Construction of an homology and a cohomology theory associated to a first order formula

D ^IAGRAMMES

R ENÉ G UITART

Construction of an homology and a cohomology theory associated to a first order formula

Diagrammes, tome 23 (1990), p. 7-13

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© Université Paris 7, UER math., 1990, tous droits réservés.

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D I A G R A M M E S V O L U M E 2 3 A C T E S D E S J O U R N E E S E . L . I . T .

<LHMIV. P A R I S 7, 2 7 J U 1 N - 2 J U I L L E T 1 9 8 8 )

CONSTRUCTION OF AN HOMOLOGY AND A COHOMOLOGY THEORY

ASSOCIATED TO A FIRST ORDER FORMULA

by René GUITART

RESUME - On m o n t r e comment chaque formule $ d'un langage £ détermine une t h é o r i e d'homologie (et une t h é o r i e de cohomologie) sur la c a t é g o r i e d e s i n t e r p r é t a t i o n s de JE, dont la valeur s u r chaque i n t e r p r é t a t i o n I de JE e s t une obstruction à I \= <p "à des c o - é q u a t i o n s p r è s " ( et "à des é q u a t i o n s près").

T h i s p a p e r is a sequel of [7].

O. Let ï be a f i r s t o r d e r language, let I be an i n t e r p r é t a t i o n of JE, and let 0 be a formula of JE. The aim of t h i s note is to indicate a way in which it is possible t o measure p a r t i a l l y and t o compute how I is f a r from t h e models of ¢.

In t h e p a p e r s [7] and [8] it is shown how t h i s q u e s t i o n is connected w i t h t h e possibility of a geometrical study of a l g o r i t h m s and a m b i g u ï t i e s .

1. PROPOSITION. Let ix(x ,...,x ) be a first order formula of JE, and

1 n

let Mod <f> be the category with objects the models of ¢, and with morphisms from M to M* the morphisms (of models of $) m : M > M' such that

V x , . . . , x [ /i(m(x ) m(x )) > ^ ( x , . . . , x ) ]

I n I n i n Then there is a small mixed sketch <r such that Mod <f> = Modcr.

The existence of cr is proved by the juxtaposition of proposition 3 p. 8 of [6], théorème 2.1 p.26 of [5], and proposition 3 p.301 of [7],II. In fact, this juxtaposition shows more than our proposition hère.

2. For C a category, let BC = | NC | be the géométrie realization of the nerve of C. BC is a cw-complexe, and n BC = C[C ] (the category of fractions of C). Of course if C is a class, BC is a class too. But, if C = Modo* for a small sketch cr, then in BC we can construct a set gtr such that the inclusion ger » BModtr is an équivalence of homotopy. In particular we get

PROPOSITION. Mod(r[(Mod(r) ] is a small groupoïd, up to équivalence. We call it the fundamental groupoïd of <r, and we dénote it by n go\

The existence of the set g<r cornes from [7].

3. Let Mod 0/1 be the category with objects the morphisms (of interprétations of JE) f : M > I where M is a model of 0, and with morphisms, from f : M » I to f : M' » I, the morphisms of models ( morphisms of Mod 0) g : M » M' such that f'.g = f.

Then

PROPOSITION. There is a small sketch <r = <r{ JE, I, 0, fi) such that Mod 0/1 = Modcr.

So we get a small cw-complexe g<r( JE, I, 0, ^x), which is a géométrie description of the position of I with respect to Mod 0.

4. Let Ab be the category of small abelian groups, and let F : Mod 0 > Ab be a functor. (In particular F could be the constant functor on a fixed abelian group A, or it could be a "canonical"

functor if JE is a language over the language of abelian groups, etc).

The André*s homology measures "how I is far from Mod 0, from the point of view of F".In order to do that we consider the chain complexe

which is

d d d

-> C ( I , F ) — ^ - > C ( I , F ) —î—> C ( I , F ) — ^ > 0

2 1 0

d d d

-> y FM — ? — > y FM — — - > y FM — ? - > o

U Z U 1 ^ c ⁰

M ^ M -* M -> I M -> M -» I M ^ I

2 1 0 1 0 0

w i t h d = s - s,, w h e r e î î 1

Id Inc

F ( a ) inc s1 : (FM ) ,w a w /3 n > (FM ) ,w £ n > V FM

î 1 (M —» M !-* I) o (M - î — > I) ^u

1 0 0 W T

M -> I and so on, and we define

HA I, F) = ker d /Im d = coker d , H,( I, F) = ker d / I m d , and, f o r

0 0 1 i l 1 2

every n ^ 0, H ( I, F) = ker d / I m d .

J n n n+1

PROPOSITION. H ( I, F) is a function of F, I, /i, 0, which in fact dépends only of the homotopy type of Mod 0/1 and of F and could be denoted by H (Mod 0/1,F).

J n il

see [1], [2], [3] and [4].

Let IntJE be t h e c a t e g o r y of i n t e r p r é t a t i o n s of JE, let J : Mod0 > IntJE be t h e canonical inclusion. Then t h e inductive Kan e x t e n s i o n of F along J is given by

[ E x t . F K D = Lim F(M )

- ^ ^J M — » 1 °

and we hâve o

H ( I, F) =[Ext F](I).

If If= 0, then H (Mod 0/1,F) = <

' ^ n /ut

f F ( I ) if n = 0 0 if n > 0

5. Now, t h e point is t h a t , because of t h e r e s u l t s h e r e o v e r (§§ 1 t o 4), we get

PROPOSITION. The tools of [1] and of [3], available in t h e s i t u a t i o n w h e r e a full and small c a t e g o r y M (called a c a t e g o r y of "models") lives inside a big c a t e g o r y of "spaces", are also available in the situation where a (possibly big and not necessarlv full) category Mod 0 of models of a theory lives inside a big category of interprétations of a language JE ( c o m p a r e w i t h t h e idea of "paires a d é q u a t e s " p. 43 of [1]).

Precisely hère we get the fact that the H (Mod 0/1,F) are small.

6. After t h e e x i s t e n c e of g<r proved in [7], the t h e o r e m h e r e u n d e r §9 is j u s t a second s t o n e f o r a work t o be pursed. T h e o r e t i c a l l y the computation of our H is based on the effective c o n s t r u c t i o n of a

"locally c o f r e e d i a g r a m " , and more precisely on t h e c o n s t r u c t i o n of a

"relatively c o f i l t e r e d locally cofree diagram" (r.cf . l . c f . d . H s e e [5]

and [61) (in t h e c a t e g o r y Mod 0) g e n e r a t e d by I. This r . c f . 1.cf.d.

c o n t a i n s ail t h e i n f o r m a t i o n we need, and it will be t h e s t a r t i n g point of an a b s o l u t e c a l c u l u s . But for concrète s i t u a t i o n s we need a r e l a t i v e calculus, by t h e way of comparaisons between various H . For t h a t it will be e s s e n t i a l t o go t o w a r d effective r e l a t i v e c a l c u l a t i o n of thèse small H , and especially we need a description of t h e link between t h è s e c a l c u l a t i o n s and t h e t h e o r y of d é m o n s t r a t i o n s . For e x a m p l e we need r e l a t i o n s among H (Mod 0/1,F), H (Mod y / I , G ) , H (Mod [0A?]/I,K),

n /i n JI n ^i

H (Mod [0=>y]/I,D (for conveniant K and L).

n M

For t h a t it will be n e c e s s a r y to describe t h e c a t e g o r y For(JE) of f o r m u l a s of t h e language JE. At f i r s t t h i s will be usefull t o p r é c i s e t h e f u n c t o r i a l i t y of t h e H (Mod 0/1,F) with r e s p e c t t o 0 and fi.

7. The f i r s t p u r p o s e of t h i s paper w a s to show precisely how each classical f i r s t o r d e r f o r m u l a 0 of a language JE d é t e r m i n e s a "small"

homology t h e o r y on t h e c a t e g o r y of i n t e r p r é t a t i o n s of JE.

Now, the c o n t i n u a t i o n of t h i s r e s e a r c h p a s s t r o u g h t h e d e s c r i p t i o n of For(JE). With r e s p e c t t o t h a t , I would like t o make t h e following r e m a r k : w h a t hâve t o be morphisms between formulas ? it is not so c l e a r a p r i o r i ; they hâve t o be "démonstrations" or "proofs", but t h e r e is no

10

canonical idea of w h a t is a d é m o n s t r a t i o n .

But if we décide t o s t a y in (or to corne back to) the s t y l e of s k e t c h e s , a f i r s t p i c t u r e is e a s y t o give. In f a c t JE "is" a s k e t c h <r (i.e. t h e c a t e g o r y of i n t e r p r é t a t i o n s of JE is isomorphic t o Modo* ), t h e f o r m u l a 0 (or 0) is a sketch o\ and t h e inclusion of the c a t e g o r y of models of 0 (of 0) in the c a t e g o r y of i n t e r p r é t a t i o n s of JE is induced by a morphism of sketches P : cr > a. This P is t h e "proof " t h a t a model of 0 (of 0) is an i n t e r p r é t a t i o n of JE. In f a c t P is not a g ê n e r a i morphism of sketches, but d é t e r m i n e s <r as a <r - s k e t c h (see [6] p. 10 for t h e p r é c i s e définition). So we choose to say now t h a t a f o r m u l a f o r cr (in the place of a JE-formula) is nothing but such a P, a cr - s k e t c h . In [6] the boolean c a l c u l u s of <r - s k e t c h e s (conjonctions, d i s j o n c t i o n s , compléments) is exposed a s construction in t h e c a t e g o r y of s k e t c h e s . Then we can defined t h e c a t e g o r y For(cr ) a s being t h e c a t e g o r y of

<T - s k e t c h e s , as o b j e c t s , w i t h morphisms from P t o P' t h e m o r p h i s m s of sketches f : <r » <rf which détermine cr* a s a tr-sketch, such t h a t f.P = P \

At t h i s level of l a n g u a g e , we can change our n o t a t i o n s , r e p l a c i n g Mod 0 by Modcr, or even, m o r e precisely, by P, and t h e H (Mod 0/1,F) will be denoted by H ( P / I , F ) . Of course for gênerai mixed s k e t c h e s (and not only for those a s s o c i a t e d t o f i r s t order formulas) t h e r e s u i t in §5 w o r k s , and the a b e l i a n g r o u p s H (P/I,F) a r e smalls. Now PROPOSITION. The functoriality of thèse H , with respect to P, I and F are trivial facts.

8. In a dual way, given a functor F : Mod 0 > Ab and an i n t e r p r é t a t i o n I of JE, t h e cohomology of I w i t h coefficient in F is defined by considering t h e cochain complexe

1 o

d d

C2(I,F) < 0*0,F) < C°(I,F) which is

11

1 0 d d

n

< n ™

<

™

M <- M *- M <- I M <- M <- I M <- I

2 1 0 1 0 0

w i t h

dl(x)

(l4 M *-> M ) " ™<*(,_î ,) - x ^

0 1 0 1

and so on, and we define H ( I, F) = ker dⁿ/Im dⁿ .

F o r t h è s e cohomology g r o u p s , t h e same r e s u i t is t r u e , t h a t is t o say t h a t they a r e small. But now, t h e computation is based on t h e effective c o n s t r u c t i o n of a "relatively f i l t e r e d locally f r e e d i a g r a m "

( r . f . l . f . d . ) (in the c a t e g o r y Mod 0) g e n e r a t e d by I. Thèse cohomolgy g r o u p s will be denoted by H ((I/Mod 0 )o p, F ) .

9.Collecting the r e s u l t s of §5, §7 and §8, we get :

THEOREM :The abelian groups H (Mod 0/1,F) and Hⁿ((I/Mod 0)°^P,F) are small, i.e. they are éléments of the category Ab, they are functorial with respect to I, F, [i and 0, and if l [ = 0, then H (Mod 0/1,F) = 0, for every n > 0, and H ((I/Mod 0)^{o p},F) = 0, for n > 0, In fact, more precisely, we hâve H (Mod 0/1,F) = 0, for every n > 0, if there is a cofree model generated by l, and we hâve H ((I/Mod 0)°^P,F) = 0, for n > 0, if there is a free model generated by I. So they are small obstructions to the satisfaction of 0 in I "up to co-equations" and "up to "équations".

REFERENCES

[1] M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, S.L.N. 32, 1967.

[2] M. Evrard, Homotopie d'un espace topologique relativement à un recouvrement. Applications à l'homotopie des préschémas, Thèse d'Etat, U n i v e r s i t é P a r i s 7, 1973.

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[3] M. Evrard, Fibrations de petites catégories, Bull. Soc. Math.

F r a n c e 103, p.241-265, 1975.

[4] D. Quillen, Higher algebraic K-theory,I, S.L.N. 341, p. 85-147, 1973.

[51 R. G u i t a r t et C. L a i r , Calcul syntaxique des modèles et calcul des formules internes, Diagrammes, vol. 4, 106 p . , 1980.

[6] R. G u i t a r t et C. Lair, Limites et co-limites pour représenter les formules, Diagrammes, vol. 7, 24 p . , 1982.

[7] R. G u i t a r t , On the geometry of computations, I and 17, C a h i e r s Top.

Géo. Diff. Cat., vol. XXVII-4 (1986), p. 107-136, et vol. XXIX-4 (1988), p. 297-326.

[8] R. G u i t a r t , L'unité de la théorie des modèles et de l'algèbre homologique (conf. a t t h e Summer meeting on Category Theory, Louvain-la-Neuve, July 26 - Aug 1, 1987), in Cahiers Top. Géo. Diff.

Cat. XXX-3 (1989).

U.F.R. Maths, Tours 45-55, 5è m c e t . , Université. PARIS 7, 2 place Jussieu, 75005, FRANCE.