism of *-definable groupoids is a *-definable set of definable maps thatsatisfies conditions (A) and (B). Again, the types defining the sets of maps mayhave infinitely many variables, but each map is defined with a finite concrete part.
Remark 7.2.
A morphism of *-definable groupoids yields a collection of morphisms of0-definable groupoids between the projections of the two groupoids, by projection.47 .1 Type-definable binding simplicial groupoids
Theorem 7.3.
Let U ′ = ( U , S ) be a 1-analysable cover of U over A that satisfies thehypothesis above. Then there exist simplicial groupoids G and G ′ , over the set A , thatare *-definable in U and U ′ eq respectively, with the following properties: • The simplicial groupoid G ′ is an extension of G with the extra object S c in theconnected component over c . Moreover, the set-theoretic inclusions between the S c belong to the simplicial groupoid. • The automorphism groups
Aut G ′ ( S c ) , along with their action on S c , are isomorphicto the groups Aut( S c / U ) .Proof. Let n be an integer. We wish to build the degree n groupoid.First, by compactness, internality and elimination of imaginaries in U , there exista 0-definable set B ⊆ U , a 0-definable surjective map B → A = n and 0-definable sets X ⊆ U , F ⊆ B × S × X, such that, for all c ∈ A = n , for all b ∈ B c , the set F b is the graphof an embedding F b : S c ֒ → X .Let R ( x ) be a 0-definable relation in U ′ . Let c be an element of A = n . For thesame reasons as in theorem 2.8, there exists a formula φ ( z, α c ), where α c ∈ U such that | = ∃ z ∈ B c , φ ( z, α c ) and | = ∀ z, t ∈ B c , ( φ ( z, α c ) ∧ φ ( t, α c )) → ” F − t ◦ F z ∪ id U preservesthe relation induced by R ( x ) on ( U , S c ) .”Now, by compactness, we can find a formula φ R ( z, w R ) that satisfies the above state-ment uniformly for c ∈ A = n . Then, keeping the same notion of ”kinds of bijections” asin theorem 2.8, we define the set of codes of the objects of G n as the set of infinite tuples( u, c, ( w R ) R ) that satisfy { c ∈ A = n } ∪ {∃ t ∈ B c , φ R ( t, w R ) ∧ ... ∧ φ R k ( t, w R k ) ∧ ” u is thecode of the set Im ( F t )” | k ∈ N , R , ..., R k : 0-definable relations } The set defined by such an infinite tuple ( u, c, ( w R ) R ) is merely the set coded by u .Note that here, two distinct objects in the groupoid may have the same underlying set.Now, the set of morphisms from S c to the object corresponding to the infinite tuple( u, c, ( w R ) R ) is coded by the set of tuples ( t, c, ( w R ) R ) realizing the type { t ∈ B c } ∪{ φ R ( t, w R ) | R : 0-definable relation } . The bijections coded by such tuples ( t, c, ( w R ) R )are just the maps F t .Note that these sets are non-empty by saturation, and uniformly *-definable. Moreover,the construction gives precisely a projective limit of groupoids, where the covering canbe indexed by the finite collections of 0-definable relations.Now, the rest of the construction follows just as in theorem 4.7. For instance, theinclusions are conjugates of the set-theoretic ones S c ֒ → S d by morphisms in the com-ponents over c and d . They can be coded by pairs of codes of morphisms, and are thus*-definable. Again, just as in theorem 4.7, condition (A) for the inclusions is given bythe technical stable embeddedness hypotheses. Remark 7.4.
The simplicial groupoids built that way are finitely faithful and have the disjoint union property , just as those given by theorem 4.7.48 .2 The cover associated to a type-definable simplicial groupoid
Proposition 7.5.
Let G be a *-definable simplicial groupoid over A . Then there existsa commuting system of inclusions in G .Proof. The same inductive construction as in proposition 4.17 works, for the types in-volved in the saturation arguments still have few enough variables and parameters.
Proposition 7.6.
Let G , G be *-definable simplicial groupoids over A with the disjointunion property. Let H : G → G be a morphism of simplicial groupoids. Let ι = ( O i c → O i d ) c ⊂ d ∈ A <ω and ι = ( O j c → O j d ) c ⊂ d ∈ A <ω be commuting systems of inclusions in G and G respectively.Then there exists a coherent family of maps ( m a : O i a → O j a ) a ∈ A belonging to themorphism of simplicial groupoids H .More generally, for all finite subsets B ⊂ A , any coherent family of maps ( m a : O i a → O j a ) a ∈ B can be extended into a coherent family of maps ( m a : O i a → O j a ) a ∈ A .Proof. The same inductive construction as in proposition 4.22 works. The only detail totake care of is removing the existential quantifiers that express the existence of higherdegree ”coherence morphisms”, and replacing them with more variables to account forthese uniquely determined coherence morphisms. Since the types involved still have fewenough parameters and variables, they can be realized.
Theorem 7.7.
Let G be a finitely faithful *-definable simplicial groupoid in U over A ,with the disjoint union property. Then there exists a cover ( U , O ∗ ) of U , along with anextension G ′ of G that is *-definable in ( U , O ∗ ) , such that : • The cover ( U , O ∗ ) is 1-analysable over A , and satisfies the technical hypothesisgiven at the beginning of section 7. • The simplicial groupoid G ′ is an extension of G with the extra object O ∗ ,c in theconnected component over c . Moreover, the set-theoretic inclusions between the O ∗ ,c belong to the simplicial groupoid. • The automorphism groups
Aut G ′ ( O ∗ ,c ) , along with their action on O ∗ ,c , are iso-morphic to the groups Aut( O ∗ ,c / U ) .Proof. Just as in the case of 0-definable simplicial groupoids, we wish to extend G bypicking a commuting system of inclusions spanning all the connected components, andcreating copies of the objects that appear in this system.We use proposition 7.5. We can thus pick a commuting system of inclusions ι in thesimplicial groupoid G .Then, just as before, we are able to create copies of objects using maps f c , in acoherent way : 49 ∗ ,a ∪ ... ∪ O ∗ ,a n O ∗ ,a O ∗ ,a n ... ...O i a O i an O i a ...an f a ...an ι ι f a f an ι ι Here, the ι are the set-theoretic inclusions, and the ι are in the commuting systempicked earlier. Note that we only consider the case of pairwise distinct a i .Again, apart from the new object sort O ∗ , we create new morphism sorts M ( n ) ∗ and N ( i,j ) ∗ , by creating copies of the concrete parts of the tuples defining the morphisms andinclusion morphisms.Now, we use the projective limit property of G . We pick an appropriate covering ofthe sets of coordinates by finite subsets. To fix notations, let I n be the set of variablesin the types defining the degree n groupoid. We cover I n with finite subsets F i,n , for i belonging to some set J . We let G i denote the 0-definable simplicial groupoid given bythe restrictions to the F i,n . If i, j ∈ J , we let i ∨ j ∈ J denote the index correspondingto the unions F i,n ∪ F j,n .We thus get back to the case of theorem 4.23, by looking at each i ∈ J . This way, weadd definable relations to the structure ( U , O ∗ ) that still yield a cover of U , and buildthe only 0-definable extension G ′ i of G i for which the copying maps f c : O ∗ ,c → O i c wealready picked are isomorphisms. To be clear, the new relations define which ”concreteparts” of morphisms, i.e. elements of the M ( n ) ∗ or N ( i,j ) ∗ , are morphisms between givenobjects of G ′ i .Then, we combine all these structures together. We still have a 1-analysable coverof U . In fact, because of the finite union property, any relation definable in the wholestructure is definable using only the structure of one of the G ′ i .We show that the projective limit of the G ′ i is an extension G ′ of G that is *-definableand satisfies the conditions of the theorem.The inclusion Aut( O ∗ ,c / U ) ⊆ Aut G ′ ( O ∗ ,c ) is proved the same way as in theorem 4.23.Let σ ∈ Aut G ′ ( O ∗ ,c ). We wish to extend σ into an automorphism of the full struc-ture ( U , O ∗ ). This amounts to extending σ into an automorphism of the simplicialgroupoid G ′ . Applying proposition 7.6, we find a coherent family of automorphisms, inthe groupoid sense, of the O ∗ ,d , for d ∈ A <ω , that extends σ . Such a family defines anautomorphism of the full structure ( U , O ∗ ).Finally, let us show that the required stable embeddedness conditions hold. Let c ⊂ d ∈ A <ω . Let σ be an automorphism of ( U , O ∗ ,c ), which we may assume to fix U σ ∈ Aut G ′ ( O ∗ ,c ), and the previous paragraph shows that σ can be extended into an automorphism of ( U , O ∗ ,d ). Proposition 7.8.
The cover built from the simplicial groupoid G , up to isomorphism,does not depend on the choices made.Proof. The only fact not shown before is that two choices of coverings define the samestructure. Let J , J be index sets for coverings of the set of variables that satisfy the twoconditions given in definition 7.1. Since the underlying sets of the new sorts O ∗ , M ( n ) ∗ can be assumed to be fixed, we only have to show that the definable relations are thesame.Let i ∈ J . We want to show that the extended simplicial groupoid G ′ i is definable withthe structure induced by some G ′ j , for some j ∈ J . Since the set defined by i is finite,and J defines a covering, it can be ”covered” with finitely many elements j , ..., j k ∈ J of the covering associated to J . So, let j = j ∨ j , ..., ∨ j k ∈ J be the index of a subsetcovering that of i . It remains to notice that the simplicial groupoid G ′ i is the projectionof the simplicial groupoid G ′ j on the coordinates coreesponding to i .By symmetry, we conclude that the structure thus defined does not depend on thecovering. From now on, we let V = ( U , S, p : S → A ) be a 1-analysable cover satisfying theconditions given at the beginning of section 7. The simplicial groupoids G ( V ) and G ′ ( V )are the binding simplicial groupoids that are *-definable in U and V eq respectively.As in the case of 0-definable simplicial groupoids, we shall also work with the cover C G ( V ) = ( U , O ∗ ) and the simplicial groupoids G ( C G ( V )) and G ′′ ( C G ( V )). Theorem 7.9.
There exists an isomorphism of covers η V : V ≃ CG ( V ) . Proof.
Pick a coherent systems of inclusions ι = ( O i c → O i d ) in G ( V ). Then pickfamilies of morphisms ( g a : S a → O i a ) a and ( h a : O i a → O ∗ ,a ) a that belong to G ′ ( V ) and G ′′ ( C G ( V )) respectively, and which are coherent , with respect to ι and the set-theoreticinclusions. By proposition 7.6, such families exist. Then, just as in theorem 4.32, thestructure ( V ∪ CG ( V ) , S a ∈ A h a g a ) defines an isomorphism of covers. Remark 7.10.
This theorem shows that the *-definable groupoids built in theorem 7.3indeed capture the whole structure of the 1-analysable covers.
Corollary 7.11.
Assume that the binding groupoid G of the cover V over U is canonical,and that there exists a commuting system of inclusions which is -definable. Then thecover V is interpretable in U . roof. Here, the reconstruction of the cover V from its binding groupoid, as described intheorem 7.7, can be done inside U itself. Indeed, adding a new object in each connectedcomponent is the same as duplicating the whole type-definable simplicial groupoid, sincethe latter is canonical. Note that the copying maps in all degrees can be chosen to be0-definable (over A , of course), precisely because we can choose the commuting systemof inclusions in G to be 0-definable.Now, for each 0-definable projection of the type-definable simplicial groupoid G , thecorresponding extended simplicial groupoid is 0-definable, using conjugation by the 0-definable copying maps. Thus, the structure of the cover C G ( V ) is interpretable in U .Since isomorphisms of covers are bi-interpreations, the structure V is therefore inter-pretable in U . From now on, we let AC ∗ A denote the category of 1-analysable covers over A that satisfythe conditions given at the beginning of section 7, and SF CG ∗ A denote the category ofthe *-definable finitely faithful simplicial groupoids over A that satisfy the disjoint unionproperty. Theorem 7.12.
The categories
Iso ( AC ∗ A ) and Iso ( SF CG ∗ A ) are equivalent.Proof. The definitions of the functors C : Iso ( SF CG ∗ A ) → Iso ( AC ∗ A ) and G : AC ∗ A → SF CG ∗ A are the same as in section 5. The natural isomorphism ε : GC ≃ id Iso ( SF CG ∗ A ) is also built as in definition 5.11. References [CH99] Z. Chatzidakis and E. Hrushovski. “Model theory of difference fields”. In:
Transactions of the American Mathematical Society (1999).[HM18] L. Haykazyan and R. Moosa. “Functoriality and uniformity in Hrushovski’sgroupoid-cover correspondence”. In:
Annals of Pure and Applied Logic (2018).[Hru12] E. Hrushovski. “Groupoids, imaginaries, and internal covers”. In: