Dominique Bourn

Professeur Université du Littoral

Laboratoire LMPA, BP 699 , 62228 Calais CEDEX, FRANCE.

my mail

Téléphone : 33 3 21 46 55 92

Fax: 33 3 21 46 36 69

__Abstracts of the main papers.__

- 1.-Natural anadeses and catadeses. Cahiers Top et Géom. Diff. XIV, 4 (1973) 371-480.
- 2.-(with J. Penon). 2-categories reductibles -Preprint Amiens (1978).
- 3.-Une autre propriété universelle pour le champ associé. Cahiers Top . et Géom. Diff.XXI,4 (1980) 403-409.

- 4.-(with J.M. Cordier) Distributeurs et théorie de la forme. Cahiers Top et Géom. Diff XXI,2(1980) 161-188.
- 5.(with J.M. Cordier) A general formulation of homotopy limits .Journal of Pure and Applied Algebra 29 (1983) 129-141.
- 6.A canonical action on indexed limits. Category Theory, Springer LN 962 (1982) 23-32.

- 7.-La tour de fibrations exactes des n-categories. Cahiers Top. Et Géom. Diff. XXV 4 (1984) 327-351.
- 8.-The shift functor and the comprehensive factorization for internal groupoids. Cahiers Top. Et Géom. Diff. XXVIII, 3 (1987), 197-226.
- 9.-A right exactness property for internal n-categories. Cahiers Top. Et Géom. Diff. XXIX, 2 (1988), 109-155.
- 10.-Another denormalization theorem for abelian chain complexes. Journal of Pure and Applied Algebra 66 (190) 229-249.
- 11.-The tower of n-groupoids and the long cohomology sequence. Journal of Pure and Applied 62 (1989) 137-183.

- 12.Pseudo functors and non abelian weak equivalences. Categorical algebra and its applications. Springer LN 1348 (1988), 55-70.
- 13.-Produits tensoriels coherents de complexes de chaine. Bulletin de la Soc. Math. de Belgique (Série A), XLI,2 (1989), 219-248.
- 14.-Normalization equivalence, kernel equivalence and affine categories category theory, Springer LN 1488 (1991) 43-62.

- 15.-Low dimensional geometry of the notion of choice. Category theory 1991 Canadian Math. Society Conference Proceedings, 13, (1992), 55-73.
- 16.-Polyhedral monadicity of n-groupoids and standardized adjunction. Journal of Pure and Applied Algebra, 99 (1995) 135-181.
- 17.-The structural nature of the nerve functor for n-groupoids. To appear in Applied categorical structures
- 18.-n-Groupoids from n-truncated simplicial objects as a solution ot a universal problem. Cahiers du Langal n°17 (1997) Preprint Univ-Littoral. To appear in Journal of Pure and Applied Algebra.

- 19.-Mal'cev categories and fibration of pointed objects, Applied Categorical structures 4,(1996), 307-327.
- 20.-(with G. Janelidze) Protomodularity, descent and semi-direct products. Theory and Applications of Categories, 4, n°2, (1998), 37-46.

__I.Aspects of internal categories
and 2-categories__

__1.-Natural anadeses and catadeses__. Cahiers Top
et Géom. Diff. XIV, 4 (1973) 371-480.

It is observed, in the appendix, that the three first
cohomology groups of a group in the sense of Eilenberg-MacLane, can be
successively calculated in terms of a limit for H_{0}, a lax limit
for H_{1}, a 2-lax limit for H_{2}. It is explicitely based
on the fact that a given abelian group A has a canonical structure of n-category
for each integer n, and it is the occasion of the general definition of
n-category in the internal way (vs the " enriched category "
way). The specific link between the abelian cohomology and this typical
property of the abelian groups will be elucidated in [11].

2.-(with J. Penon).__2-categories reductibles__-Preprint
Amiens (1978).

The 2-categories which are comonadic, via a left exact comonad, on a 2-category of the form Cat(E) (the 2-category of the internal categories in E) are characterized.

3.-__Une autre propriété universelle pour
le champ associé__. Cahiers Top . et Géom. Diff.XXI,4
(1980) 403-409.

It is shown here that the stack completion __C__ of
an internal category C (in the sense of M. Bunge : Stack Completions
and Morita equivalence for categories in a topos, Cahiers Top et Géom.
Diff. XX,4, (1979)401-436) has another universal property : it classifies,
the weakly representable profunctors with codomain C, ie the profunctors
with codomain C representable up to weak equivalence.

The condition (BA) on a category E is considered :
for every internal category C, the stack completion __C__ is still internal,
or equivalently the stack completion __C__ is representable.

__II Profunctors, shape theory
and homotopylimits__.

4.-(with J.M. Cordier) __Distributeurs et théorie
de la forme__. Cahiers Top et Géom. Diff XXI,2(1980) 161-188.

The notion of shape category was introduced by Borsuk (1968) in order to study the homotopy properties of compact spaces. Axiomatic approaches of this notion were given by Holsztynski (1971) and Bacon (1975), as well as categorical ones by Deleanu and Hilton (1976). It is shown here that the axioms determined by these two ways of approach are actually universal properties (in terms of Kan extensions) with respect (not to ordinary functors but) to profunctors in the sense of Benabou . In this context, the category S is the shape category related to the functor K : Wà H if and only if it is the Kleisli category of the codensity promonad generated on H by the functor K. In the same way, further aspects of the Cech approach of the shape theory can be elucidated in terms of profunctor.

5.(with J.M. Cordier) __A general formulation of homotopy
limits__ Journal of Pure and Applied Algebra 29 (1983) 129-141.

Let å be the category of simplicial sets. The definition of the homotopy limit of a functor F :Ià å has been introduced by Bousfield and Kan (1972), taking into account the higher order homotopy coherences involved in certain constructions. This paper shows that this notion is a particular case of the general notion of indexed limits (Borceux, Kelly (1975)) in the context of the categories enriched in å , and that the Bousfield Kan replacement scheme holds in this extended context.

6.__A canonical action on indexed limits__. Category
Theory, Springer LN 962 (1982) 23-32.

Given a V-category A and a profunctor j : A--->1,then this profunctor is an indexation for a certain type of V-limits (Borceux, Kelly (1975)). It is shown here that the jûindexed limit of the profunctor j itself is a V-monoïd and that this V-monoid has a canonical action on each j-indexed limit.

The particular situation which induced the idea of this
result is the following classical one. Let D be the 2-category with a simple
objet * and generated by a 1-morphism t :* à*,
and two 2-morphism l :* =>t et m
:t^{2}
=>t such that m .l
t=t=m .tl and m
.m t=m .tm
. It is clear that a monad on a category C is a 2-functor P
from D to Cat, and well known that the lax limit of P
is the category of algebras C^{T} of this monad. Now the indexation
P(D) :D ---> 1 determining this laxlimit
is nothing but the laxlimit of the Yoneda embedding, and a simple
but tedious computation shows that the Cat-monoid determined, following
our result, by this indexation is D^{co} (the dual of D for the
2-cells). That this monoid acts on C^{T} just emphasized
that there is a comonad on C^{T}. An application of the existence
of such an action is then given in homotopy theory. Let us denote by H
the sub 2-category of D obtained by cancelling the 2-cell l
. As any 2-category, H is a simplicial category.Now let us call coherent
homotopy idempotent a simplicial functor F :Hà
Top. The action determined by H allows to show that the associated homotopy
idempotent (i.e. the associated idempotent in the homotopy category) splits,
although, in general, the homotopy idempotent do not split.

__III Internal n-categories and
cohomology__ .

__7.-La tour de fibrations exactes des n-categories__.
Cahiers Top. Et Géom. Diff. XXV 4 (1984) 327-351.

A general setting is introduced here in order to control, in the case of the internal n-categories, the passage from n to n+1 in a way which allows to use proofs of inductive type (see 9 and 10 for applications).

Once defined the category CatE of internal categories
in E, with the associated fibration ()_{0} : CatEàE
assigning to each internal category its object of objects, the categorie
(n+1)-CatE of internal (n+1) categories in E is defined by the following
inductive process : suppose defined n-CatE and the fibration ()_{n-1} :
n-CatEà (n-1)-CatE, then (n+1)-CatE
is the full subcategory of Cat(n-CatE) whose objects are the categories
internal to the fibres of ()_{n-1}.

It is then shown that when E=Sets, this definition coincides with the enriched version of n-categories.

__8.-The shift functor and the comprehensive factorization
for internal groupoids__ Cahiers Top. Et Géom. Diff. XXVIII, 3
(1987), 197-226.

It is actually a preliminary paper to the following one. It contains the fact that the category GrdE of internal groupoids in E is monadic above the category SplE whose objects are the split pairs (an epimorphism with a given splitting) and morphisms the commutative squares between such pairs, as soon as E is left exact. This result allows to proof that if furthermore E is Barr exact then any internal functor between internal groupoids can be factorized, in a unique way up to isomorphism, into the composite of a discrete fibration with a final functor. It is called, according the result of Street and Walters (1973) when E=Sets , the comprehensive factorization.

9.-__A right exactness property for internal n-categories__
Cahiers Top. Et Géom. Diff. XXIX, 2 (1988), 109-155.

When E is Barr exact, then the category SimplE of simplicial
objects in E is still Barr exact. It is no more the case for the category
n-CatE of internal categories in E. However it is possible to show that
this category has the same kind of right exactness but only with respect
to a certain class å _{n}
of regular epimorphisms. This class å
_{n} is defined by induction. Firstly it is
easy to show that when E is Barr exact, the fibration ()_{n-1} :n-CatEà
(n-1)-CatE is Barr exact : each fiber is Barr exact and each change
of base functor is Barr exact.

The class å _{1}
is defined in the following way : the internal functor f_{1}
is in å _{1}
if f_{0} is a regular epimorphism in E and its decomposition f_{1}=g_{1}h_{1}
with respect to the fibration ()_{0} :Cat Eà
E (h_{1} in the fibre, and g_{1} cartesian) is such that
h_{1} is a regular epimorphism in the fibre.

The class å _{n}
is then defined inductively : the internal n-functor f_{n}
is in å _{n}
if f_{n-1 }is in å _{n-1}
and its decomposition f_{n}=g_{n}.h_{n} with respect
to ()_{n-1} : n-CatEà
(n-1)-CatE is such that h_{n} is a regular epimorphism in the fibre.

10.-__Another denormalization theorem for abelian chain
complexes__. Journal of Pure and Applied Algebra 66 (190) 229-249.

Once the general setting of [7] introduced, it was very
easy to show that for each additive category A and each integer n the category
C^{n}A of Complexes of length n in A is equivalent to the category
n-GrdA of internal n-groupoids in A, what is an alternative of the classical
Dold Kan equivalence between C^{¥}A
and SimplA.

Moreover these new denormalization equivalences exchange chain homotopies with what are expected to be n-lax natural transformations. Thus n-groupoids appear as possible substitutes for n-complexes in the non abelian context.

__11.-The tower of n-groupoids and the long cohomology
sequence__ Journal of Pure and Applied 62 (1989) 137-183.

Given a Barr exact category E, then the fibration ()_{0} :Grd
EàE is Barr exact [9]. Now if A is
an abelian group object in E, the classical H^{0} (E,A) is nothing
but the group E(1,A) of global elements of A. But A being abelian, it is
also an abelian group, under the notation K_{1}(A), in the fibre
above 1 of the fibration ()_{0}. Now given any object X in E with
global support t :Yà
1, then the inverse image t *(K_{1}(A))
determines an abelian group object in the fibre [X] above X. The starting
point of this paper is the following remark : the abelian group H^{1}(E,A)
defined classically by the classes of A-torsors is nothing but the co-limit
of the abelian groups H°([X], t *(K_{1}(A))),
each X having a global support.

This is clearly the beginning of an inductive process
which defines H^{n+1}(E,A) as the co-limit of the H^{n}([X],t
*(K_{1}(A))). This process describes H^{n+1} (E,A) as the
set of connected components of the category whose objects are the aspherical
internal n-groupoids in E equipped with a structure of K_{n}(A)-torsors
with respect to the fibration ()_{n-1} : n-GrdEà
(n-1)-GrdE, where K_{n}(A) is precisely the structure of n-groupoid
in E associated with the abelian group A in E (as defined in the paper
[1]) which is naturally an abelian group object in n-Grd(E).

These H^{n} has the property of the long cohomology
sequence. Furthermore when E=A is abelian, this description of the H^{n}
coincides with the classical YonedaÆs description of Ext^{n}, via
the denormalization of [10]. When E=Gp the category of abstract groups,
this description is equivalent to those given by Holt (1979) and Huebschmann
(1980) by means of crossed n-fold extensions.

__IV Some applications of the n-categorical
denormalization__.

12.__Pseudo functors and non abelian weak equivalences__.
Categorical algebra and its applications. Springer LN 1348 (1988), 55-70.

The usual Dold-Kan equivalence between the category C^{¥}A
of positive chain complexes in the additive category A and the category
SimplA of simplicial objects in A did not strictly exchange chain homotopies
with simplicial homotopies, while the new denormalization of [10] do strictly
exchange chain homotopies with what is expected to be higher lax natural
transformations. In that sense, the two notions of n-complexes and n-groupoids
seem to be more closely connected than the two notions of n-complexes and
n-truncated simplicial objects.

The aim of this paper is to make use of this strong relationship
between C^{n}A and n-Grd A as guide in exploring certain aspects
of the categories n-GrdE, for any E, and especially in elucidating what
could be the notion of weak n-equivalence between internal n-groupoids.

Indeed there are too many choices of possible definitions. The paper, here, is devoted to describe the notion of weak n-equivalence which, in the additive case, is in one to one correspondance, via the denormalization theorem, with the notion of weak equivalence between (n-truncated) chain complexes. We shall suppose, from now on, that the basic category E is Barr exact.

Of course, at level 1, the definition is clear and classical :
a functor f_{1} :X_{1}à_{
}Y_{1} is a weak equivalence when it is fully faithful (i.e.
cartesian with respect to the fibration ()_{0} :GrdEà
E) and essentielly surjective (i.e. P _{0}(f_{1})
is an isomorphism). More generally a functor f_{n} : X_{n}à_{
}Y_{n} is a weak n-equivalence when it is cartesian with respect
to ()_{n-1} : n-GrdEà
(n-1)-GrdE (see [7]) and when P _{n-1}(f_{n})
is a weak (n-1)-equivalence, where P _{n-1} :
n-GrdEà (n-1)-GrdE is the left adjoint
to the discrete functor dis : (n-1)-GrdEà
n-GrdE, left adjoint to ()_{n-1}.

In the case E=Sets, a weak 2-equivalence f_{2} :X_{2}à_{
}Y_{2} between the two 2-groupoids X_{2} and Y_{2}
is a 2-functor such that :

1)f_{2} is essentially surjective, 2)f_{2}
is " hom by hom " a weak equivalence.

Now, thanks to the axiom of choice, it is possible to
characterize weak 2-equivalences in the category 2-Grd of 2-groupoids as
those 2-functors which have (not an inverse equivalence as at level 1 but)
a pseudo-inverse equivalence. Consequently every pseudo-functor :
X_{2 }--->Y_{2} can be represented by a pair of 2-functors :
X_{2}ß_{ }V_{2}à_{
}Y_{2} where the left hand part is a weak 2-equivalence.

It is then possible to define abstract n-pseudofunctor
in any n-Grd(E) as a pair of internal n-functors : Xnß
Vnà Yn where the left hand part is
a weak n-equivalence. This is all the more interesting than the H^{n}(E,A)
of [11] appears then as the classes of n-pseudo-functors 1- - - >K_{n}(A).

__13.-Produits tensoriels coherents de complexes de chaine__.
Bulletin de la Soc. Math. de Belgique (Série A), XLI,2 (1989), 219-248.

There are at least two different structures of closed monoidal category on the category 2-Grd of 2-groupoids : the cartesian closed structure and that which comes from the Gray tensor product and the lax natural transformations. It is clear that we may expect n different closed monoidal structures on the category n-Grd of n-groupoids, corresponding to the different level k, 1£ k£ n, where we stop the introduction of kÆ-cells, 1£ kÆ£ k, in the possible definitions of higher order lax natural transformations. But it seems very difficult to explicit them.

Now, in the additive situation, thanks to the new denormalization
[10], n-Grd(Ab) is equivalent to C^{n}Ab. It is obviously much
easier to work in C^{n}Ab, and the aim of this paper consists in
expliciting the corresponding expected n different closed monoidal structures
on C^{n}Ab.

One interesting point is that, for any additive category
A, the category C^{n}A is enriched in any of these n different
closed monoidal structures on C^{n}Ab.

Thus the category C^{¥}
Ab of positive chain complexes is endowed with infinitely many different
closed monoidal structures and the category C^{¥}A
of positive chain complexes in A is endowed with infinitely many different
enriched category structure.

__14.-Normalization equivalence, kernel equivalence and
affine categories category theory__, Springer LN 1488 (1991) 43-62.

The starting point here is the caracterisation by Carboni (1989) of the categories of affine spaces by means of a " modularity " condition. Actually the denormalization theorem [10] leads naturally to an equivalent caracterisation. Let us denote by Pt(E) the category whose objects are the split pairs (an epimorphism with a given splitting) and morphisms the commutative squares between such pairs, and by p :Pt(E)à E the functor assigning to each pair the codomain of the epimorphism. This p is obviously a fibration as soon as E is left exact. It is called the fibration of the pointed objects in E.

Now a category E is modular if and only if

1)the fibration p is trivial (i.e. every change of base is an equivalence)

2)the functor -+1 (sum with the terminal object) satisfies a certain " modularity " condition.

The main result is that the condition 1 implies that the fibration p is additive, meaning that each fiber is additive and each change of base functor is additive. We shall call essentially affine a category satisfying the condition1, and consequently a category C is additive if and only if it is essentially affine and has a zero object.

Now a functor is an equivalence if and only if 1) it reflects the isomorphisms and 2) its left adjoint is a right inverse. So in presence of a zero object, the fact that C is essentially affine means that : 1)the short five lemma holds for split exact sequences and 2) the sequence 0à Aà A+Bà Bà 0 is exact. But the (split) short five lemma still holds for non abelian groups, and if we denote by Gp the category of groups then the fibration p :Pt(Gp)à Gp is such that every change of base functor reflects the isomorphisms.

Whence the definition : a category is protomodular when it is left exact and the fibration p :PtCà C is such that every change of base functor reflects the isomorphisms. Given any left exact category E, then the category Gp(E) of internal groups in E is protomodular as well as any fibre of the fibration

()_{o} : GrdEà
E (defined in [7]).

In any protomodular category C, as it is the case in Gp, the pullback functors reflect the monomorphism, any internal category is a groupoid. If furthermore C has a zero object and is Barr exact, any regular epi is the cokernel of its kernel.

__V. Relationship between n-groupoids
and n-truncated simplicial objects in the non abelian situation :
the nerve functor for n-groupoids.__

15.-__Low dimensional geometry of the notion of choice__.
Category theory 1991 Canadian Math. Society Conference Proceedings, 13,
(1992), 55-73.

In the same way as the category Grd E of internal groupoids in E is monadic above the category PtE of the split epimorphisms (see [8] and [14], the change of notation is due to the importance of the fibration p : PtEà E), the category 2-Grd E of internal 2-groupoids in E is shown to be monadic above the category N-GrdE of internal groupoids equipped with a normalization ; i.e. with the choice of an object in each connected component and a morphism between this object and any other object of this component. The underlying normalized groupoid of a 2-groupoid is the groupoid of the oriented triangles. This monadicity theorem yields a natural comparison functor N:2-GrdEà SimplE which is a synthetic version of the nerve functor for 2-groupoids and which, allowing to get free from the combinatorial geometric aspect of the pasting used in the case of 2-categories, has a meaning in the purely internal situation.

__16.-Polyhedral monadicity of n-groupoids and standardized
adjunction__. Journal of Pure and Applied Algebra, 99 (1995) 135-181.

This paper is the natural continuation of the previous
one, which, by the way, contained already inductive definition of normalized
internal n-groupoids. Considering the fibration ()_{n-1} :n-GrdEà
(n-1)-GrdE (see[7]), a normalized n-groupoid is a n-groupoid X_{n}
whose underlying (n-1)-groupoid X_{n-1} is normalized and
such that the terminal map t _{n} :X_{
n} à G_{ n} (X_{n-1}),
in the fibre above X_{ n-1} , has a given splitting s
_{n}.

It is devoted to the proof that the category (n+1)-GrdE
of internal (n+1)-groupoids is monadic above the category N-n-GrdE of normalized
n-groupoids. The definition of the functor U_{n} : (n+1)-GrdEà
N-n-GrdE is inductively clear. On the contrary the proof of the monadicity
is not easy and the geometric intuition is lost. More abstract and technical
notions (such as those of standardized adjunctions and monads) are needed.

17.-__The structural nature of the nerve functor
for n-groupoids__. To appear in Applied categorical structures

The notion of the nerve of a category is well known since Grothendieck. From that time, the question of the nerve for n-categories has been studied by various authors, mainly at the Australian (Street and all) and the Welsh (Brown and all) schools. It is always presented in a geometrical combinatorial way which is descriptively growing in a rather discouraging complexity as the level n grows up, and always defined only in the set-theoretical context.

The previous paper about the monadicity theorem for n-groupoids, using a kind of polyhedral presentation of them, led here to an intrinsic and synthetic definition of the nerve, in the case of n-groupoids. It has the advantage to use always the same inductive construction when going from the level n to the level n+1, and to make sense in any internal context.

The whole process in question is based upon the observation
that the categories S-Simpl_{1}E, S-Simpl_{2}E,...,S-Simpl_{n}E
of the split 1,2,..., n-truncated simplicial objects in E can be obtained
successively, one from the previous one, by a unique iterative construction
(E_{1},T_{1}),(E_{2},T_{2}),...,(E_{n},T_{n})
from a category E_{i} endowed with a monad T_{i} to the
category E_{i+1 }with the monad T_{i+1}, the starting point
being E with the identity monad.

More precisely starting from a left exact fibration t
:Eà B and a monad (T,l
,m ) on E, no relationship between t
and (T,l ,m ) being
required, the category TE of the T-elements of E is introduced, whose objects
__X___{1} are defined as triples (X_{0},d,t) of an object
X_{0} of E, a map d :X_{1à
}TX_{0} in the fibre of TX_{0} and a map t :X_{0à
}X_{1} in E such that d.t=l X_{0}.
Such a T-element must be thought of as a map in the fibres of t
which is split " modulo the monad T ". Moreover, there
is on TE a monad (T_{1}, l _{1},m
_{1}) with T_{1}__X___{1} = (X_{1},dÆ,tÆ)
where the map dÆ is the pullback of d along m
X_{0}.Td, and the map tÆ=[1, l X_{1}].

When the monad (T, l ,m
) is left exact, then the functor t _{o} :TEà
E defined by t _{o}(__X___{1})=X_{o}
is again a left exact fibration and the monad (T_{1}, l
_{1},m _{1}) is again left exact.
It is therefore the beginning of an inductive process.

The relationship of this construction with the simplicial situation is the following one. The functor F :S-SimplEà SimplE from the split simplicial objects in E to the simplicial objects, forgetting the splittings, has a right adjoint U which determines a monad (T,l ,m ) on S-SimplE.

Actually this monad is stable on any subcategory S-Simpl_{n}E
of n-truncated split simplicial objects, and we denote by (T_{n},
l _{n},m
_{n}) the monad at this level. Our main remark is that S-Simpl_{n+1}E=T_{n}
(S-Simpl_{n}E), i.e. the category of T_{n}-elements of
S-Simpl_{n}E, with respect to the fibration

t _{n-1} :S-Simpl_{n}Eà
S-Simpl_{n-1}E, which forgets the last level. The new fibration
and the new monad on S-Simpl_{n+1}E determined by the inductive
process are, of course, t _{n} and (T_{n+1},
l _{n+1},m
_{n+1}).

Now this remark is the heart of the inductive definition
of the nerve functor for the n-groupoids through the following result :
given a category E with a left exact fibration t
:Eà B and a left exact monad
(T,l ,m ) on E there
is always a comparison functor nT :GrdtAlgTà
AlgT_{1} where GrdtAlgT is the full
subcategory of the category GrdAlgT of those internal groupoids in AlgT
which are actually groupoids in the fibres of t
.

Thus, it is the fact that the category S-Simpl_{n}E
possesses the structure of the iterated T-elements which determines entirely
the existence of the nerve functor in the case of the n-groupoids.

18.-__n-Groupoids from n-truncated simplicial objects
as a solution ot a universal problem__. Cahiers du Langal n°17 (1997)
Preprint Univ-Littoral. To appear in Journal of Pure and Applied Algebra.

The previous paper [17] shew that the tower of fibrations :

Eß PtEß
S-Simpl_{1}Eß S-Simpl_{2}E...S-Simpl_{n}Eß
S-Simpl_{n+1}Eà was inductively built by the construction of the
iterated categories of T-elements from the final fibration Eà
1 and the identity monad.

On the other hand, the paper [17] built inductively another tower of fibrations :

Eß PtEß
N-GrdEß N-2-GrdE ... N-n-GrdEß
N-(n+1)-GrdE..., also equipped with monads (T_{n}, l
_{n},m _{n}) on N-nGrdE whose
category of algebras was the category (n+1)-GrdE of internal (n+1)-groupoids
in E .

These two towers were related at the level of their algebras
by the nerve functor [17]. The aim of this paper is the study of the direct
link which related them. This link is actually based on a very simple observation.
Let t :Eà
B be a left exact fibration, then any fibre has a terminal object. Now
given a left exact monad (T,l ,m
) on E, it is always possible to associate with the fibration t
, a left exact fibration __t__ :__E__à
B in which the terminal objects in the fibres are also initial, with respect
to the given monad (T,l ,m
).

If we denote by __TE__ the combination of the two constructions :
first, from t , the construction of the T-elements
[17] t _{0} :TEà
E, then __t___{o} :__TE__à
E, we get a new inductive process, which, starting from the final fibration
Eà 1 and the identity monad, produces
the tower of the normalized n-groupoids.

Thus, and it is the claim of this paper, the construction which forces the terminal objects in a fibre to be also initial, determines the normalized n-groupoids from the n-truncated split simplicial objects. There is hidden, here, as intuitively felt in [15], a kind of 1,2,...,n-geometry of the notion of choice.

As a consequence, there is a characterization of the nerve
of n-groupoids. This characterization simplifies these of Dakin and Ashley
(1988) and has a meaning even in the internal context. Let us call q
-complex in E a pair (X,T) where X is a simplicial object and T=(T_{i}),1£
i, a graded subobject of X, with T_{iÌ
}X_{i}. The elements of T are called thin and satisfy the
following axioms, which can be simply expressed in terms of limits :

-All degenerate elements of X are thin

-Any __initial__ horn is X has a unique thin filler

-If all but the __last one__ of the faces of a thin
element of X are themselves thin, then also is the last face thin.

Then (X,T) is the nerve of an internal groupoids as soon
as (X,T) is of rank n (T_{m}=X_{m}, "
m>n).

The method of this characterization, based upon an inductive
algebraicization of the definition of the nerve leaves aside of course,
but rather frustratingly, the geometric description for the benefit of
a decisive algebraic property : the nerve functor n
_{n+1} is cofibrant on the hypomorphisms.

__VI Protomodularity and related
topics.__

__19.-Mal'cev categories and fibration of pointed objects__,
Applied Categorical structures 4,(1996), 307-327.

The observation in [14] that when the fibration p :PtEà E of pointed objects in E is trivial (each change of base functors is an equivalence), then it is additive (each fibre is additive as well as any change of base functor), led here to some classifying properties for this fibration p.

-It is additive if and only if the category E is Naturally MalÆcev in the sense of Johnstone (1989)

-It is unital if and only if the category E is Mal'cev in the sense of Carboni, Lambek, Pedicchio (1990).

A fibration is unital when each fiber is unital and each
change of base functor is left exact, a category C being unital when it
is left exact, has a zero object and is such that for any pair of objects
X,Y, the pair of maps i_{x} :Xà
XxY, i_{y} :Yà XxY is
jointly strongly epic, which means that any product XxY is generated by
Xx1 and 1xY.

Actually other characterizations of Mal'cev and Naturally Mal'cev are given which show that :

modular | Þ |
essentially affine |
Þ |
Naturally MalÆcev |

ß |
ß |
|||

protomodular |
Þ |
MalÆcev |

20.-(with G. Janelidze) __Protomodularity, descent and
semi-direct products__ Theory and Applications of Categories, 4, n°2,
(1998), 37-46.

An exact category with a zero object is protomodular [14] if and only if its satisfies the " split short five lemma ". This paper is devoted to two new observations about protomodular categories :

-using descent theory, one can generalize various form of the short five lemma to non-exact categories, which again gives conditions equivalent to protomodularity.

-the change of base functors of the fibration p :PtEà E, in many important cases as in the case of the category Gp of groups, is not only conservative but also monadic. This has an important application : a categorical definition of a semi-direct product which generalizes the algebraic notion of semi-direct product.