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
<http://www.numdam.org/item?id=DIA_1990__23__7_0>

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DIAGRAMMES

VOLUME
ACTES DES JOURNEES
<LHMIV. P A R I S

23

E.L.I.T.

7, 2 7 J U 1 N - 2 J U I L L E T

1988)

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 théorie de cohomologie) sur
la
catégorie
des
interprétations
de JE, dont
la
valeur
sur
chaque
i n t e r p r é t a t i o n I de JE est une obstruction à I \= <p "à des co-équations
p r è s " ( et "à des équations près").

This paper is a sequel of [7].

O. Let ï

be a f i r s t

order

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 partially

and

t o compute

how I is

question

connected

f a r from t h e models of ¢.
In t h e p a p e r s
with

the

[7] and

possibility

[8] it

is shown how t h i s

of

geometrical

study

of

algorithms

ix(x ,...,x ) be a first

order

formula

of

the

models

models

of

a

is

and

ambiguïties.

1. PROPOSITION.

Let

1

let

Mod <f> be

morphisms
such

the

from

category

with

objects

M to M* the morphisms

(of

of

¢,

$) m : M

that
V x ,...,x
I

Then

JE, and

n

there

[ /i(m(x )

n

is a small

I

mixed

sketch

m(x ))

> ^(x ,...,x ) ]

n

i

<r such

that

n

Mod <f> = Modcr.

and

with
> M'

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
such that the inclusion ger

construct

a

set

gtr

» BModtr is an équivalence of homotopy. In

particular we get
PROPOSITION. Mod(r[(Mod(r) ] is a small groupoïd,
call it the fundamental

groupoïd

up to équivalence.

We

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

interprétations of JE) f : M

with

objects

the

morphisms

(of

> I where M is a model of 0, and with

morphisms, from f : M

» I to f

models ( morphisms of Mod 0) g : M

: M'

» I, the morphisms of

» 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
description
4.

Let

of the position
Ab

F : Mod 0
functor

cw-complexe

on

be

the

g<r( JE, I, 0, ^x), which is a

of I with respect

category

of

small

géométrie

to Mod 0.
abelian

groups,

and

let

> Ab be a functor. (In particular F could be the constant
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

d

d

d

-> C ( I , F ) — ^ - > C ( I , F ) —î—> C ( I , F ) — ^ > 0
2

1

0

which is
d

-> y FM
U

Z

d
U

M ^ M -* M -> I
1

2

d

— ? — > y FM — — - > y FM 0 — ? - > o
1

^

M -> M -» I

0

1

M ^ I

0

c

0

with d = s - s,, where
î
î
1
Inc

Id

F(a)
s1 : (FM ) , w a w /3 n
î
1 (M —» M !-* I)
1

inc
> (FM ) , w £
o (M - î — > nI)

> Vu

0

0

W

FM
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, for
0

0

i

1

l

1

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

PROPOSITION.

n

H ( I, F) is

a function

only of the homotopy
byJ H (Mod 0/1,F).
n
il

see

[2],

[1],

Let

IntJE

J : Mod0

n+1

n

dépends
denoted

[3]

and

be

the

2

.

type

of

F, I, /i, 0,

of

Mod 0/1 and of

of

interprétations

which

in

F and could

fact
be

[4].

category

> IntJE be the canonical

inclusion.

Then

the

of

JE,

let

inductive

Kan

extension of F along J is given by
[ E x t . F K D = Lim
F(M )
-^J
M —»1
°
o

and we hâve

H ( I, F) =[Ext F](I).
f F(I) if n = 0
If

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

^

n

/ut

0

if n > 0

5. Now, t h e point is t h a t , because of the r e s u l t s hereover
we get

(§§ 1 to 4),

PROPOSITION.

The

tools

of

[1] and

of

[3],

available

in t h e

situation

where a full and small category M (called a category of "models")
inside a big c a t e g o r y

of

"spaces", are

also

available

where

a (possibly big and not necessarlv full) category

of

theory

a

lives

inside

a

big

category

of

in

lives

the

situation

Mod 0 of

models

interprétations

of

a

language

JE (compare with the idea of "paires adéquates" p. 43 of [1]).

Precisely

hère

we get the fact

that the H (Mod 0/1,F) are

small.

6. After the e x i s t e n c e of g<r proved in [7], the theorem h e r e u n d e r §9 is
just

a

second

stone

for

computation of our H
"locally cofree
"relatively
and

[61)

diagram",

the

work

is based

cofiltered
(in

a

and

to

on

the

pursed.

effective

more precisely

locally

category

cofree

Mod 0)

contains ail t h e information

be

construction

a

on the c o n s t r u c t i o n

of

(r.cf . l . c f . d . H s e e

by

I.

This

we need, and it will be t h e s t a r t i n g

calculus, by t h e way of comparaisons between various H .

and

t o go t o w a r d

small

H ,

especially

thèse

calculations

need

relations

For

that

it

will

be

formulas

of

the

language

and

we

the

effective

For

[5]

point

relative

that

it

of

thèse

relative calculation

need

a

description

of

the

link

theory

of

démonstrations.

For

example

among

H (Mod 0/1,F), H (Mod y/I,G),
n
/i
n
JI
H (Mod [0=>y]/I,D (for conveniant K and L).
n
M
necessary

to

describe

JE. At f i r s t

this

the
will

a

r.cf. 1.cf.d.

of an absolute calculus. But for concrète situations we need a

will be e s s e n t i a l

the

of

diagram"

generated

Theoretically

between
we

H (Mod [0A?]/I,K),
n
^i
category

be usefull

For(JE)
to

of

précise

the functoriality of the H (Mod 0/1,F) with respect to 0 and fi.
7. The

first

purpose

classical f i r s t

of

this

o r d e r formula

paper
0

was

to

show

precisely

how

of a language JE d é t e r m i n e s a

each

"small"

homology t h e o r y on t h e category of i n t e r p r é t a t i o n s of JE.
Now, the continuation

of

this

research

pass trough

the

description

For(JE). With r e s p e c t to t h a t , I would like to make the following

of

remark

: what hâve t o be morphisms between formulas ? it is not so c l e a r a
priori

; they

hâve to be "démonstrations" or "proofs",

10

but t h e r e

is no

canonical idea of w h a t is a démonstration.
But if we décide t o s t a y in (or to corne back to) the s t y l e of

sketches,

a

(i.e.

first

picture

is easy

t o give.

In fact

JE "is" a sketch

<r

the

category 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 the inclusion of the category of models of 0

(of

0)

in

the

category

of

interprétations

morphism of sketches P : cr
of

0

(of

0) is an

of

JE is

induced

by

a

> a. This P is t h e "proof " t h a t a model

interprétation

of

JE. In f a c t

P is

not

a

gênerai

morphism of sketches, but détermines <r as a <r -sketch (see [6] p. 10 for
t h e précise définition).

So we choose to say now t h a t a f o r m u l a for cr

(in the place of a JE-formula) is nothing but such a P, a cr - s k e t c h .
[6]

the

boolean

calculus

compléments)

is

Then

defined

we

can

exposed

of
as

the

<r -sketches

construction
category

(conjonctions,

in t h e

For(cr ) a s

In

disjonctions,

category

of

being

category

the

sketches.
of

<T - s k e t c h e s , as objects, with morphisms from P to P' t h e morphisms of
sketches

f

» <rf which

: <r

détermine cr* a s

a tr-sketch,

such

that

f.P = P \
At this level of language, we can change our notations, replacing Mod 0
by Modcr, or even, more precisely,
denoted
only

by

for

works,

H (P/I,F).

those
and

Of

course

associated

to

abelian

groups

the

PROPOSITION. The functoriality
are trivial

8.

In

a

by P, and t h e H (Mod 0/1,F) will be
for

first

of

gênerai
order

formulas)

H (P/I,F)
thèse

mixed

are

H , with

sketches

(and

the

resuit

smalls.

Now

respect

not

in

§5

to P, I and F

facts.

dual

way,

given

interprétation

I of

JE, the

a

functor

F

cohomology

of

:

Mod 0

> Ab

I with

coefficient

defined by considering t h e cochain complexe
1

C 2 (I,F) <

o

d

d

0*0,F) <

which is

11

C°(I,F)

and

an

in

is

F

1

n FM2 <
M <- M *- M <- I
1

2

0

d

d

n ™i <

n ™0

M <- M <- I

M <- I

0

1

0

0

with
dl(x)

(l4 M *-> M ) " ™<*(,_î
0

1

,) - x

^

0

1

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

thèse cohomology groups,

t h e same r e s u i t

is t r u e ,

that

is t o

say

t h a t they a r e small. But now, t h e computation is based on the

effective

construction
(r.f.l.f.d.)

of

a

the

category

(in

"relatively

filtered

locally

free

diagram"

generated

by

Thèse

cohomolgy

Mod 0)

I.

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

small,

i.e.

are

with

respect

for

every

precisely,

they

éléments

H (Mod 0/1,F) and
of

the category

to I, F, [i and 0, and if

H n ((I/Mod 0)° P ,F)

Ab, they

are

are

functorial

l [ = 0, then

H (Mod 0/1,F)

n > 0, and H ((I/Mod 0) ,F) = 0, for

n > 0, In fact,

more

we hâve

n > 0, if

is a

op

cofree

model

n > 0,

if

H (Mod 0/1,F) = 0, for

generated

there

obstructions
to

groups

is

by

l, and we

a free

model

to the satisfaction

of

every

hâve

generated

=

there

P

H ((I/Mod 0)° ,F) = 0,
by

I. So

they

are

0 in I "up to co-equations"

0,

for
small

and "up

"équations".

REFERENCES
[1] M. André, Méthode
commutative,

en algèbre

homologique

et

algèbre

relativement

à un

S.L.N. 32, 1967.

[2] M. Evrard,
recouvrement.

simpliciale

Homotopie
Applications

d'un

espace

à l'homotopie

Université P a r i s 7, 1973.

12

topologique
des

préschémas,

Thèse d'Etat,

[3]

M.

Evrard,

Fibrations

de

petites

catégories,

Bull.

Soc.

Math.

France 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. Lair, Calcul
des formules

internes,

Diagrammes, vol.

[6] R. G u i t a r t et C. Lair,
formules,

syntaxique

Limites

et

calcul

4, 106 p., 1980.
co-limites

pour

représenter

les

Diagrammes, vol. 7, 24 p . , 1982.

[7] R. G u i t a r t , On the geometry
Géo.

et

des modèles

Diff.

Cat.,

vol.

of

XXVII-4

computations,
(1986),

I and 17, C a h i e r s Top.

p.

107-136,

et

vol.

XXIX-4

des

modèles

et

de

l'algèbre

Category

Theory,

(1988), p. 297-326.
[8]

R.

Guitart,

homologique

L'unité

(conf.

Louvain-la-Neuve,

at

de

la

the

July 26 -

théorie
Summer

meeting

Aug 1, 1987),

on

in Cahiers Top.

Cat. XXX-3 (1989).

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

Géo.

Diff.