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.

I.Aspects of internal categories and 2-categories II Profunctors, shape theory and homotopylimits III Internal n-categories and cohomology IV Some applications of the n-categorical denormalization V. Relationship between n-groupoids and n-truncated simplicial objects in the non abelian situation : the nerve functor for n-groupoids. VI Protomodularity and related topics.

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 H0, a lax limit for H1, a 2-lax limit for H2. 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 jűindexed 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  :t2 =>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 CT 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 Dco (the dual of D for the 2-cells). That this monoid acts on CT  just emphasized that there is a comonad on CT. 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 f1 is in å 1 if f0 is a regular epimorphism in E and its decomposition f1=g1h1 with respect to the fibration ()0 :Cat Eà E (h1 in the fibre, and g1 cartesian) is such that h1 is a regular epimorphism in the fibre.

The class å n is then defined inductively : the internal n-functor fn is in å n if fn-1 is in å n-1 and its decomposition with respect to ()n-1 : n-CatEà (n-1)-CatE is such that hn 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 CnA 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 H0 (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 K1(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 *(K1(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 H1(E,A) defined classically by the classes of A-torsors is nothing but the co-limit of the abelian groups H°([X], t *(K1(A))), each X having a global support.

This is clearly the beginning of an inductive process which defines Hn+1(E,A) as the co-limit of the Hn([X],t *(K1(A))). This process describes Hn+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 Kn(A)-torsors with respect to the fibration ()n-1 : n-GrdEà (n-1)-GrdE, where Kn(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 Hn has the property of the long cohomology sequence. Furthermore when E=A is abelian, this description of the Hn coincides with the classical YonedaĂs description of Extn, 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 CnA 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 f1 :X1à Y1 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(f1) is an isomorphism). More generally a functor fn : Xnà Yn 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(fn) 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 f2 :X2à Y2 between the two 2-groupoids X2 and Y2 is a 2-functor such that :

1)f2 is essentially surjective, 2)f2 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 : X2 --->Y2 can be represented by a pair of 2-functors : X2ß V2à Y2 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 Hn(E,A) of [11] appears then as the classes of n-pseudo-functors 1- - - >Kn(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 kĂ-cells, 1££ 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 CnAb. It is obviously much easier to work in CnAb, and the aim of this paper consists in expliciting the corresponding expected n different closed monoidal structures on CnAb.

One interesting point is that, for any additive category A, the category CnA is enriched in any of these n different closed monoidal structures on CnAb.

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 Xn whose underlying (n-1)-groupoid Xn-1  is normalized and such that the terminal map t n :X n à G n (Xn-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 Un : (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-Simpl1E, S-Simpl2E,...,S-SimplnE 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 (E1,T1),(E2,T2),...,(En,Tn) from a category Ei endowed with a monad Ti to the category Ei+1 with the monad Ti+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 X1 are defined as triples (X0,d,t) of an object X0 of E, a map d :X1à TX0 in the fibre of TX0 and a map t :X0à X1 in E such that d.t=l X0. 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 (T1, l 1,m 1) with T1X1 = (X1,dĂ,tĂ) where the map dĂ is the pullback of d along m X0.Td, and the map tĂ=[1, l X1].

When the monad (T, l ,m ) is left exact, then the functor t o :TEà E defined by t o(X1)=Xo is again a left exact fibration and the monad (T1, 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-SimplnE of n-truncated split simplicial objects, and we denote by (Tn, l n,m n) the monad at this level. Our main remark is that S-Simpln+1E=Tn (S-SimplnE), i.e. the category of Tn-elements of S-SimplnE, with respect to the fibration

t n-1 :S-SimplnEà S-Simpln-1E, which forgets the last level. The new fibration and the new monad on S-Simpln+1E determined by the inductive process are, of course, t n and (Tn+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 AlgT1 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-SimplnE 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-Simpl1Eß S-Simpl2E...S-SimplnEß S-Simpln+1EÓ 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 (Tn, 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 :Eà 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 to :TEà 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=(Ti),1£ i, a graded subobject of X, with TiÌ Xi. 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 (Tm=Xm, " 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 MalĂcev 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 ix :Xà XxY, iy :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


Naturally MalĂcev







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.