Model theory and the Tannakian formalism

MODEL THEOR Y AND THE T ANNAKIAN FORMALISM MOSHE KAMENSKY Abstract. W e draw the con nection bet we en th e model theoretic notions of int ernality and the binding group on one hand, and the T annakian form alism on the other. More precisely , w e deduce the fundamen tal results of the T annakian formal- ism by asso ciating to a T annakian categ ory a first order theory , and applying the results on internalit y there. W e also formulate the notion of a di fferen- tial tensor cate gory , and a ve rsion of the T annakian formalism for different i al linear gr oups, and show how the same tec hniques can be used to deduce the analogous r esults in that con text. Introduction The aim of this paper is to exhibit the analogy and relationship betw een tw o seemingly unrelated theo ries. On the one hand, the T anna kian formalis m, g iving a duality theor y betw ee n affine group schemes (o r, more generally , gerbs) and a certain type of categor ies with a dditional structure , the T annakia n categ ories. On the other hand, a general no tion of internality in m o del theory , v alid for an a rbitrary first o rder theor y , that gives r ise to a definable Galois group. The analog y is made precise b y deriving (a weak v e r sion of ) the fundamen tal theor em of the T annakian duality (Theorem 2.8) using the mo del theo retic in ternality . The T annak ia n formalism ass igns to a group G over a field k , its catego ry of representations R ep G . In the version w e ar e ma inly interested in, due to Saav edra (Saav edra -Riv ano [24]), the gro up is an affine group scheme ov er a field. A similar approach works with groups in other categories, the first due to Krein and T annak a, concerned with lo cally compact to po logical gro ups. Another exa mple is provided in Section 4. In the algebra ic case, the categor y is the catego r y o f algebra ic finite- dimensional representations. This is a k -linear catego r y , but the catego ry structur e alone is not sufficient to r ecov er the gro up. O ne therefore considers the additiona l structure given by the tensor pro duct. The T annakian formalism says that G can be recov ered fr om this s tr ucture, together with the forgetful functor to the categor y o f vector-spaces. The other half of the theo r y is a descr iption o f the tensor ca tegories that ar is e a s categor ies o f representations: any tensor ca tegory satisfying suitable axioms is of the form R ep G , pr ovided it has a “fibre functor” into the categor y of vector spaces. Our main refer e nc e s for this sub ject a re the firs t thr ee sections of Deligne and Milne [4] and Deligne [3]. In mo del theory , in ternality was dis cov er e d by Zilb er as a to ol to study the s tr uc- ture o f s trongly minimal structure (Zil 1 ber [27]). Later, Poizat rea lised (in Poizat [23]) that this no tion ca n b e used to treat the Galo is theory of differe ntial equations. The definable Galo is co rresp ondence outlined in Theorem 1.6 ha s its origins there. Later, the theory was gener alised to large r class es o f theorie s (Hrushovski [10], Hart 1 2 M. KAMENSK Y and Shami [7], etc.), and applied in v ar ious con texts (e.g., Pillay [19] extended the differential Galois theor y to arbitrar y “ D -g roups” definable in DCF ). In app endix B of Hr us hovski [8], internality was reformulated in a wa y that holds in an a rbitrar y theory . One is interested in the group of auto morphisms G of a definable set Q over a nother definable set C . A set Q is internal to ano ther set C if, after e xtending the base parameters , any element of Q is definable ov er the elements of C . The idea is that under this condition, Q is close enough to C so that the G has a chance to b e definable, but the requirement that a base extension is required preven ts it from b e ing trivial. The theor em is that indeed G is the gr oup of p oints of a (pro-) defina ble gr oup (see Theor em 1.6 ). This theory is refor mulated again in Hrushovski [9 ], where the sa me constr uction is describ ed as an abstract duality theory between definable gr oup oids in a theory T , and certain expans io ns of it, called in ternal cov er s . It is this formulation that we use. The main results of the pap er appea r in Sections 3 and 4. In Section 3 we apply int ernality to pr ov e the fundamen tal result on T annakian ca tegories (Theor em 2.8). This is done b y constructing, fo r a tensor categ ory C an internal cov er T C of ACF k . Mo dels of this theory corre s po nd, r o ughly , with fibre functors on C . The theo ry of int ernality provides a definable group in ACF k , and this is the group corres p o nding to C . The other par ts of the theory follow from the Galois theory , a nd from the abstract duality theor y o f Hrushovski [9]. The r esult we prove is weak er than the or iginal Theor e m 2.8 in the fo llowing ways. First, 2.8 (1) states that a certa in functor is r epresentable b y a n affine a lgebraic group, but we only pr ov e that its r estriction to the categor y of fields a grees with such a gr o up (s e e also Question 0.1 .3 b elow). Second, o ur pro of works o nly in characteristic 0 . Finally , 2.8(2) is cov ered, in part, by the genera l mo del theo r etic statement 1.10 , but the r est of the pro o f is only sk etched, since it is not sig nificantly different from the pro of fo und in Deligne and Milne [4]. On the other ha nd, the pro of has the adv antage that it is s imple and mo re “geometric” than the algebr aic one. It a lso has the adv antage tha t the metho d is applicable in a more genera l con text. One suc h a pplication, concerning usua l Galois theo ry , is briefly discussed in Remark 3 .14. A more de ta iled a pplication app ears in Section 4, where we define the notio n of a differential tensor ca tegory , and explain how the s ame mo del theoretic approach gives an analogo us theor em (Theorem 4.5.5) there (though the metho d aga in gives a w eaker r esult, in the same wa y a s in the a lgebraic ca se). A similar result, using a somewhat different lang uage, was first obtain using a lgebraic metho ds in Ovc hinnik ov [17, 18 ]. It se ems obvious that s imilar formalisms are p ossible in other cont exts (e.g., difference fields, real closed fields ). W e note tha t, though mo del theory provides a gener al metho d of pro of of such res ults, the right notio n of a “ tensor ca teg ory” is ne w, and cannot b e deduced directly from this metho d. The fir st t w o sections co nt ain background material. Section 1 contains some general notions from mo del theory , as well a s a statement o f internalit y result, and some aux ilia ry r emarks on it. Sectio n 2 g ives a sho r t r eview of T annak ian categ o ries, with enough termino lo gy and results to state the main theor em. Both sections are provided with the hop e that the pap er will be acce s sible to a wide a udience (b oth within mo del theory and o utside it), and that the information con tained there will provide the reader with enough informa tion to at least get a feeling for the na tur e of the results. MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 3 W e mention a ls o tha t there is a result in the o ther dir ection: It is p o s sible to define the a nalogue of in ter nal covers a nd recov er a group ob ject in a general categoric al co nt ext, and recov er the mo del theoretic result from it. This r e sult will app ear separately . 0.1. Ques ti ons. Several questio ns (so me v ag ue ) remain unanswered in the pap er. The main ones, from my p oint o f view, are the following. 0.1.1. As describ ed in Remarks 3.12 and 3.13, the main results of Deligne [3] hav e very natura l model theoretic transla tions. It see ms re asonable to ex p ect that there is a mo del theo retic pro o f, esp ecially of the ma in theor em (which translates to having no new structure on ACF k ), but I could not find it. The c ase when C is neutral a ppe ars to b e easier fr om the algebr aic p oint of view (and is prov e n already in Deligne a nd Milne [4]), but mo del theor etically I don’t know how to do even this case. 0.1.2. Conv ers ely , the r e s ults and metho ds of Deligne and Milne [4 ] and Delig ne [3] seem to s uggest that one ca n do “mo del theory” inside a T annakian categor y (and p erhaps more g enerally in a tenso r catego ry). F or example, as ca n b e seen from 3 .10, the statement of 2.8(2) ca n b e viewed as saying that C has elimination of quant ifiers and elimination of imaginar ies. I think it would b e interesting to find out if this makes sense , and make it precise. 0.1.3. What is a genera l mo del theoretic machinery to prove results as in 3 .7, but in full generality? (ra ther than just for fields ). This can also b e asked pur ely in terms of functors (though in this context there is proba bly no a ns wer): Assume there is a functor F from k - algebras to sets (or gro ups ), and a sc heme (or g r oup scheme) G with a map G Ý Ñ F that is a bijection on fields. What ar e conditions on F that a llow us to deduce that this map is an isomorphis m? Ac knowledgemen ts. I would like to tha nk Ehud Hrushovski for sug gesting this approach, and for help along the wa y . I would als o like to thank Ana nd Pillay for his in ter est and useful remar ks. 1. Mod el theor y and internality In this section, w e recall some notions from mo del theory . So me elements of the mo del theor etic la ng uage a r e briefly sketc hed in 1.1 through 1.3. These app ear in most texts on mo del theory , for exa mple Marker [13] or Pillay [2 0]. In the rest of the section, we reca ll the main mo del theoretic to o l that we use, namely internality , and the definable Galois g roup. W e s hall, for the mos t pa rt, follow the presentation a nd terminology from Hrushovski [9 ]. The only new res ults in this sectio n app ear after the statement o f the theorem (Theorem 1.6), though most of them are implicit in Hrushovski [9]. 1.1. Basic termi nology. W e briefly recall that a the ory is collectio n of statements (axioms) written in a fix ed formal la nguage, a nd a mo del of the theory T is a structure c o nsisting of a n in terpr etation of the sy mbo ls in the langua g e of T , in which all the axioms of T hold. F or e xample, the theory T  ACF of a lg ebraically clos ed fields can be wr itten in a lang uage co ntaining symbols p 0, 1,  ,  , q , and a mo del of this theory is a particular algebra ically clos ed field. 4 M. KAMENSK Y A formula is written in the same formal languag e, but has fr e e variables , into which elemen ts of the mo del can b e plugged. F or the example o f ACF a bove, an y finite collection of p olyno mial equations and ineq ua lities c an be viewed as a for mula, but there are other fo rmulas, inv olving quantifiers. An y such formula φ p x 1 , . . . , x n q (where x 1 , . . . , x n contain all fre e v a riables of φ ) th us determines a subset φ p M q of M n , for any mo del M , namely , the set of all tuples ¯ a for which φ p ¯ a q holds. Two formulas φ and ψ a re e quivalent if φ p M q  ψ p M q for all mode ls M . An eq uiv alence class under this r elation is called a definable set (So, in this pap er, “definable” is without parameter s.) If M is the mo del of some theory , the set of all statements (in the underlying language) that are true in M is a theor y T p M q , and M is a mo del of T p M q . W e w ill assume all our theor ies to b e multi-sorted, i.e., the v ariables o f a for mula can take v alues in any num b er of disjo int sets. In fact, if X and Y ar e sorts, we view X  Y as a new s ort. In particula r, a ny definable set is a subs e t of some so rt. By a statement such as “ a P M ” we will mean that a is an element o f one o f the sorts, in terpreted in M . 1.2. Definable clos ure and automorphis ms. A definable function f fro m a definable set X to a nother definable set Y is a definable subset of X  Y that determines the graph o f a function f M : X p M q Ý Ñ Y p M q for each mo del M . If M is a model, A  X p M q any subset, and b P Y p M q , then b is definable ov er A if there is a definable function f : X n Ý Ñ Y for some n , such that f p ¯ a q  b for some ¯ a P A n (note that even though we a re working in M , it is T that should think that f is a function, rather than just T p M q ). A subset is definably close d if it is closed under definable functions. The definable closur e dcl p A q of a s ubset A  M is the smallest definably closed subset of M co n taining A . W e denote by Y p A q , the set of all element s in Y p M q definable ov er A . W e note that if T is complete, then the definable closure X p 0 q of the empty set do es not dep end on the mo del. F o r gene r al T , we deno te by X p 0 q the se t o f definable subsets of X containing o ne element . More generally , a form ula over A is a formula φ p x, a q , where φ p x, y q  X  Y is a regular formula, and a P Y p A q . It defines a subs et of X p M q for a ny mo del M containing A , in the sa me w ay a s regular formulas do. An automor phism of a mo del M is a bijection ϕ fr om M to itself such that the induced bijection ϕ n : M n Ý Ñ M n preserves a ny definable subset X p M q  M n . If A  M is a set o f parameter s , the automor phism ϕ is over A if it fixes all elements of A . The gr oup of a ll auto mo rphisms of M over A is denoted by Aut p M { A q . It is clea r that X p A q is fixed p o int wise by any automorphis m ov er A . The mo del M is called homo gene ous if, conv er sely , for any A  M of cardinality smalle r than the cardinality of M , for any definable set X , any element of X p M q fixed by all of Aut p M { A q is in X p A q (computed in T p M q ). Homoge ne o us mo dels are known to exist (cf. Sacks [25 , Ch. 2 0]). 1.3. Imaginaries and in terpretations . A definable e quivalenc e r elation on a de- finable set X is a definable s ubs et of X  X that determines a n equiv a lence relatio n in any mo del. The theory eliminates imaginaries if any equiv alence relation has a quotient, i.e., a ny equiv alence relatio n can b e r epresented as f p x q  f p y q for s ome definable function f on X . MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 5 If T eliminates imagina ries, an interpr et ation i o f another theory T 0 in T specified by g iving, fo r each sort X 0 of T 0 a definable set X  i p X 0 q of T , and fo r each atomic relation Y 0  X 0 of T 0 a defina ble subs et Y  i p Y 0 q of X , such that for any mo del M of T , the sets X p M q form a mo del M 0 of T 0 (when in ter pr eted a s a T 0 structure in the obvious wa y ). It follows tha t an y definable set Z 0 of T 0 determines a defina ble set i p Z 0 q in T , and M 0 can b e viewed as a s ubset of M (there is a universal int erpretation of any theor y T in a theory T eq that eliminates imaginaries. An int erpretation of T 0 in a gener al theory T is then defined to b e a n interpretation of T 0 in T eq in the sense alrea dy defined.) If M is a mo del of a theory T , and A  M , w e denote by T A the theo ry obtained by adding constants for A to the language, and the axioms s atisfied for A in M (in particular, T A is complete). Ther e is an obvious interpretation of T in T A . The mo del M is in a natural wa y a mo del of T A , and for any defina ble set X of T , X p A q in the sense of T is identified with X p 0 q in T A . 1.4. Int ernal cov ers. An interpretation i of T 0 in T is stably emb e dde d if any subset of the sorts of T 0 definable in T with pa rameters fr om T is also definable in T 0 with parameters from T 0 . More precisely , given definable se ts Y 0 in T 0 , and X and Z  X  i p Y 0 q in T , there are defina ble sets W 0 and Z 0  W 0  Y 0 in T 0 , such that T implies that for each x P X , Z x  i p Z 0 q w for some w P i p W 0 q . Definition 1. 5. A stably embedded in terpretation i of T 0 in T is an int ern al c over (of T 0 ) if there is a stably embedded interpretation π o f T in T 0 , such that π  i is the identit y . The theory T is also ca lled an in ter nal cov er of T 0 (if i is understo o d). If Q is a defina ble set in T (an int ernal cov er of T 0 ), the identit y ma p on π p Q q comes, by assumption, from some bijection fro m Q to i p π p Q qq in T , defina ble with parameters . Hence there is a set of pa rameters A , such tha t dcl p M 0 Y A q  M for any model M of T . Such a set A will b e ca lled a set o f internality p ar ameters , and will a lwa ys b e taken to be definably clo sed. W e denote by A 0 the restriction of A to T 0 . (In the lang ua ge of Hrus hovski [9], this implies that the co rresp onding group oid in T 0 is equiv alent to a group oid with ob jects ov er A 0 .) W e note that b y compa ctness, a finite num ber o f elements of A suffices to define all ele ments of a given sort of T ov er M 0 . Thus, if T has a finite num b er of so rts that do not come from T 0 ov er which everything is definable, then A can b e taken to b e g e nerated by one tuple a . In g eneral, A c a n b e thought of as an e lement of a pro-definable set. The following for mulation of the int ernality theor em is closest in langua ge a nd strength to the one in Hrushovski [9, prop. 1.5]. This is the language we would like to use later, but in the strength we actually r e quire (na mely , for ω - stable theo r ies), it was already prov ed in Poizat [23]. Theorem 1.6 (Hrushovski [8 , App endix B],Hrushovski [9 , prop. 1 .5]) . L et T b e an internal c over of T 0 . Ther e is a pr o-gr oup G in T , to gether with a definable action m Q : G  Q Ý Ñ Q of G on every definable set Q of T , such that for any mo del M of T , G p M q is identifie d with Aut p M { M 0 q thr ough this action (with M 0 the r estriction of M to T 0 ). F urthermor e, given a set of internality p ar ameters A (whose r estriction to T 0 is A 0 ), ther e is a Galois c orr esp ondenc e b etwe en A - definable pr o-sub gr oups of G and definably close d subsets A 0  B  A . Su ch a su bset B H for a sub gr oup H is 6 M. KAMENSK Y always of the form C p A q , wher e C is an A -definable ind-set in T , and H is t he sub gr oup of G fi xing C p M q p ointwise. If H is normal, t hen G { H p M q is identifie d with Aut p C p M q{ M 0 q . Here, a pr o-gr oup is a filtering inv erse system of definable g roups. 1.7. In fact, the result is slightly stro nger. With notation as ab ov e, the as sumption that T 0 is stably embedded implies that any automor phism of M 0 fixing A 0 can be extended (uniquely) to an automorphism of M fixing A . In other w ords, we hav e a split exact sequence 0 Ý Ñ G p M q Ý Ñ Aut p M { A 0 q Ý Ñ Aut p M 0 { A 0 q Ý Ñ 0 (1) where G is as provided by Theorem 1.6. More genera lly , we ha ve the following interpretation of G p B q . Prop ositi o n 1.8. L et B 0  M 0 b e a definably close d set c ontaining A 0 , and let B  dcl p A Y B 0 q . Then G p B q  Aut p B { B 0 q . Pr o of. An y g P G p B q acts as an automor phis m, pr eserves B as a se t (since B is definably closed), and fixes B 0 po int wise, so G p B q  Aut p B { B 0 q . Conv ers ely , since B 0 is definably closed a nd contains A 0 , B X M  B 0 , and so any automo rphism of B ov er B 0 extends to an automor phism of B ov e r M 0 . Since M 0 is stably embedded, this automo rphism e xtends to an automo rphism of M . Thus, a ny elemen t of Aut p B { B 0 q is r epresented by some g P G p M q . Since it is fixed b y any automorphis m fixing B point wis e, it is in fact in G p B q .  1.9. W e make the ass umption that a set A as above can b e found (in s o me mo del) such that the corresp onding set A 0 is (the definable closure o f ) the e mpty set. This is not a real assumption in the cur rent context, since the results will ho ld in genera l for the theory with para meters from A . Likewise, the interpretation of T 0 in T factors through a maxima l extension T 1 of T 0 (i.e., T 1 is the theor y , in T , of the definable sets coming from T 0 ). W e assume from now on that T 1  T 0 . Combining the tw o assumptions, we get that B 0 ÞÑ B  dcl p A Y B 0 q gives an equiv alence b etw ee n definably clos ed sets B 0 of T 0 and definably clos e d se ts B containing A of T . In par ticular, we hav e a definable group G A in T 0 . The following pro p o sition s ays that all defina ble gr o up actio n of G A in T 0 come fro m the canonical action of G in T . Prop ositi o n 1 .10. Assume 1.9, and let a : G A  D Ý Ñ D b e a definable gr oup action in T 0 . Ther e is a definable set X D in T , and an A - definable isomorphism of G A actions fr om X D to D . If D and E ar e two such G A -sets, and f : D Ý Ñ E is a definable map of G A sets in T 0 , it also c omes fr om a definable map F : X D Ý Ñ X E . Pr o of. The pro of of Theore m 1.6 pro duces a defina ble G -torso r X in T , which is 0 - definable with our assumptions. There is a n A - definable isomo rphism f b : G Ý Ñ G A , which, comp osed with a , defines a n A -definable action c b : G  D Ý Ñ D . W e set X D  X  G D , i.e., X D is the set of pair s p x, d q P X  D , up to the equiv alence p gx, d q  p x, c b p g, d qq for g P G . Since G A is in T 0 , f hb p hgh  1 q  f b p g q for all h, g P G , s o the equiv a lence relatio n is inv aria nt under the action of G , a nd hence X D is definable without para meters. If b P X p A q is any element (known to ex ist), the map r b, d s ÞÑ d is an isomor- phism of G A actions from X D to D . The pro of for maps is similar .  MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 7 2. T annak ian ca tegories In this se c tio n we review , without pro ofs, the definitions and ba sic pr op erties of T annakian categ o ries. The pro ofs, as w ell as more details (or ig inally from Saavedra- Riv ano [24]) can b e found in the fir st tw o sections o f Delig ne and Milne [4]. The section contains no new res ults (though the terminology is slightly different, a nd follows in part Mac Lane [12, ch. VII]). Let G b e an affine algebra ic g roup (or, more generally , an affine gro ups scheme) ov er a field k . The categor y R ep G of finite-dimensiona l repr esentations of G over k , admits, in addition to the catego ry structure, a tenso r pro duct op eration o n the ob jects. With this structure, and with the forgetful functor in to the category o f vector space s ov er k it satisfies the axioms of a (neutr a lised) T annakian categor y (see Example 2.7 ). The main theorem of Saav edra- Riv ano [24] (Theor em 2.8 ) asser ts that categor ies satisfying these axioms, summa r ised be low, a re in fact precisely of the form R ep G , a nd that further more, the g roup G ca n be recov e red from the (tensor) categor y str ucture. 2.1. Mono i dal categories. Recall that a symmetric monoidal c ate gory is given by a tuple p C , b , φ, ψ q , where: (1) C is a ca tegory (2) b is a functor C  C Ý Ñ C , p X , Y q ÞÑ X b Y (in other words, the op e ration is functorial in each co o rdinate separa tely). (3) φ is a co llection of functorial isomor phisms φ X , Y , Z : p X b Y q b Z  Ý Ñ X b p Y b Z q one for each triple X , Y , Z of o b jects of C . It is c a lled the asso ciativity c onstr aint . (4) ψ is a collection of functoria l isomorphisms ψ X , Y : X b Y  Ý Ñ Y b X , called the c ommutativity c onstr aint . The commutativit y constr aint is r equired to satisfy ψ X , Y  ψ Y , X  id X b Y for all ob jects X , Y . In addition, φ and ψ are r equired to s atisfy ce r tain “p entagon” and “hexago n” identities, which ensure that any tw o tenso r expressio ns computed from the same set of ob jects are c anonic al ly isomorphic. Finally , C is r e quired hav e an identity obje ct : this is an ob ject 1 , to gether with an isomorphism u : 1 Ý Ñ 1 b 1 , such that X ÞÑ 1 b X is a n equiv a lence of categories. It follows (Deligne a nd Milne [4, prop. 1.3]) that p 1 , u q is unique up to a unique isomorphism with the prop erty that u can b e uniquely extended to isomorphisms l X : X Ý Ñ 1 b X , commuting with φ and ψ . W e will usually dro p φ and ψ from the no ta tion, and cho ose a particular identit y ob ject 1 . W e will als o dr op the adjective “sy mmetr ic ”. 2.2. Rig idity . A mono idal ca tegory p C , bq is close d if for any tw o ob jects X and Y , there is an ob ject Hom p X , Y q and a functorial isomorphism Hom p Z b X , Y q  Ý Ñ Hom p Z , Hom p X , Y qq . In this case, the dual q X of X is defined to b e Hom p X , 1 q . The symmetry constraint and the functoriality determine, for each o b ject X , a ma p X Ý Ñ q q X and for any four ob jects X 1 , X 2 , Y 1 , Y 2 , a map Hom p X 1 , Y 1 q b Hom p X 2 , Y 2 q Ý Ñ Hom p X 1 b X 2 , Y 1 b Y 2 q 8 M. KAMENSK Y C is s aid to be rigid if it is close d, and all these maps are isomo rphisms (The latter requirement can b e viewed as a formal analo g ue of the ob jects b eing finite- dimensional.) W e no te that setting Y 1  X 2  1 in the ab ov e map, we ge t an isomo r phism q X b Y Ý Ñ Hom p X , Y q . In particular , a map f : X Ý Ñ Y cor resp onds to a glo bal section 1 Ý Ñ Hom p X , Y q , hence to a map q f : 1 Ý Ñ q X b Y . 2.3. The rank of an ob ject. Let f : X Ý Ñ X b e an endomo rphism in a rig id category . As above, it corresp o nds to a morphism q f : 1 Ý Ñ q X b X . Co mpo sing with the c o mmut ativity followed by the ev aluatio n we get a morphism T r X p f q : 1 Ý Ñ 1 called the tr ac e o f f . The tra ce map T r X : End p X q Ý Ñ k :  End p 1 q is m ultiplica tive (with r esp ect to the tensor pro duct), and T r 1 is the identit y . The r ank of an ob ject X is defined to be T r X p id X q . Thus it is an element of k . 2.4. T ensor categories. A t ensor c ate gory is a rig id monoidal catego ry C which is ab elian, with b additive in each co ordinate. It follows that b is exact in each co ordinate, that k  End p 1 q is a c ommut ative ring, that C has a na tural k - linear structure, where b is k -bilinear, and T r X is k -linear. Given an ob ject X of a tenso r ca tegory C , let C X be the full sub categ o ry of C whose ob jects a re isomorphic to sub-quotients of finite sums of tensor powers of X (including X b  1  q X a nd its p owers). The ob ject X is called a tensor gener ator for C if any ob ject of C is isomorphic to a n ob ject of C X . F or most of wha t follows we will restrict our attention to ca tegories that hav e a tensor g enerator. In gener al, C is a filtered limit of catego ries of this form. 2.5. T ensor functors. Let C and D be mo no idal categ o ries. A tensor fun ctor from C to D is a pair p F , χ q , where F : C Ý Ñ D is a functor, and χ is a collection o f isomorphisms χ X , Y : F p X q b F p Y q  Ý Ñ F p X b Y q , compatible with the constr aints, such that p F , χ q takes identit y ob jects to identit y ob jects. When C and D a re a b elian, we assume F to b e a dditive. In this case, F ma kes k D int o a k C -algebra , and F is automatically k C -linear, and in this sense prese r ves the trace. In pa rticular, rk p X q  rk p F p X qq . If p F , χ q and p G , δ q are t wo tens o r functors fro m C to D , a map from p F , χ q to p G , δ q is a map of functors that commutes with χ and δ . W e denote by Aut p F q the group of automorphisms of F as a tensor functor. 2.6. T annakian categories. A neutr al T annakian c ate gory ov er a field k is a ten- sor catego ry C with End p 1 q  k , which admits an exact tensor functor ω into the category V ec k of finite-dimensional k -v ector spaces (with the usual tensor struc- ture). Such a functor is called a fibr e fun ctor . It is a utomatically faithful, and is said to neutralise C . Given a fibre functor ω and a (commutativ e ) k -alg ebra A , the functor ω b k A (with p ω b k A qp X q  ω p X q b k A ) is ag ain an exact tenso r functor (into a tensor sub c ategory of the categor y V ec A of pro jective finitely pres ent ed A -mo dules ). W e th us get a functor A ÞÑ Aut p ω b k A q from the categor y of k -algebr as to gr oups. This functor is denoted by Aut b k p ω q . Example 2.7 (representations of a group) . If G is an affine gro up scheme ov er k , the c a tegory C  R ep G of (finite-dimensional) representations of G ov er k , with the usual tens o r pro duct (and constra int s) forms a monoidal c a tegory . It is rigid, with MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 9 Hom p X , Y q the space of all linear maps b etw een X and Y , and is a be lian, hence it is a tensor c ategory . The trace of a map coincides with the image in k of the usual trace. Fina lly , C is neutralised by the for getful functor ω . If A is a k -algebra , and g P G p A q , the ac tion of G exhibits g as an automo r phism of ω b k A . This deter mines a map (of functors) G Ý Ñ Aut b k p ω q . The main theo rem states that this map is an is omorphism, and that this example is the most genera l one: Theorem 2.8 (Saavedra) . L et C b e a neut r al T annakian c ate gory, and let ω : C Ý Ñ V ec k b e a fibr e functor. (1) The funct or A ut b k p ω q is r epr esentable by an affine gr oup scheme G over k . (2) The fibr e fun ctor ω factors thr ough a tensor e qu ivalenc e ω : C Ý Ñ R ep G . (3) If C  R ep H for some affine gr oup scheme H , t hen the natur al map H Ý Ñ G given in Example 2.7 is an isomorphism. In the following sec tion w e pr e sent a pro of of a slightly weaker s tatement (as explained in the introduction), using the mo de l theor etic to ols of Section 1. 3. The theor y a ssocia ted with a tenso r ca tegor y In this section, w e asso cia te a theory with any T annakian category , and use Theorem 1 .6 to prove the main Theor em 2.8 , in the case that the base field ha s characteristic 0 (and with the other caveats mentioned in the intro duction). 3.1. Let p C , b , φ, ψ q be a tensor ca teg ory over a field k of characteris tic 0 , suc h that the rank of every ob ject is a natural num b er. Let K be an extens ion field of k . The theories T C and  T C (depending on K , which is omitted from the notation) are defined as follows: 3.1.1. T C has a s ort L , a s well as a so rt V X for every ob ject X of C , and a function symbol v f : V X Ý Ñ V Y for any morphism f : X Ý Ñ Y . There a re a binary o p e r ation symbol  X on ea ch V X , and a function symbol  X : L  V X Ý Ñ V X for e ach ob ject X . The sor t L con tains a constant symbol for each elemen t o f K . 3.1.2. The theor y says that L is an algebr aically closed field, whos e field op erations are given by  1 and  1 . The r estriction of the o p er ation to the constant symbo ls is given by the field s tructure on K . It also says that  X and  X determine a vector space structure over L on every V X , and V X has dimension rk p X q ov er L . Each v f is an L -linear map. 3.1.3. The k -line a r ca teg ory structur e is reflected in the theory: v id X is the identit y map for each X , v f  0 if and only if f  0 , v f  g  v f  v g , if a P k is viewed a s an element of End p X q , then v a is multiplication by a P K . Also, v f  g  v f  v g , and v f is injectiv e or surjective if a nd only if f is. 3.1.4. F or any tw o ob jects X and Y ther e is a further function s ymbol b X , Y : V X  V Y Ý Ñ V X b Y . The theor y sa ys that b X , Y is bilinear, and the induced map V X b L V Y Ý Ñ V X b Y is an iso morphism. Here b L is the usual tensor pro duct of vector spaces o ver L (in a given mo del). This statement is first order, sinc e the spaces are finite-dimensional. The theory also says that v ψ X , Y p b X , Y p x, y qq  b Y , X p y, x q , and similar ly for φ . If u P V X and v P V Y (either ter ms or elements in some mo del), we write u b v for b X , Y p u, v q . 10 M. KAMENSK Y 3.1.5. The theor y  T C is the ex pansion of T C by an extra so rt P X , for every ob ject X , together with a surjective map π X : V X Ý Ñ P X ident ifying P X with the pro jective space asso ciated with V X . It als o includes function symbols p f : P X Ý Ñ P Y and d X , Y : P X  P Y Ý Ñ P X b Y for any ob jects X , Y and mo rphism f of C , and the theory says that they are the pro jectivisations of the c o rresp onding maps v f and b X , Y . Prop ositi o n 3.2. The the ory T C is stable. In p articular, L is stably emb e dde d. In e ach mo del, L is a pur e algebr aic al ly close d field (p ossibly with additional c onstants) . Pr o of. A choice of ba sis for each vector space identifies it with a power o f L . All linear and bilinear ma ps are definable in the pure field structure on L . Since L is stable and stability is not a ffected b y parameters , so is T C .  Prop ositi o n 3. 3. The the ories T C and r T C eliminate quantifiers (p ossibly after n am- ing some c onstants). The the ory r T C eliminates imaginaries. Pr o of. That r T C eliminates ima ginaries is precisely the sta tement of Hrushovski [9, prop. 4.2] (see also the pro of of P rop osition 4.5.6 ). The pro o f o f quantifier elim- ination is simila r: since all sorts are interpretable (with par ameters) in ACF , a ll definable sets a re bo olean combinations of Zaris ki-closed s ubsets. A Z ariski-c lo sed subset of V X or P X is the set of zer o es o f p o lynomials, which ar e elements of the symmetric algebra on | V X . B ut | V X is identified with V q X , and the tensor a lgebra o n it, as well as the actio n on V X or P X is definable without quantifiers.  3.4. Mo dels and fibre functors. Mo dels of T C are ess ent ially fibre functors ov er algebraic ally close d fields. More precisely , we have the following sta tement s. Prop ositi o n 3.4.1 . L et M b e a mo del of T C , and let A b e a subset of M such that for any obje ct X , V X p A q c ontains a b asis of V X p M q over L p M q . Then X ÞÑ V X p A q determines a fi bre functor over the field ex tension L p A q of K . Pr o of. W e first note that since (by definition) L p A q is definably closed, it is indeed a field. Likewise, since each V X p A q is definably clo sed, it is a vector space ov er L p A q , and the V X p A q are closed under the line a r ma ps that c ome from C . Th us X ÞÑ V X p A q defines a k -linear functor from C to the category of vector spaces ov er L p A q . The map b X , Y determines a ma p c X , Y : V X p A q b L p A q V Y p A q Ý Ñ V X b Y p A q . Since the map V X p A q b L p A q V Y p A q Ý Ñ V X p M q b L p M q V Y p M q is injective, s o is c X , Y , and so the whole statement will follow from k nowing that the dimension o f V X p A q ov er L p A q is r k p X q . Since V X p A q contains a basis ov er L p M q , w e know that it is at least rk p X q . F or any a 1 , . . . , a n P V X p A q , the set o f all p x 1 , . . . , x n q P L n such tha t ° x i a i  0 is a n A -defina ble subspace o f L n , Since L is stably embedded, it is L p A q definable. Hence it ha s an L p A q -definable ba s is (Milne [16, L e mma 4.10 ]). This s hows tha t the dimension is corr ect.  Prop ositi o n 3. 4.2. L et A b e a subset of a mo del M of T C such that dcl p A Y L p M qq  M , and such that L p A q is algebr aic al ly close d. Then dcl p A q is a mo del. In p articular, X ÞÑ V X p A q determines a fibr e functor for C over L p A q . Pr o of. W e only need to prove that each V X p A q contains a basis ov er L p M q . By assumption, there is an A -definable map f from a subset U of L m to V n X , where MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 11 n  rk p X q , such that f p u q is a bas is of V X p M q for all u P U p M q . Since U is L p A q - definable, and L p A q is alg ebraically closed, U p A q contains a p oint u , a nd f p u q is th us a basis in V X p A q .  Prop ositi o n 3.4.3. Conversely, if K 1 is a field extension of K , ω : C Ý Ñ V ec K 1 is a fibr e functor, and ¯ K is an algebr aic al ly close d fi eld c ontaining K 1 , t he assignment V X ÞÑ ω p X q b K 1 ¯ K determines a mo del M ω of T C , such that A  ω  M satisfies dcl p A Y L p M ω qq  M ω , and L p A q  K 1 . In p articular, T C is c onsistent if and only if C is has a fibr e functor. Pr o of. Since ¯ ω  ω b K 1 ¯ K is itself a fibre functor, it is clear tha t M ω is a mo del ( b X , Y is defined to be the comp osition o f the canonica l pairing ¯ ω p X q  ¯ ω p Y q Ý Ñ ¯ ω p X q b ¯ K ¯ ω p Y q with the structure map c X , Y : ¯ ω p X q b ¯ ω p Y q Ý Ñ ¯ ω p X b Y q ). Also, it is clear that dcl p A Y L p M ω qq  M ω . T o prov e that L p A q  K 1 , we note that a ny automorphism o f ¯ K ov e r K 1 extends to an automorphism o f the mo de l by acting the second term.  3.5. F rom now on we assume that C ha s a fibre functor ω , and denote the theory of the mo del M ω constructed in P rop osition 3.4.3 by T ω (so T ω is a completion of T C ). W e a lso assume that the ca tegory C has a tensor gener ator (2.4 ). The genera l results a re o btained as a limit o f this case, in the us ual w ay . (This is not really necessary , since, as men tioned in 1 .4, the theor y works in genera l, but it makes it easier to apply standar d mo del theoretic results ab o ut definable , rather than pro-definable set.) W e fix a tensor gener ator Q . Claim 3.6. The the ory T C is an internal c over of L , viewe d as an interpr etation of ACF K . Pr o of. L is stably embedded in T C since T C is sta ble. Given a mo del M o f T C , any element of V X p M q is a linear c o mbination o f elements o f a basis a for V X p M q ov er L p M q , hence is definable ov er a and L p M q . W e note that if Q is a tensor g enerator , then a basis for V Q determines a basis for a ny other V X (This is clear for tensor pro ducts, direct s ums, duals and quotients. F or a sub-o b ject Y Ý Ñ X , the set o f all tuples x in L suc h that ° x i a i P V Y is a -definable, hence has a po int in L p a q .)  3.7. Pro of of 2.8 (1 ) . Let G be the definable g roup in T C corres p o nding to it as an as a n internal cov e r of L (Theorem 1.6). By P rop osition 3.4.3, ω determines a definably clos e d subset A of an arbitr a rily larg e mo del M , such tha t dcl p A Y L p M qq  M and L p A q  k . Given a field extension K , we may assume that L p M q contains K . F or each ob ject X , V X p A Y K q  ω p X q b k K (again b y Pr op osition 3.4.3), It follows by Prop ositio n 1.8 that for such a field K , G p dcl p A Y K qq  Aut p dcl p A Y K q{ K q , and since by the definition of the theory , suc h a utomorphisms are the same as a utomorphisms of ω b k K as a tenso r functor, we get that G p dcl p A Y K qq  Aut b p ω qp K q . On the other hand, A satisfies Assumption 1.9, he nc e we get a definable g r oup G A in L (i.e., in ACF ), such that G A p K q  G p dcl p A Y K qq . By Poizat [2 2, § 4.5], any such g r oup is algebr a ic.  R emark 3.8 . By insp ecting the pro o f of P rop osition 1.8 a nd the cons truction o f G in this particular cas e , it is ea s y to extend the ab ove pr o of to the cas e when K is 12 M. KAMENSK Y an integral domain. Howev er, I don’t know how to use the same argument for more general K . 3.9. Pro of of 2.8 (3) . Let C  R ep H , where H is an algebraic group ov er k , which we view as a definable group H in ACF k . Le t G be the defina ble in ter nality gro up in T ω and A the de fina ble subse t cor resp onding to the (forgetful) fibre functor ω , all as in 3.7. W e obta in a definable group G A , and the action of H o n its representations determines a map of g roup functors from H to G A , whic h by Beth definability is definable a nd therefore algebra ic. This homomorphism is injective (on points in any field extension) since H has a faithful representation. W e identify H with its image in G A . By the Galois corres po ndence (Theo rem 1.6), to prove that H  G A it is enoug h to s how tha t any A -definable element in T eq C fixed by H p M q is also fixed by G A p M q , i.e., is 0 -definable. By Pro p o sition 3.3, T eq C  r T C . If v P V X p A q is fixed by H , then the map L Ý Ñ V X given b y 1 ÞÑ v is a map of H repres e n tations, hence is given by a function sy m bo l f , so v  f p 1 q is 0 -definable. Likewise, if p P P X p A q is fixed by H , then it co rresp onds to a sub-repr esentation l p  V X (ov er k ), hence is given by a predicate, so again p is 0 -definable. This prov es tha t the map from H to G A is bijective on p oints in any field. An y such a lg ebraic map (in characteristic 0 ) is an iso morphism (W aterhouse [26, § 11.4]).  3.10. Pro of of 2.8 (2 ) (s k etch). Let T C , G , A and G A be as ab ov e , and let V ec L be the c a tegory of definable L -vector spaces interpretable in T C , with defina ble linea r maps b etw een them as mor phisms. T his is c learly a tenso r catego ry , and X ÞÑ V X is an exact faithful k -linear tensor functor fro m C to V ec L . On the other ha nd, a r epresentation of G A is given by some definable action in ACF , and by Pr o p osition 1.10 , this action comes from some action of G on a definable set in T C , which must ther efore b e a v ector space ov er L . Conv ers ely , each linear action of G on an o b ject in V ec L gives a representation of G A after ta king A -p oints, and simila rly for mor phisms. Thus, we g et an equiv a lence (of k -linear tensor categories) b e tween V ec L and R ep G A . It remains to show that the functor X ÞÑ V X from C to V ec L is full and sur jective on isomor phism classes , i.e., that any definable L -vector space is essentially of the form V X , and that any map b etw ee n is of the form v f . T o prov e the first, we co nsider the gr oup G a nd its to r sor X . The s e are defined by some 0 -defina ble subspa ces of powers of V q Q (where Q is a tensor ge nerator.) By examining the explicit definition of G and X , o ne concludes that these 0 -definable subspaces in fact come from C . W e omit the pro of, s ince it is very similar to the original a lgebraic one. Once this is know, given any o ther repres e n tation D , the set X D in Pr op osition 1 .1 0 is ex plicitly given by tensor o p erations, and so comes fro m C a s well. Then pro o f for morphisms is again similar.  Corollary 3.11. Quant ifier elimination in T C holds without p ar ameters: any de- finable s et is quant ifier- fr e e without p ar ameters. Pr o of. This holds anywa y with para meter s fro m the definable closure of the e mpt y set. W e now k now that any s uch parameter is given b y a term in the languag e.  R emark 3.12 . In Hrushovski [9 ], an equiv a le nce is constructed b etw een internal cov er s of a c omplete theory T , and connected definable gro upo ids in T . It follows MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 13 from Theo rem 2.8 there that if T C is cons istent and the induced structure on L is precisely ACF k , then the asso ciated group oid is connec ted. It also follows that T C itself is complete. It then follows from le mma 2.4 in the sa me pap er that any t w o mo dels of T C are isomo rphic. Given tw o fibre functors of C (o ver some ex tension field), ex tending b oth to a mo del of T C th us shows that the fibre functor s ar e lo cally isomorphic. Conv e rsely , if T C induces new structure on L , it is easy to construct fibre functors which ar e no t lo cally isomor phic (cf. Deligne and Milne [4, § 3.15]). The main re s ult of Deligne [3 ] can b e viewed as stating that the induce d structure on L is indeed ACF k . More precisely , g iven a fibre functor ω ov er an (affine) scheme S ov er k , Deligne constr ucts a g roup oid scheme Aut b k p ω q , with ob ject s cheme S . The main result of Deligne [3 ] (Theorem 1 .12) states that this gro upo id sc heme is faithfully flat ov er S  S (hence connected). A K -v alued p o int o f S (for some field K ) determines a fibr e functor ω K ov er K . Viewing this fibre functor as subset of a mo del, a choice of a basis for a tensor gener ator determines a type ov er L , and therefore an ob ject o f the group o id corre sp o nding to T C (ch o osing a different basis amounts to cho osing a differ ent ob ject which is iso mo rphic ov er K in the gro up oid). It is clear from definition o f Aut b k p ω q that this pro cess gives an equiv alence from this gr oup oid to the g roup oid a sso ciated with the internal cov er T C . Th us, Deligne’s theorem implies that this gr oup oid is connected, and therefore that the induced structure o n L is that of ACF k . It seems plaus ible that there should b e a direct mo del theoretic pro of of this result, but I could not find it. W e note that the non- existence of new structure also directly implies (by the existence of prime mo dels) Deligne [3, corolla ry 6.20], which states that C has a fibre functor over the algebraic closure of k . W e note also that an y definable group o id in ACF k is equiv alent to a group oid scheme (any such g roup oid is equiv a lent to one with a finite set of ob jects, a nd the automorphism gro up of each ob ject is algebra ic). The category C of repr esentations of the (ger b asso cia ted with the) group oid is a tensor c a tegory sa tisfying End p 1 q  k (Deligne and Milne [4, section 3]), and it is easy to see (using the s a me metho ds as in 3.9) tha t the g roup oid asso cia ted with the internal c ov er T C is equiv alent to the or iginal one. Therefore, we obtain that any internal cov er of ACF k has (up to equiv alence) the form T C for some T annakian categ ory over k . R emark 3.13 . W e interpret mo del theoretically tw o additional result of Deligne [3]. In se c tion 7, it is shown tha t in our co ntext (i.e., char p k q  0 and rk p X q is a natural num b er), C alwa ys has a fibre functor . In ligh t of Pro p o sition 3 .4.3, this result asserts that T C is consistent for any s uch C . In section 8, Deligne defines, for a T anna kian c a tegory C (and even so mew ha t more gener ally), a C -gro up π p C q , called the fundamental gr oup of C (a C -group is a commut ative Hopf alg ebra ob ject in Ind p C q ; similarly for affine C -schemes). The basic idea is that the concepts a sso ciated with neutral T anna k ian catego r ies, also make sense for “ fibre functors” into other tensor c a tegories , and π p C q is obtain by reconstructing a group from the identit y functor. Thu s, π p C q comes with a n a ction on each ob ject, commuting with a ll morphisms. It thus repr esents the tensor automorphisms o f the ident it y functor from C to itself, in the sense that for a ny affine C -scheme X , the action identifies π p C qp X q with the group of tenso r a utomorphisms of the functor A ÞÑ A b X (fro m C to the tensor category of vector bundles over X ). It has the prop erty that for any “re a l” fibr e 14 M. KAMENSK Y functor ω over a scheme S , applying ω to the action identifies ω p π p C qq with the group scheme Aut b S p ω q constructed. Using either the explicit co nstruction, or the prop erties a b ove, it is clea r that in terms of the theory T C , π p C q is nothing but the definable automor phism group (more precisely , the Hopf algebra ob ject that defines it maps to an ind-definable Hopf a lgebra in T C , which is definably isomo rphic to the ind-definable Hopf algebr a of functions on the int ernality gr oup). R emark 3.14 . One can recover in a similar manner (and somewhat more easily) Grothendieck’s appr oach to usual Galois theor y (cf. Grothendieck et al. [6, Ex - po s´ e V.4]). Brie fly , given a ca tegory C with a “fibre functor” F in to the c a tegory of finite sets, satisfying conditions (G1)–(G6), o ne constr ucts as ab ov e a theor y T C with a sort V X for each o b ject, and a function symbo l for each morphism. T C is then the theory of F viewed a s a structure in this language. Since every s ort is finite, they a re a ll internal to 2 (the co-pro duct o f the terminal ob ject 1 with itself ). Conditions (G1)–(G5) ens ure that any defina ble set in fact c omes from C , and (G2), (G5) e ns ure tha t T C has e limination of imaginaries . The Ga lois ob jects P i that app ea r in the pro of are precis ely the 1 -types o f int ernality parameters that app ear in the mo del theoretic constr uc tio n of Ga lois group. 4. Differential T annak ian ca tegories Our purp os e in this section is to define differential tensor c ategories , a nd to give a mo del theoretic pro of o f the basic theorem, cor resp onding such ca tegories, endow ed with a s uitably defined fibre functor, with linear differential alg ebraic groups. The metho d is completely analo gous to that in the previous section. Throughout this sectio n, k is a field of character istic 0 . 4.1. Prolongatio ns of ab el ian categories . W e assume that in a tensor ca tegory p C , bq , the functor b is exact; this is automatic if C is rigid (see Deligne a nd Milne [4, prop. 1.16].) Definition 4.1. 1 . Let C b e a k - linear category . The pr olongation P p C q of C is de- fined as follows: The ob jects ar e exact sequences X :  0 Ý Ñ X 0 i X Ý Ñ X 1 π X Ý Ý Ñ X 0 Ý Ñ 0 of C , and the mor phisms b e t ween such o b jects are morphisms of exa ct sequences whose tw o X 0 parts coincide. An ex act functor F : C 1 Ý Ñ C 2 gives rise to a n induced functor P p F q : P p C 1 q Ý Ñ P p C 2 q . W e denote by Π i ( i  0, 1 ) the functors from P p C q to C assigning X i to the o b ject 0 Ý Ñ X 0 i X Ý Ñ X 1 π X Ý Ý Ñ X 0 Ý Ñ 0 of P p C q (th us there is an exa c t sequence 0 Ý Ñ Π 0 i Π Ý Ñ Π 1 π Π Ý Ý Ñ Π 0 Ý Ñ 0 .) Π i p X q is also abbreviated as X i , and X is said to b e ov er X 0 (and similarly for morphisms.) R emark 4.1.2 . W e note that P p C q can b e viewed as the full sub category of the category of “differential ob jects” in C , consisting of o b jects whose homology is 0 . A differential ob ject is a pa ir p X , φ q where X is an ob ject of C a nd φ is an endomorphism of X with φ 2  0 . A morphis m is a mor phism in C that commutes with φ , and the homology is ker p φ q{ im p φ q . This is the same as the categor y of k r ǫ s -mo dules in C , in the sense of Deligne and Milne [4, p. 155] (wher e ǫ 2  0 ). The adv antage o f this categor y is that it is again k -linea r. How ever, I don’t k now how to extend the tensor structur e (defined below) to this whole ca tegory (in pa r ticular, the tensor structure defined there do es not s eem to coincide with ours). MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 15 4.1.3. Let A and B be t wo ob jects ov er X 0 . Their Y one da su m A Æ B is a new ob ject over X 0 , defined a s follows (this is the additio n in Y oneda’s des cription of Ext 1 p X 0 , X 0 q ): the combined map X 0  X 0 Ý Ñ A 1  B 1 factors through A 1  X 0 B 1 , and to gether with the map X 0 1,  1 Ý Ý Ý Ñ X 0  X 0 gives rise to a ma p f : X 0 Ý Ñ A 1  X 0 B 1 . Let W 1 be the co-kernel o f this map. The map f comp osed with the pro jection from A 1  X 0 B 1 to X 0 is 0 , so we o btain an induced map p : W 1 Ý Ñ X 1 . The diagonal inclus ion ∆ of X 0 in W 1 together with p g ive r ise to an exac t s equence 0 Ý Ñ X 0 ∆ Ý Ñ W 1 p Ý Ñ X 0 Ý Ñ 0 , which is the required ob ject. F or any ob ject A o f P p C q , we deno te by T p A q the ob ject obtained by neg a ting all arrows that app ear in A . 4.1.4. T ensor s tructur e. Let p C , b , φ 0 , ψ 0 q be a tensor category . An ob ject X 0 of C gives rise to a functor from P p C q to itself, by tensoring the exact s equence p oint wise . Since we assumed b to b e e x act, this functor also commutes with Y oneda sums : p A Æ B q b X 0 is ca nonically iso morphic with p A b X 0 q Æ p B b X 0 q . Also, T p A q b X 0 is isomorphic to T p A b X 0 q . W e endow P p C q with a monoida l structure. The tensor pro duct A b B of the t wo P p C q ob jects A a nd B is defined a s follows: After tenso ring the first with B 0 and the second with A 0 , we obtain tw o ob jects ov er A 0 b B 0 . W e now take their Y oneda sum. W e shall make use of the following exac t sequence. Lemma 4. 1.5. F or any two obje cts A and B of P p C q , t her e is an exact se quenc e 0 Ý Ñ p A b T p B qq 1 i Ý Ñ A 1 b B 1 π Ý Ñ p A b B q 1 Ý Ñ 0 (2) wher e π is the quotient of the map obtaine d fr om the maps π A b 1 and 1 b π B , and i is the r est riction of the map obtaine d fr om t he maps i A b 1 and  1 b i B . Pr o of. Exactness in the middle follows direc tly from the definitions. W e prov e that π is surjective, the injectivity of i being similar. W e shall use the Mitc hell embedding theo r em (cf. F r eyd [5]), which r educes the questio n to the cas e of a b elian groups. W e in fact prove that already the map A 1 b B 1 π Ý Ñ A 0 b B 1  A 0 b B 0 A 1 b B 0  : U is surjective. Let y be an element of U , and let y 1 and y 2 be its tw o pro jections to the comp onents of U . Since the map A 1 b B 1 π A b 1 Ý Ý Ý Ý Ñ A 0 b B 1 is sur jective, y 1 can be lifted to an element  y 1 of A 1 b B 1 . W e hav e that p π A b 1 qpp 1 b π B qp  y 1 qq  p 1 b π B qpp π A b 1 qp  y 1 qq  p 1 b π B qp y 1 q  p π A b 1 qp y 2 q Let z  p 1 b π B qp  y 1 q  y 2 . Since z is killed by π A b 1 , it comes from a n element, also z , of A 0 b B 0 . Let r z b e a lifting of z to A 0 b B 1 , and denote by r z a lso its image in A 1 b B 1 under the inclusion i A b 1 . Then  y 1  r z is a lifting of y .  4.1.6. Let A , B , C b e three ob jects of P p C q . The asso cia tivity constra int φ 0 of C gives rise to an isomo rphism o f p A b B q b C with the quotient o f A 1 b B 0 b C 0  A 0 b B 0 b C 0 A 0 b B 1 b C 0  A 0 b B 0 b C 0 A 0 b B 0 b C 1 that identifies the three natur al inclusions o f A 0 b B 0 b C 0 , and similarly for A b p B b C q . W e th us get an asso ciativity cons tr aint φ on P p C q , o ver φ 0 . 16 M. KAMENSK Y Likewise, the commutativit y constraint ψ 0 induces a commutativit y constr aint ψ on P p C q ov er ψ 0 . Prop ositi o n 4. 1.7. The data p P p C q , b , φ, ψ q as defin e d ab ove forms a symmetric monoidal c ate gory, and Π 0 is a monoidal functor. It is rigid if C is rigid. Pr o of. W e define the a dditional data. V er ification of the axioms reduces, a s in Lemma 4.1.5, to the case of ab elian groups , where it is easy . Let u : 1 0 Ý Ñ 1 0 b 1 0 be an identit y o b ject of C . W e set 1  0 Ý Ñ 1 0 Ý Ñ 1 0 ` 1 0 Ý Ñ 1 0 Ý Ñ 0 . F or a ny ob ject A of P p C q , 1 b A is identifi ed via u with 0 Ý Ñ A 0 Ý Ñ p A 1  A 0 p A 0 ` A 0 qq{ A 0 Ý Ñ A 0 Ý Ñ 0 This ob ject is ca no nically isomorphic (ov er C ) to A , and so 1 acquire s a structur e of an iden tity ob ject. Assume that C is rigid. F or an ob ject A of P p C q , w e set q A to b e the dual exact sequence 0 Ý Ñ q A 0 i | A Ý Ý Ñ q A 1 π | A Ý Ý Ñ q A 0 Ý Ñ 0 . W e define a n ev aluation ma p A b q A Ý Ñ 1 as follows: W e need to define t w o maps fro m A 0 b | A 1  A 1 b | A 0 to 1 0 , that agree on the tw o inclusions of A 0 b | A 0 , and such that the resulting map restricts to the ev aluation on A 0 b | A 0 . T o c o nstruct the first map, we consider the exa ct seq uenc e (2), for B  q A . W e claim that the ev aluatio n ma p on A 1 b | A 1 restricts to 0 when compos ed with i . T o prove this, it is enoug h to show that the pair of maps obtained from ev A 1 by comp osition with i A b 1 a nd  1 b i q A comes from a ma p A 0 b | A 0 Ý Ñ 1 0 . How ever, under the adjunction, this pair of maps corr esp onds to p i A ,  π A q , and so comes from the iden tity map on A 0 . It follows that ev A 1 induces a map on p A b q A q 1 , which is the required ma p. The second map is obtained by pro jecting to A 0 b | A 0 , and using the ev alua tion map on A 0 . By definition, this second map commutes with the pro jections to A 0 b | A 0 and the second coo rdinate of 1 , restricting to the ev aluation on A 0 . T o prov e that the first map r estricts to the ev aluation as well, we note that there is a commutativ e diagram p A b q A q 1 i / / π A b | A   A 1 b | A 1 π   A 0 b | A 0 i A b | A / / p A b q A q 1 where i is the (restriction of the) map obtained from the tw o maps i A b 1 and 1 b i q A . Since π A b q A is surjective, it is therefore enough to prov e that the maps ev A 1  i and ev A 0  π A coincide. This is indeed the case, s ince they bo th corresp ond to the inclusion of A 0 in A 1 .  4.2. Diffe rential tensor categories. Definition 4. 2.1. A differ ential structur e o n a tensor categ o ry C is a tensor functor D fr om C to P p C q which is a section of Π 0 . If D 1 and D 2 are tw o differential structures on C , a morphism from D 1 to D 2 is a morphism of tensor functors that induces the identit y mor phism under Π 0 . A differ ential tens or c ate gory is a tens or category together with a differential structure. MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 17 Let D b e a differential structur e on C . Since D is a section of Π 0 , it is determined by B  Π 1  D . In o ther words, on the ab elian le vel, it is given by a functor B : C Ý Ñ C , together with an exa ct sequence 0 Ý Ñ Id Ý Ñ B Ý Ñ Id Ý Ñ 0 . How ever, this description do es not include the tensor structure. W e als o no te that B is necessarily exact. 4.2.2. Let p C , D q be a differential tensor category , let B  Π 1  D , and let A  End p 1 q . Rec a ll tha t for any ob ject X , End p X q is an A -algebra . The functor B defines another ring homomor phism B 1 : A Ý Ñ End pBp 1 qq . Given a P A , the mor phis m B 1 p a q  a in End pBp 1 qq restricts to 0 on 1 , and thus induces a mo rphism from 1 to Bp 1 q . Similar ly , its comp osition with the pro jection Bp 1 q Ý Ñ 1 is 0 , so it factors through 1 . W e thus get a new element a 1 of A . Claim 4. 2 .3. The map a ÞÑ a 1 of 4.2.2 is a derivation on A . Pr o of. W e need to show that given elements a, b P A , the maps Bp ab q  ab and pBp a q  a q b  a pBp b q  b q coincide on 1 . This follows fr o m the formula Bp ab q  ab  Bp a qpBp b q  b q  pBp a q  a q b , together with the fact that Bp a qpBp b q  b q induces a pBp b q  b q on 1 .  Example 4.2.4 . Let C b e the tenso r category V ec k of finite dimensional vector spa ces ov er a field k . Given a deriv a tive 1 on k , w e co nstruct a differen tia l structure on C as follows: F o r a vector space X , define d p X q  D ^ b X , where D is the vector space with basis 1, B , and ^ b is the tensor pro duct with resp ect to the right vector space structure on D , given b y 1  a  a  1 a nd B  a  a 1  1  a  B . The ex act sequence D p X q is defined by x ÞÑ 1 ^ b x , 1 ^ b x ÞÑ 0 and B ^ b x ÞÑ x , for any x P X . If T : X Ý Ñ Y is a linea r map, d p T q  1 ^ b T . W e s hall write x for 1 ^ b x and B x for B ^ b x . The structure o f a tensor functor is obtained by sending Bp x b y q P d p X b Y q to the image of Bp x q b y ` x b Bp y q in p D p X q b D p Y qq 1 . Claim 4 .2.5. The c onstruct ions in 4.2. 4 and in 4.2.2 give a bije ctive c orr esp on- denc e b etwe en derivatives on k and isomorphism classes of differ ent ial structur es on V ec k . Pr o of. If D 1 and D 2 are tw o differential s tr uctures, then D 1 p 1 0 q and D 2 p 1 0 q are bo th identit y ob jects, and are therefore cano nically iso morphic to the same ob ject 1 . If D 1 and D 2 are isomorphic, then the maps d i : End p 1 0 q Ý Ñ End p 1 q are conjuga te, and therefore equal, since End p 1 q is commu tative. It is clea r from the definitio n that the deriv ative on k obta ined from the differ- ent ial structure asso ciated with a deriv ative is the origina l one. Con versely , if D 1 and D 2 are tw o differe ntial structures that give the same de r iv ative on k , then we may identify D 1 p 1 0 q and D 2 p 1 0 q . Under this ident ification, we get that the maps d i are the same. But the functors D i are determined by d i .  4.2.6. W e now come to the definitio n of functors b etw een differential tenso r c a te- gories. F o r simplicit y , we shall only define (and use) exact such functors. Let ω : C Ý Ñ D b e an exact functor b etw een ab elian ca tegories. There is an induced functor P p ω q : P p C q Ý Ñ P p D q , given by applying ω to each term. If C and D are tens or categor ies, the structure o f a tens o r functor o n ω g ives rise to a similar structure on P p ω q (again, since ω is exa ct.) If t : ω 1 Ý Ñ ω 2 is a (tensor ) mo rphism of functors, w e likewise g e t an induced mor phism P p t q : P p ω 1 q Ý Ñ P p ω 2 q . Definition 4. 2.7. Let p C 1 , D 1 q and p C 2 , D 2 q be tw o differen tial tensor categor ie s. 18 M. KAMENSK Y (1) A differ ential ten s or functor from C 1 to C 2 is an ex a ct tensor functor ω from C 1 to C 2 , together with an isomorphism o f tensor functors r : P p ω q  D 1 Ý Ñ D 2  ω , such that the “restrictio n” Π 0 d r : ω Ý Ñ ω , obtained b y comp os ing with Π 0 on P p C 2 q , is the identit y . (2) A morphism b etw een tw o such differential tensor functor s p ω 1 , r 1 q and p ω 2 , r 2 q is a morphism t b etw een them as tensor functors such that the following diagra m (of tensor functor s and tensor maps b etw een them) com- m utes: P p ω 1 q  D 1 r 1 / / P p t qd D 1   D 2  ω 1 D 2 d t   P p ω 2 q  D 1 r 2 / / D 2  ω 2 (3) where D 2 d t is the map from D 2  ω 1 to D 2  ω 2 obtained by applying D 2 to t “p oint wise” . 4.2.8. Given a differential tensor functor ω , we denote by Aut B p ω q the group of automorphisms of ω . If C is a differential tensor ca tegory , and k  End p 1 q is a field, a k -linear dif- ferential tenso r functor ω into V ec k (with the induced differential structure) is called a differ ential fibr e functor . Analogous ly to the alge br aic ca se, we will say that C is a neutr al differ ential T annakian c ate gory if suc h a fibre functor exists, and that ω ne utr alises C . Since we will not define more general differential T anna k ian categorie s, the adjective “ neutral” will be dropp ed. As in the a lgebraic case, g iven a differential fibre functor ω on C , we deno te by Aut B p ω q the functor from differential k -a lgebras to groups assig ning to a differential algebra A the group Aut B p A b ω q . 4.2.9. Given a k v ector space V , the map d : V Ý Ñ D ^ b V given by v ÞÑ B v is a deriv ation, in the sens e that d p av q  a 1 v  ad p v q (where V is iden tified with its image in D ^ b V .) It is universal for this pro per ty: an y pair p i, d q : V Ý Ñ W , where i is linear, and d is a deriv ation with res pec t to i factor s through it. Therefore, a fibre functor on p C , D q is a fibre functor ω in the sense of tensor categorie s, together with a functorial deriv atio n d X 0 : ω p X 0 q Ý Ñ ω pB X 0 q (where B X 0  D p X 0 q 1 ), s atisfying the Leibniz rule with resp ect to the tensor pro duct (and additio na l conditions). The condition that the restriction to ω is the identit y corres p o nds to the deriv ation being relative to the canonic a l inclusio n of ω p X 0 q in ω pB X 0 q given by the differ e ntial structure. Similarly , a differential automorphis m of ω is an a utomorphism t of ω as a tensor functor, with the additional condition that for any ob ject X 0 , the diagra m ω p X 0 q d X 0 / / t X 0   ω pB X 0 q t B X 0   ω p X 0 q d X 0 / / ω pB X 0 q (4) commutes. Thus the condition (3) really is ab out preser v ation of the differentiation. MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 19 4.2.10. Derivations. More genera lly , we define a derivation o n an ob ject X 0 of C to be a mo rphism d : X 1 :  Bp X 0 q Ý Ñ X 0 such that the comp osition X 0 Ý Ñ X 1 d Ý Ñ X 0 is the iden tit y . Given tw o deriv ations d X : X 1 Ý Ñ X 0 and d Y : Y 1 Ý Ñ Y 0 , we define the co m bined deriv ation d  d X b d Y : p X b Y q 1  p X 0 b Y 0 q 1 Ý Ñ X 0 b Y 0 to b e the re s triction of d X b Id ` Id b d Y to p X b Y q 1 (this makes se ns e, since both are the identit y on X 0 b Y 0 ). In V ec k this cor resp onds to a r e al deriv ation, in the sense that it gives a map d 0 : X 0 b Y 0 Ý Ñ X 0 b Y 0 which is a der iv ation ov er k a nd also d 0 p x b y q  d X p x q b y  x b d Y p y q . There is a canonic a l deriv ation d 1 on 1 0 given by the first pro jection 1 1  1 0 ` 1 0 Ý Ñ 1 0 . It has the prop er ty that d 1 b d  d b d 1 for any der iv ation d on any X 0 , under the canonical identification of 1 b X 0 and X 0 b 1 with X 0 . 4.3. Diffe rential algebraic groups. In this sub-section, we reca ll a nd s ummarise some definitions a nd bas ic facts fro m differential algebra ic geo metry (developed by K olchin [11]) and linear differential alg ebraic groups (studied by Cas s idy [1 ]). W e show that the c a tegory of differential repre sentations of such a gro up is a differential tensor category in our sense. 4.3.1. Let K be a differe ntial field (i.e., a field endo w ed with a deriv ation). W e recall (Kolchin [11]) that a Kolchin close d subs e t (o f a affine n -spac e ) is given by a collection of p olynomial (o rdinary) differe n tial equa tions in v ariables x 1 , . . . , x n , i.e., p olynomial equatio ns in v aria ble s δ i p x j q , for i ¥ 0 . Suc h a collectio n determines a set of po int s (solutions) in any differ e ntial field extension of K . As with a lgebraic v arieties, it is p oss ible to study these sets by considering p oints in a fixed field, provided that it is differ ential ly close d . The Kolchin clo sed sets form a basis of closed sets for a No etherian top olo gy . Morphisms are also given by differential po lynomials. A differential algebr aic group is a group ob ject in this categor y . More g e nerally , it is p ossible to c onsider a differential K -a lg ebra, i.e., a K -a lgebra with a vector field extending the deriv a tion on K , a nd develop these notions there. 4.3.2. By a line ar different ial algebraic g roup, we mean a differential algebraic group which is represented by a differential Hopf a lgebra. A differential a lg ebraic group which is affine as a differential algebra ic v ariety need not b e line a r in this sense, since a mo r phism of affine differential v arieties need not corr esp ond to a map of differential algebra s. Any linear differential a lgebraic group has a faithful representation. All these r esults app ear in Cassidy [1 ], along with an exa mple o f an a ffine non-linea r gr oup. In Cassidy [2] it is shown that a ny representation o f a line ar gro up (and mor e gener a lly , a ny mor phism of linear groups ) do es c orresp o nd to a map of differen tial algebr as. 4.3.3. Differ en tial r epr esentations. Let G be a linear differential algebra ic group ov er a differential field k . A representation of G is given by a finite dimensional vector space V over k , tog e ther with a mor phism G Ý Ñ GL p V q . A map of represe n- tations is a line a r transfo rmation that gives a map of g r oup representations for ea ch differential k -alg e bra. The categor y o f all such repres e n tations is deno ted R ep G . W e endow R ep G with a differential str uc tur e in the sa me way as for vector spaces. If V is a representation of G , assigning gv to p g, v q , then the a ction o f G on D ^ b V is given by p g, x ^ b v q ÞÑ x ^ b gv . With this differential structure, the for getful functor ω in to V ec k has an obvious structure of a differential tensor functor. 20 M. KAMENSK Y A differential automor phis m t of ω is g iven by a collection o f vector spa c e au- tomorphisms t V , for any repr esentation V of G . The co mmutativit y condition (4) ab ov e translates to the condition that t D ^ b V  1 ^ b t V . In particula r, g iven a differential k -algebra A , and g P G p A q , action by g gives an automor phism of A b ω as a different ial tenso r functor, since the a ction of g on D ^ b V is deduced fro m its action on V . Thus w e ge t a ma p G Ý Ñ G ω . W e shall prov e in Theorem 4.5.5 that the map is an isomo rphism. Example 4.3 .4 . Let G m be the (differential) multiplicativ e group, and let  G m be the m ultiplicative group of the constants (th us, as differential v arieties, G m is g iven by the equation xy  1 , and  G m is the subv ar iety g iven by x 1  0 .) There is a dif- ferential a lgebraic group homomor phism dlog from G m to G a (the a dditive group), sending x to x 1 { x , and x ÞÑ x 1 is a differential alge br aic gr o up endomorphism of G a . Let V b e the standard 2 -dimensional algebra ic repr esentation of G a . Using dlog and the deriv ative, w e get for any i ¥ 0 a 2 -dimensional irreducible represe n tation V i of G m , which ar e a ll unrelated in terms of the tensor str ucture (and unr elated with the non-trivia l 1 -dimensional algebr aic r epresentations of G m .) How ever, if X is the G m representation c orresp o nding to the identit y map on G m , an easy calculatio n shows that V 0 is isomor phic to B X b q X . Similarly , V i  1 is a quotient of B V i . The inclusio n of  G m in G m gives a functor from R ep G m to R ep  G m . But in R ep  G m , V 0 is isomor phic to 1 ` 1 (and B X to X ` X .) 4.4. Diffe rential s chemes in C . W e define affine differential s chemes in a differ- ent ial tensor ca teg ory , and show that any ob ject can b e view ed a s a “differential affine space”. This is analogues to the notion of C -schemes for tensor catego ries that a ppe a rs in Deligne [3]. The main application is the pro of of elimination of imaginaries in Prop ositio n 4.5 .6. 4.4.1. W e recall that the b op eration on C extends canonica lly to Ind p C q , mak ing it again a n abelia n mono ida l ca tegory . The prolo ngation P p Ind p C qq can be identified with Ind p P p C qq , and a differential s tructure on C thus extends cano nically to a differential structure on Ind p C q . Recall (Deligne [3 , § 7.5 ]) that if C is a tensor categ ory , a r ing in Ind p C q is an ob ject A of Ind p C q together with maps m : A b A Ý Ñ A and u : 1 Ý Ñ A sa tisfying the usual axioms. Definition 4.4.2. Let p C , D q be a differential tensor catego ry , a nd let A be a commutativ e ring in Ind p C q . A ve ctor field on A is a deriv ation on A in the sense of 4.2.10 , which commutes with the pro duct, and w hich res tr icts to the ca nonical deriv ation d 1 on 1 . A differ ential algebr a in Ind p C q is a commutativ e ring in Ind p C q together with a vector field. An (affine) differ ent ial scheme in C is a differen tia l algebra in Ind p C q , view ed as an ob ject in the opp osite categor y . 4.4.3. Higher derivations. Let X 0 be an ob ject of a differential tensor categor y p C , D q . As explained ab ov e, D p X 0 q can b e viewed as r epresenting a universal deriv a- tion on X 0 . W e now construct the analog ue of higher deriv atives. More precisely , we define, by induction fo r e a ch n ¥ 0 , the following data: (1) An ob ject X n (of C ) (2) A map q n : X n  1 Ý Ñ X n (3) A map t n : B X n  1 Ý Ñ X n MODEL THEOR Y AND THE T ANNAKIAN FORMALISM 21 such tha t q n  1  t n  t n  1  Bp q n q . In the context o f V ec k , the data ca n be thought of as follows: X n is the space of expressio ns v 0  B v 1      B n v n with v j P X 0 ; q n the inclusion of element s as a bove; the map t n the linear map corresp onding to the deriv ation. F or the base, setting X  1  0 determines all the data in an obvious wa y . Given X n , t n  1 : Bp X n q Ý Ñ X n  1 is defined to b e the co-equaliser of the following tw o maps: X n i " " F F F F F F F F Bp X n  1 q t n : : v v v v v v v v v Bp q n q / / Bp X n q (5) Where i is part of the str ucture of D p X n q . The map q n  1 is the c o mp o sition t n  1  i . Clearly the co mm utativity condition is satisfied. W e note that the t wo ob ject we denote by X 1 coincide, and the map q 1 coincides with the ma p i for D p X 0 q . The map t 1 is the iden tit y . Definition 4.4. 4 . Let X 0 be an ob ject of of C . The differ ential s cheme asso ciate d with X 0 , denoted A p X 0 q , is a differential sch eme in C defined as follows: Let D b e the ind-ob ject defined by the system q X i , with maps q i (as in 4.4.3 ). The maps t i there define a deriv ation t on D . This deriv atio n induces a deriv ation on tensor powers o f D (as in 4 .2 .10), which descends to the symmetric p ow ers. It is ea s y to see that this determines a differential algebr a s tructure on the symmetric algebra on D . W e let A p X 0 q be the asso cia ted sc heme. A morphism in C clea rly deter mines a morphism of schemes on the as so ciated schemes, making A pq int o a functor. 4.5. Mo del theory of di fferen tial fibre functors. W e now wis h to prov e state- men ts analo gous to the ones for algebr a ic T annakian ca tegories, using the sa me approach as in Sectio n 3. W e work with a fixed differential tensor catego r y p C , D q , with k  End p 1 q a field. W e view k as a differential field, with the differential structure induced from D , as in 4.2.2. W e will be using the theory DCF of differentially closed fields. W e refer the reader to Marker [14] or Marker e t al. [15] for more information. 4.5.1. The the ory asso ciate d with a differ ential tensor c ate gory. The theor y T C as- so ciated with the data a bove, as well as a differ ent ial field extension K of k is an expansion of the theory T C as defined in 3.1 by the following structure: (1) L has an a dditional function symbol 1 , and the theory says that 1 is a deriv ation, and that L is a different ially clos ed field (and with the restriction of 1 to K as given). (2) F or every ob ject X , ther e is a function s ymbol d X : V X Ý Ñ V Bp X q . This function is a deriv ation, in the se ns e that for any a P L and v P V X , d X p av q  a 1 V i X p v q  ad X p v q The theor y furthermore s ays that d X ident ifies V Bp X q with D ^ b V X (in any mo del), in the sense of 4.2.9 (explicitly , it says that V p X  d X is the identit y map.) 22 M. KAMENSK Y (3) The ma ps d a nd b (from the tensor structure ) ar e c ompatible with the structure of tensor functor of D : given ob jects X and Y of C , let c X , Y : Bp X b Y q Ý Ñ pBp X q b Bp Y qq 1 be the isomorphism supplied with D . Then we r equire that V c X , Y  d X b Y  b X , Y coincides with b Bp X q , Y  d X  1  b X , Bp Y q  1  d Y . 4.5.2. Let ω b e a differential fibre functor on C , a nd let L  M 1 be a differentially closed field containing K . As in Se c tion 3, we expand M 1 to a mo del M of T C by tenso r ing with L . The differential str uctur e o f ω gives (as in 4.2.9 ) a universal deriv ation ω p X q Ý Ñ ω pBp X qq , which extends uniquely to a (universal) deriv ation p d X q M on M X . As in the algebr aic case, we get: Prop ositi o n 4.5.3. Assume that C has a differ ent ial fibr e functor. Then T C is c onsistent, and ( in a mo del) dcl p 0 q X L  k . The the ory T C is stable, and L is a pur e differ ent ial ly close d field. Pr o of. Same as in Prop ositio n 3.2 and Prop o s ition 3 .4 .3.  4.5.4. Int ernality. Since the differential T C is an expansion o f the alg ebraic one with no new s o rts, it is again an internal c ov er of L . F urthermore, if B is a basis for some V X , then B Y d X p B q is a basis for V Bp X q . Therefor e, if C is generated a s a differen tial tensor categ o ry by one ob ject (in the sense that the ob jects B i X genera te C as a tensor category), then all of the so rts are internal us ing the same finite parameter. As usual, the genera l cas e is obtained b y taking a limit of such. Theorem 4 .5.5. L et C b e a differ ential T annakian c ate gory over k , and let ω : C Ý Ñ V ec k b e a differ ential fibr e functor (4.2.8). (1) The functor Aut B p ω q (r estricte d to differ ential fields) is r epr esent e d by a line ar differ ential gr oup G over k . (2) The fibr e functor ω factors thr ou gh a differ ential tensor e quivalenc e ω : C Ý Ñ R ep G . (3) If C  R ep H for some line ar differ ential gr oup H , t hen the natur al map H Ý Ñ G given in 4.3.3 is an isomorphism. Pr o of. The pro o f is co mpletely ana logous to the pro of of the corresp o nding state- men t 2 .8, which is given, resp ectively , in 3.7, 3.10 and 3.9. W e only mention the differences. F or (1), the main p oint is again that Aut B p ω qp K q is isomor phic to G ω p K q (func- torially in K ), where G ω is a definable copy of the in ter na lity group inside L . The sort L is now a pur e differentially closed field, so the res ult follows fr om the fact that an y definable group in DCF is differential a lgebraic (Pillay [2 1 ]). F or (2), the arg ument in 3.10 go es through without a change. F or (3 ), again the pro of in 3.9 applies, once we clas s ify the imaginaries in T C . This is the conten t of Prop os ition 4.5 .6.  Prop ositi o n 4.5.6. T C eliminates imaginaries t o the level of pr oje ctive sp ac es. Pr o of. Both the statement and the pro o f are a na logous to Hrushovski [9, Pro p o si- tion 4.2]. W e need to show that a ny definable set S over parameters c a n b e defined with a cano nical para meter . Since, by a ssumption, no new structure is induced o n L , and any set is internal to L , every such set is Kolchin constructible. By Noether ian induction, it is enough to consider S Ko lchin closed. REFERENCES 23 W e c laim that the algebra of differential p olyno mials on a sort U is ind-definable in T C . Indeed, it is pr ecisely given by the interpretation o f scheme structure asso- ciated with U (Definition 4.4.4 ). W e only mention the definition of the ev alua tion map (using the notation there): it is enough to the ev aluation on (the interpre- tation of ) D , since the rest is as in the alg ebraic case. W e define the e v a luation e n : q U n  U Ý Ñ L by induction o n n : e 0 is the usual ev aluation. If u P U , the map d ÞÑ e n p d, u q 1 is a der iv ation on q U n , a nd so defines a linear map from Bp q U n q to L . Insp ection of the definition of q U n  1 (for vector s paces) reveals that this map descends to a linear map e n  1 p , u q on q U n  1 . 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