M. BARR & H. KLEISLI, Topological balls, 3-20 (and Correction in XLII-1, 227-228).

This paper shows how the use of the "construction of Chu" can simplify the rather complicated construction of the *-autonomous category of reflexive z-z*-complete set up by the first author in the original papers and lecture notes on *-autonomous categories (these Cahiers XVII, 1976, 335-342; and Springer Lecture Notes in Math. 752, 1979).

M. DOUPOVEC & I. KOLAR, On the jets of fibered manifold morphisms, 21-30.

The (r,s,q)-jet of a morphism of fibred manifolds f is determined by the r-jet of the map f, by the s-jet of the restriction of f to the fibre and by the q-jet of the base map induced by f, for s > r < q. This paper shows that the (r,s,q)-jets are the only homomorphic images of finite dimension of germs of morphisms of fibred manifolds satisfying two natural conditions.

J. PENON, Approche polygraphique des  ¥ -catégories non strictes, 31-80.

The Author gives a new definition of a non-strict ¥ -category, called Prolixe. The prolixes are algebras for some monad built on the category of reflexive ¥ -graphs. That monad can be defined by a universal property which expresses the universal relaxation of the axioms of ¥ -categories. On the other hand, the building of this monad uses a 'polygraphical' material, obtained by logical technics.

M. GARCIA ROMÁN, M. MÁRQUEZ HERNÁNDEZ, P. JARA & A. VERSCHOREN, Uniform filters, 82-126.

The aim of this paper is to unify and develop some of the main properties of uniform filters, emphasizing their functorial nature and their quantale structure.

A. KOCK & G. REYES, A Note on frame distributions, 127-140.

In the setting of the constructive theory of locales or frames (i.e., in the theory of locales over a base locale), the Authors study some aspects of the "frame distributions", i.e. of the maps from a frame with values in a base frame preserving all suprema.. They derive a relationship with some results of Jibladze-Johnstone and of Bunge-Funk. Moreover, descriptions are given of the interior closure operator defined on the opens of a locale in terms of frame distributions and in terms of generalized double negation operation.

P. DAMPHOUSSE & R. GUITART, Liftings of Stone's monadicity to spaces and the duality between the calculi of inverse and direct images, 141-157.

In this paper, two categories Qual⁺ et Qual^- are introduced, the dual of each of which is algebraic (up to a natural equivalence) on the other, thus lifting the classical algebraicity of Ens^op over Ens. Moreover Qual⁺ is cartesian closed. The calulus of inverse (resp. direct) images is presented as the data of Qual^- (resp. Qual⁺) and of a monad on this category lifting the "monad of Stone" on Ens. The concept of duality between categories is extended in the duality between monads, and in this way the calculus of direct and inverse images are dual. A consequence is the algebraicity on Qual⁺ (up to a natural equivalence) of the dual of the category Top of topological spaces and of the dual of the category of sets equipped with an equivalence relation.

M. GRANDIS & R. PARE, Limits in double categories, 162-220.

In the setting of double categories, the Authors define the (horizontal) double limit of a double functor from I to A and give a theorem to construct these limits from double products, double equalizers and tabulators (double limit of a vertical morphism). The double limits lead to important tools; for instance the Grothendieck construction for a profunctor is its tabulator in the "double category" of categories, functors and profunctors. If A is a 2-category, it gives back the construction of Street for indexed limits; if I has only vertical arrows, it gives back the construction of Bastiani-Ehresmann of limits relative to double categories.

H. HERRLICH & L. SCHRÖDER, Composing special epimorphisms and retractions, 221-226.

The Authors prove that, in the category Cat of small categories (which is locally presentable), the composite of a regular epimorphism and of a retraction is generally not regular, as well as the composite of a retraction and of a regular epimorphism in the category of connected spaces, They introduce a natural category which includes Cat as a full sub-category and in which the composite of an extremal epimorphism and of a retraction is generally not extremal.

D.-C. CISINSKI, La classe des morphismes de Dwyer n'est pas stable par rétractes, 227-231.

In a paper published in these "Cahiers" (Volume XXI-3, 1980), R. Thomason claims that a retract of a Dwyer map is a Dwyer map to show that the closed model category structure he defines on the category of small categories is proper. This paper gives a counterexample to this claim and shows how to give a correct proof of the propriety axiom.

E. DAVID, Stone spaces of more partially ordered sets, 233-240

Using the definition of an ideal of a poset introduced by Doctor, the Author shows that more posets are representable as compact-open sets of Stone spaces than in the case where the definition of Frink is used (as he has done in a preceding paper). Thus he obtains a dual equivalence between the category of posets and a category related to Stone spaces, which extends the dual equivalence given in the preceding paper.

Kyung Chan MIN, Young Sun KIM & Jin Won PARK, Fibrewise exponential laws in a quasitopos, 242-260.

The Authors obtain different types of exponential laws for fibrations in a quasi-topos C, such as

C_ABD(X×Y,Z) @ C_ABD(X,C_BD(Y,Z)) and M_XD(X×_BY,Z) @ M_B(X,C_BD(Y,Z)),

with isomorphisms in C for spaces on different bases. They prove that there exists an isomorphism in C between the space of fiber preserving maps from q to r and the space of transversal sections to q · ₁ r . They discuss the examples of convergence spaces, of sequential convergence spaces and of simplicial spaces, as well as the related cases of quasi-topological spaces and of compactly generated spaces.

A. MUTLU & T. PORTER, Free crossed resolutions from simplicial resolutions with given CW-basis, 261-282.

In this paper, the Authors study the relationship between a CW-base for a simplicial group, and methods to freely generate the associated crossed complex. The case of resolutions is detailed, with a comparison between free simplicial resolutions and crossed resolutions of a group.

E. VITALE, Multi-bimodels, 284-296.

The paper studies the equivalences between multi-reflective sub-categories of categories of covariant presheaves. An appropriate notion of multi-bimodel allows to generalize the classical theorems of Eilenberg-Watts and of Morita for module categories. The motivating example is given by the multi-presentable categories, i.e., the categories which are sketchable by limit-coproducts sketches.

D. BOURN, Baer sums and fibered aspects of Malc'ev operations, 297-316.

The geometric meaning of the axioms in the laws of Malc'ev is emphasized so that, in the associative case, it leads to the associated classical group action in the general setting of exact categories in the sense of Barr, when the support is global. This action is defined through a direction functor d which is shown to be a cofibration preserving the products and the terminal object. This is the case when any group structure on an object X of the base canonically determines a monoidal closed structure on the fibre on X. It gives a conceptual approach to the Baer construction of the sum of two extensions of a group Q with abelian kernels defining the same Q-module structure. The cofibration d allows to precise the relation between Naturally Mal'cev categories and essentially affine categories. The last section studies the case when the support is not global.

S. CRANS, On braidings, syllepses, and symmetries, 2-74 et 156.

Using the tensor product of Gray-categories, the Author defines the concept of a 4-dimensional tas, which generalizes the tas of a Gray-category, and he deduces ideas for generalizing to higher dimensions. The first result is that the 4-dimensional teisi with a unique object and a unique arrow are the semistrict braided monoidal 2-categories. Combining this with the idea that a sylleptic 2-category should be a 5-dimensional tas with a unique object, a unique arrow and a unique 2-arrow, the second result shows that it leads to a notion of syllepse equivalent to that of Day and Street. Similarly, in the third result, the idea that a symmetric 2-category should be a 6-dimensional tas with a unique object, a unique arrow, a unique 2-arrow and a unique 3-arrow, leads to a notion of symmetry still equivalent to that of Day and Street. These last two results extend easily to slightly weaker braided and sylleptic monoidal 2-categories.

ALVAREZ & VILABOA, On Galois H-objects and invertible H*-modules, 75-79.

For a cocommutative Hopf algbra H in a symmetric closed category C, the Authors obtain, as a generalization of a theorem of Childs, a homomorphism between the group Gal_C(H) of isomorphism classes of H-objects of Galois and that Pic(H*) of isomorphism classes of invertible H*-modules. They show that, is Pic(H*) = 1, the group of Brauer of triple H-modules of Azumaya with an internal action coincides with the group of Brauer of triple H-modules of Azumaya defined by the second author in a preceding paper.

DUBUC & ZILBER, Infinitesimal and local structures for Banach spaces and its exponentials in a topos, 82-100.

In a preceding paper, the Authors have constructed an embeggin of the category of open sets of Banach spaces and holomorphic functions into the topos which is an analytical model of SDG. This embedding preserves finite products and is consistent with the differential calculus.

In this paper, they study in a general setting the internal topological structure inherited by an object from a topology on the set of its global sections. And they analyze the particular cases of an open set of a Banach space and of its exponential with an object of the site. In this last case, they introduce a topology which generalizes the canonical topology (considered in a preceding paper) on the set of morphisms with complex values of an analytic space. This topology is related to uniform convergence on compact subsets and the inductive limit topology on rings of germs.

D. HOLGATE, Completion and closure, 101-119 et 314.

The closure (or, in other terms, the density) has always played an important rôle in the theory of completions. Using ideas of Birkhoff, a closure is canonically extracted by a process of reflexive completion of a category. This closure characterizes the completeness and the completion itself. The closure has not only good internal properties, but also it is the largest among the closures which describe the completion.

The main theorem shows that the natural description of the closure/density of a completion is equivalent to the fact that the completion reflectors are exactly those reflectors which preserve embeddings. Such reflectors can be deduced from the closure itself. The rôle of the preservation of the closure and of the embeddings then gives a new light on other examples of completion.

PULTR & THOLEN, Localic enrichments of categories, 121-142.

A class of objects of a category C (which can be seen as the system of finite objects of C) naturally induces topologies of Hausdorff on the hom-sets (A,B). In this way, C becomes a Haus-category. Moreover there is a naturally associated Loc-category C* of which C is the spectrum; in C*, a frame C*(A,B) can be non-trivial even if C(A,B) is void.

L. SCHRÖDER, Isomorphisms and splitting of idempotents in semicategories, 143-154.

The paper shows that the categories freely generated by some systems of generators and relations, called semicategories, have no other isomorphisms than those explicitly specified by the given relations. And the condition that any idempotent splits in a category can be verified in a semicategory which generates the category.

P. BOOTH, Fibrations and classifying spaces: overview and the classical examples, 162-206.

Let G be a topological group. The construction and the properties of the Milnor principal bundle associated to G provide the main model for the development of theories of fibrations et their classifying spaces. In this paper, the Author develops such a theory for general structured fibrations. Particular cases include analog results for principal, Hurewicz and sectional fibrations.

Some preceding papers have obtained results which are as good as the Milnor construction in terms of simplicity and generality; others have done the same in terms of generality and potential for applications; still others in terms of simplicity and potential for applications. But the present results for structured fibrations and for the considered level of fibration are the first to succeed simultaneously in these three desirable features.

DUBUC & ZILBER, Inverse function theorems for Banach spaces in a topos, 207-224.

In a preceding paper, the Authors have constructed an embeffing of the category of open sets of Banach spaces and holomorphic functions in the topos which is an analytic model of SDG. This embedding preserves finite products and is consistent with the differential calculus.

In this paper, they study the arrows in the topos between open sets of Banach spaces. They prove that they can be considered as functions between these spaces, and that they become Goursat or G-holomorphic. Moreover they must be compatible, in an appropriate sense, with the congruences defined by ideals of the rings of germs which define the objects of the site. The continuity of the arrow in the topos with respect to the topology of Banach corresponds exactly to the condition that the function is holomorphic. However, this is not the case for the internal variables of exponential type. A stronger condition is given which determines a sub-object of the exponential strictly included in the one determined by the continuity condition, and which defines the correct internal notion on holomorphic maps.

These results are used to develop the infrastructure allowing to quantify on internal holomorphic variables in the topos and to prove an internal local inverse function theorem for Banach spaces.

A. PULTR & J. SICHLER, A Priestley view on spatialization of frames, 225-238.

The representation of "frames" by the duality of Priestley gives a simple spatiality criteria (in the sense to be isomorphic to a topology). This criteria allows to easily deduce the spatiality of G_d-absolute frames (Isbell), or of continuous ditributive lattices (Hofmann & Lawson, Banaschewski).

M. BARR, On *-autonomous categories of topological vector spaces, 243-254.

The Author proves that two (isomorphic) full sub-categories of the category of locally convex topological vector spaces form *-autonomous categories, namely the weakly topologized spaces and those equipped with the Mackey topology.

N. S. YANOFSKY, The Syntax of Coherence, 255-304.

This paper studies categorical coherence in the setting of a 2-dimensional generalization of the functorial semantics of Lawvere. It introduced the 2-theories which are a syntactic manner for describing categories with structure. With this approach, several results on coherence become simple assertions on the quasi-Yoneda Lemma and the morphisms of 2-theories. Given two 2-theories and a morphism between them, the Author analyzes the relation induced between the corresponding 2-categories of algebras. The strength of the induced quasi-adjoints is classified by the strength of the 2-theories morphisms. These quasi-adjoints reflect how one of the structures can be replaced by the other. A 2-dimensional analog of the Kronecker product is defined and constructed. This operation leads to the generation of new coherence laws from preceding ones.

MOENS & VITALE, Groupoids and the Brauer group, 305-313.

The Authors use bigroupoids to analyze the exact sequence connecting the group of Picard and the group of Brauer, and to give a K-theoretical description of the groups of Picard and of Brauer.

P. AKUESON, Géométrie de l'espace tangent sur l'hyperboloïde quantique, 2-50.

The author introduces the tangent space on a quantum hyperboloid. He defines an action of this tangent space on the corresponding "quantum function space" A, which converts the elements of the tangent space into "braided vector fields". The tangent space is shown to be a projective A-module and there is defined a quantum (pseudo)metric and a (partially defined) quantum connection on it.

HEBERT, ADAMEK & ROSICKÝ, More on orthogonality in locally presentable categories , 51-80.

This paper proposes a new solution to the problem of orthogonal sub-categories in locally presentable categories, different from the classical solution given by Gabriel and Ulmer. Several applications are given. In particular it allows to characterize the omega-orthogonal classes in locally finitely presentable categories, that is the full sub-categories of the form S ^ where the domains and codomains of the morphisms of S are finitely presentable. It allows also to find a sufficient condition for the reflexivity of sub-categories of accessible categories. Finally, a description of fraction categories in small finitely complete categories is given.

M. GOLASINSKI & D. LIMA GONÇALVES, Equivariant Gottlieb groups, 83-100.

The Authors study the diagram of Gottlieb groups G_n(X) et G_n(X), for n ³  1, where X is a space of diagrams and X an equivariant space respectively. They give several properties, extending those in the non-equivariant case. Then, using the universal G-fibration p¥ : E¥ ® B¥ , they obtain a relationship between G_n(F) and the connection homomorphisms determined by a G-fibration E ® B with fibre F.

M. GRANDIS, Higher fundamental functors for simplicial sets, 101-136.

This paper introduces a theory of combinatorial homotopy for the topos of symmetric simplicial sets (presheaves on positive infinite cardials), extending a theory already developed for simplicial complexes; the main interest of this extension is that the fundamental groupoid becomes the left adjoint of a symmetric nerve functor and preserves colimits, that is a strong van Kampen property. Analog results are obtained in any dimension less than infinite. The Author develops a notion of oriented homotopy pour ordinary simplicial sets, with a fundamental n-category functor left adjoint to the n-nerve. Similar constructions can be given in several categories of presheaves.

S. PICCARRETA, Rational nilpotent groups as subgroups of self-homotopy equivalences, 137-153.

Let X be a CW-complexe, E(X) the group whose elements are the homotopy classes of self-homotopy equivalences of X, and E_#(X) and E_*(X) its sub-groups whose elements induce respectively the identity for homotopy and homology. In this paper, the rational groups of nilpotence 1, of nilpotence 2 et of rank less than or equal to 6, whose commutator sub-group has a rank equal to 1, are realized as E_#(X) and E_*(X) where X is the rationalization of a finite CW-complexe.

W. RUMP, Almost abelian categories, 163-225.

The Author introduces and studies a class of additive categories with kernels and cokernels, which are more general than abelian categories and thus are called almost abelian. One of the aims of this work is to prove that this notion unifies and generalizes structures associated to abelian categories: torsion theories, adjoint functors and bimodules, Morita duality and tilting theory. Moreover it is proved that there are numerous almost abelian categories: in homological algebra, in functional analysis, in the theory of filtered modules and in the theory of representations of orders on Cohen-Macaulay rings of dimension less than or equal to 2.

P. AGERON, Esquisses inductives et presque inductives, 229-240.

The Author studies the sketches whose (projective) cones are all based on the empty diagram. He proves that the category of models of such a sketch has multilimits. This provides a canonical way to re-sketch it. As a special case, the category of models of a colimit sketch can always be re-sketched by some limit sketch. Specific examples are investigated further.

E. LOWEN-COLEBUNDERS, R. LOWEN & M. NAUWELAERTS, The cartesian closed hull of the category of approach spaces, 242-260.

This paper describes the smallest cartesian closed enlargement of the category of approach spaces AP, that is the cartesian closed hull of AP. It is constructed as a sub-category of the category of pseudo-approach spaces which two ot the authors had proved to be the topological quasi-topos hull of AP.

KLAUS, Cochain operations and higher cohomology operations, 261-284.

Extending a program initiated by Kristensen, this paper give an algebraic construction of unstable cohomology operations of higher order by simplicial cochain operations. Pyramids of cocycle operations are considered, which can be used for a second construction of cohomology operations of higher order.

M. SIOEN, Symmetric monoidal closed structures in PRAP, 285-316.

It is know that the category PRTOP (of pretopological spaces and continuous maps) is not cartesian closed, and thus the same holds for the category PRAP of pre-approach spaces and contractions, introduced by E. Lowen and R. Lowen. The aim of this paper is to prove that PRAP admits only one symmetric monoidal closed structure (up to a natural isomorphism), which is the canonical inductive monoidal structure studied (in the context of topological or initially structured categories) by Wischnewsky and Cincura. This result is proved thanks to a technique developed by J. Cincura to solve this problem in PRTOP.

4. Abstracts Volume XLIII (2002)

ALVAREZ, VILABOA, RODRIGUEZ & NOVOA, About the naturality of Beattie's Decomposition Theorem with respect to a change of Hopf algebras, 2-18.

Given a morphism between two finite commutative Hopf algebras G and H in a symmetric closed category C with projective basic object, the Authors construct an homomorphism of abelian groups between Gal_C(H) and Gal_C(G) (groups of isomorphism classes of Galois H-objects and G-objects, respectively). Its restriction gives a homomorphism between the groups of isomorphism classes of Galois H-objects and G-objects with a normal basis N_C(H) et N_C(G), thanks to two exact sequences relating these groups with G(H*) and G(G*). Finally, a commutative diagram is constructed which links the preceding morphisms to other seqences, such as a derivative of the Decomposition Theorem of Beattie.

J. PASEKA, A Note on nuclei of quantale modules, 19-34.

The aim of this paper is to give factorization theorems for Q-modules (quantale modules) similar to those known for locales. It is proved that any module nucleus associated to a module prenucleus is a meet of module nuclei of a special form.

A. FRÖLICHER, Linear spaces and involutive duality functors, 35-48.

Barr has shown that the category of locally convex spaces admits full sub-categories A with the following properties: A is complete and cocomplete; A admits bifunctors L and Ä satisfying the usual proprieties of a closed category, in particular

L(E, L(F, G)) @ L(E Ä F, G) et E Ä F @ F Ä E ;

moreover

D := L(-,R) : E |® E' := L(E,R)

is an involutive functor, that is D _° D @ Id_A. Hence any object E is reflective in the sense E @ E". This is remarkable since generally dim E = ¥ . Explicit descriptions and proofs are given. Finally an involutive duality functor is constructed for a category of projective geometries of any dimension.

V. LYUBASHENKO, Tensor product of categories of equivariant perverse sheaves, 49-80.

It is proved that the tensor product introduced by Deligne for the categories of equivariant constructible perverse sheaves is still a category of this type. More precisely, the product of the categories associated to a complex algebraic G-variety X and to a H-variety Y is the category associated to the G x H-variety X x Y - product of the constructible spaces.

J. ADAMEK, H. HERRLICH, J. ROSICKY & W. THOLEN, On a generalized small-object argument for the injective subcategory problem, 83-106.

The Authors prove a generalization of the Small Object Argument well-known in homotopy theory. It can be applied to each set of morphisms H not only in locally finitely presentable categories but also in the category of topological spaces. It says that the sub-category of H-injective objects is weakly reflective, and moreover that the weak reflections are H-cellular.

P. GAUCHER, About the globular homology of higher dimensional automata, 107-156.

This paper introduces a new simplicial nerve of parallel automata the augmented homology of which gives a new definition of globular homology. With this definition, the difficulties of the construction given in a former paper of the Author are suppressed. Moreover important morphisms which associate to each globe the corresponding branching and merging areas of execution paths become here morphisms of simplicial sets.

MOENS, BERNI-CANANI & BORCEUX, On regular presheaves and regular semi-categories, 163-190.

The Authors generalize the theory of regular modules on a ring without unit to the case of presheaves on a "category without unit" which they call a semi-category. They work in the context of enriched categories. The regularity axiom on a presheaf canonically corresponds to a colimit of representable presheaves and the semi-category itself is regular when its Hom functor satisfies this condition. The relation with Yoneda lemma is given, as well as an example of what F.W. Lawvere calls "the unity of opposites". Finally a Morita Theorem for regular semi-categories is obtained. Several examples are given related to the theory of matrices, of Hilbert-Schmidt operators and of -sets.

R. LECLERCQ, Symétries de Hecke à déterminant associé central, 191-212.

This paper gives an explicit construction of a family of quantum R-matrices of Hecke type which are not deformations of the volte and the associated determinant of which is central. This condition allows to associate to such a quantum R-matrix a braided category by the process suggested in a preceding paper of Gurevich, Leclercq and Saponov.

P. KAINEN, Isolated squares in hypercubes and robustness of commutativity, 213-220.

It is proved that, in a non-void collection of at most d-2 squares of a hypercube Q_d of dimension d there exists a 3-cube sub-graph of Q_d which contains exactly one of these squares. It follows that a diagram of isomorphisms on the scheme of the d-dimensional hypercube which has strictly less than d-1 non-commutative squares must actually be commutative. Statistical implications to verify the commutativity are deduced.

T. PIRASHVILI, On the PROP corresponding to bialgebras, 221-239.

A PROP A is a strict symmetric monoidal category with the following property: the objects of A are the natural numbers and the monoidal operation is the addition on the objects. An algebra on A is a strict monoidal functor from A toward the tensor category Vect of vector spaces on a commutative field k. The PROP QF(as) is constructed and it is proved that the algebras on it are exactly the bialgebras.

L. STRAMACCIA, Shape and strong shape equivalences, 242-256.

The concepts of shape and strong shape equivalences retain their own interest apart from Shape Tlieory itself. They can be defined in the abstract setting of a pair (C,K) of categories, where C is endowed with a generating cylinder functor. Related to their study is the problem of characterizing homotopy epimorphisms and monomorphisms in C. In order to do this, the Author makes use of the double mapping cylinder construction and introduces a strong homotopy extension property. Their connections with the previous concepts are studied.

B. TOEN, Vers une interprétation galoisienne de la théorie de l'homotopie, 257-312

Given any CW complex X, and x Î X, it is well known that p₁(X,x) @ Aut(w_x⁰), where w_x⁰ is the functor which associates to each locally constant sheaf on X its fibre at x. The purpose of the present work is to generalize this formula to higher homotopy. For this the Author introduces the 1-Segal category of locally constant (¥ -)stacks on X, and he proves that the H¥ -space of endomorphisms of its fibre functor at x is equivalent to the loop space W_xX.

Y.T. RHINEGHOST, The Boolean Prime Ideal Theorem holds iff maximal open filters exist, 313-315.

This paper proves that the following properties are equivalent in ZF Theory of Sets:

5. Abstracts Volume XLIV (2003)

B. JOHNSON & R. MCCARTHY, A classification of degree n functors, I, 2-38.

Using the theory of calculus for functors from pointed categories to abelian categories developed by the authors in a preceding paper, they prove in Part II that degree n functors can be classified in terms of modules over a particular DGA P_nxn(C). In Part I, they develop the calculus structures needed to prove this and related results. They also construct a filtration by rank for functors from pointed categories to abelian cat-egories and compare rank n functors with degree n functors.

M. MACKAAY, Note on the holonorny of connections in twisted bundles, 39-62.

Twisted vector bundles with connections have appeared in several places. In this note the Author considers twisted principal bundles with connections and studies their holonomy, which turns out to be most naturally formulated in terms of functors betwen categorical groups.

BULLEJOS, FARO & GARCÍA-MUÑOZ, Homotopy colimits and cohomology with local coefficients, 63-80.

The authors describe the structure of the generalized Eilenberg-Mac Lane simplicial sets as homotopy colimits and use this representation to provide an elementary proof of the fact that they represent singular cohomology with local coefficients

A. KOCK, The stack quotient of a groupoid, 85-104.

The Author describes a precise 2-dimensional sense in which the stack of principal G-bundles is a quotient of the groupoid G. The main tool for this is a reformulation of descent data (or coequalizing data) in terms of simplicial liftings of simplicial diagrams.

FIEDOROWICZ & VOGT, Simplicial n-fold monoidal categories model all loop spaces, 105-148.

In a preceding paper, the Authors proved that the classifying space of an n-fold monoidal category is equivalent to a C_n-space, where C_n is the little n-cubes operad. Here they show a partial converse: any C_n-space is up to weak equivalence the classifying space of a simplicial n-fold monoidal category. The main tool is a version of categorical coherence theory wliich translates directly to topological coherence theory and which is suited for extensions to higher order categories; this result has its independent interest.

KOLAR, A general point of view to nonholonomic jet bundles, 149-160.

A general r-th order jet functor on fibered manifolds is defined as a fiber product preserving subfunctor of the r-th nonholonomic prolongation containing the r-th holonomic one. The jet functors are characterized in terms of Weil algebras. Using this algebraic model, we classify all second order jet functors and deduce two geometric results for the higher order cases.

B. JOHNSON & R. MCCARTHY, A classification of degree n functors, II, 153-216.

Using the theory of calculus for functors from pointed categories to abelian categories developed in a preceding paper, the Authors prove that degree n functors can be classified in terms of modules over a particular DGA P_nxn(C). They further show that homogeneous degree n functors have natural classifications in terms of three different module categories. They use the structures developed for these classification theorems to show that ail degree n functors factor through a certain category P_nC, extending a result of Pirashvili. This paper depends on results established in Part I (above).

BARKHUDARYAN, EL BASHIR & TRNKOVÁ, Endofunctors of Set and cardinalities, 217-239.

The functors F: K ®  H which are naturally equivalent to every functor G: K ®  H for which FX is isomorphic to GX for all X are called DVO functors. The Authors discuss DVO functors in the category Set of all sets and mappings. A set-theoretical assumption (EUCE) (relatively consistent with (ZFC+GCH)) is introduced and, under (GCH+EUCE), the classes W of cardinal numbers which have the form W = {|X|; |FX| = |X|} for some F: Set ®  Set, are characterized. The presented results solve several problems raised by Rhineghost and by Zmrzlina.

BIOGRAPHIE de René LAVENDHOMME, 242-246.

GARZON & del RIO, Low-dimensional cohomology for categorical groups, 247-280.

In this article, the authors define the cohomology categorical groups Hⁱ(G,A), for i = 0,1, of a categorical group G, with coefficients in a braided categorical group (symmetric for i = 1) A equipped with a coherent left action of G These coefficients are called (symmetric) G-modules. They show that to any short exact sequence of symmetric G-modules one can associate a six-term exact sequence connecting H⁰ and H¹. Well-known cohomology groups in various contexts, as well as the exact sequences which connect them, prove to be projections of this general theory in the category of abelian groups, by considering the homotopy groups ₀ and ₁ of H¹.

M. GRANDIS, Directed homotopy Theory, I, 281-316.

Directed Algebraic Topology is emerging, from several applications. The basic structure that the author studies in this paper, called a directed space or d-space, is a topological space equipped with a suitable family of directed paths. Within this framework, directed homotopies, generally non reversible, are represented by cylinder and cocylinder functors. The existence of pastings provides a geometrical construction of the cubic sets as d-spaces, as well as the usual homotopical constructions. The autheor introduces the fundamental category of a d-space, computable with the help of a van Kampen-type Theorem; its homotopic invariance is brought back to the directed homotopy of categories.

It should be noted taht this study reveals new 'shapes' for d-spaces and for their elementary algebraic model, the small categories. Applications of these tools are suggested, in the case of objects which model a directed image, or a portion of space-time, or a concurrent system.

6. Abstracts Volume XLV (2004)

N. BALL & A. PULTR, Forbidden forests in Priestley spaces, 2-22.

The authors present a first order formula characterizing the distributive lattïces L whose Priestley spaces P(L) contain no copy of a finite forest T. For Heyting algebras L, prohibiting a finite poset T in P(L) is characterized by equations iff T is a. tree. They also give a condition characterizing the distributive lattices whose Priestley spaces contain no copy of a finite forest with a single additional point at the bottom.

J. KUBARSI & T. RYBICKI, Local and nice structures of the groupoid of an equivalence relation, 23-34.

A comparison between the concepts of local and nice structures of the groupoid of an equivalence relation is presented. It is shown that these concepts are closely related, and that generically they characterize the equivalence relations induced by regular foliations. The first concepl was introduced by J. Pradines (1966) and studied by R.Brown and O.Mucuk (1996), while the second one was given by the first author (1987). The importance of these concepts in the non transitive geometry is indicated.

DAWSON, PARE & PRONK, Free extensions of double categories, 35-80.

This paper is devoted to the study of double categories obtained by freely adding new cells or arrows to an existing double category. The authors specifically discuss the decidability of equality of cells in the new double category.

LAWSON & SIEINBERG, Ordered groupoids and etendues, 82-108.

Kock and Moerdijk proved that each étendue is generated by a site in which every morphism is monic. This paper provides an alternative characterisation of étendues in terms of ordered groupoids. Specifically, the authors define an Ehresmann site to be an ordered groupoid equipped with what they term an Ehresmann topology - this is essentially a family of order ideáls closed under conjugation - and in this way they are able to define the notion of a sheaf on an Ehresmann site. The main result is that each étendue is equivalent to the category of sheaves on a suitable Ehresmann site.

D-C. CISINSKI, Le localisateur fondamental minimal, 109-140.

In "Pursuing stacks", Grothendieck defines basic localizors as classes of weak equivalences in the category of small categories satisfying good descent properties (closely related to Quillen's Theorem A). For example, the usual weak equivalences (defined by the nerve functor) form a basic localizor W¥ . More generally, every cohomological theory on small categories defines canonically a basic localizor. In this paper, the author gives a proof of Grothendieck's conjecture that states that W¥ is the smallest basic localizor. Furthermore, we get a second characterization of W¥ involving Quiiien's Theorem B. This gives an elementary and axïomatic way to define the classical homotopy theory of CW-complexes.

CENTAZZO, ROSICKY & VITALE, A characterization of locally D-presentable categories, 141-146.

In a preceding paper, Adamak, Borceux Lack and Rosicky have generalized the locally finitely presentable categories to locally D-presentable categories, by replacing the filtered colimits by colimits which commute in Ens with limits indexed by an arbitrary doctrine D. In this paper, the locally D-presentable categories are characterized as the cocomplete categories with a strong generator consisting of D-presentable categories. This unifies results known for locally finitely presentable categories, varieties and categories of presheaves.

EBRAHIMI, TABATABAEE & MAHMOUDI, Metrizability of -frames, 147-156.

Imposing the necessary changes to the definition of a metric diameter on a frarne given by Banaschewski and Pultr, the authors get a notion of -frame, and hence the category MFrm of metric -frames with uniform -frame maps. Then they prove, among other things, the counterparts for -frames of the point-free metrization theorems proved by Barrasrhewski and Pultr. Finally, they characterize the category MFrm as the intersection of the categories of metric Lindelöf frames, and of regular -frarnes.

Y. KOPYLOV, Exact couples in a Raikov semi-abelian category, 162-178.

The author studies exact couples in semi-abelian categories of Raikov, a class of additive categories which includes many non-abelian categories in functional analysis and algebra. Using the approach of Eckmann and Hilton to the spectral sequence in an abelian category, he considers exact couples in a semi-abelian category and shows the possibility of derivation if the endomorphism of the exact couple is strict and, consequently, the existence of the spectral sequence of the couple if all its morphisms are strict. It is shown that it is also possible to derive a Rees system.

ADAMS & van der ZYPEN, Representable posets and their order components, 179-192.

A partially ordered set (poset) P is representable if there exists a distributive (0;1)-lattice in which the ordered set of prime ideals is isomorphic to P. In this paper, the authors prove that, if all the order components of P are representable, P is also representable. Moreover they prove that, though the interval topology of each component is compact, there exists a poset which is representable and which admits a non representable order component.

GRANDIS & PARE, Adjoint for double categories, 193-240.

The authors pursue their study of the general theory of weak double categories, addressing adjunctions and monads. A general 'double adjunction', which appears often in concrete constructions, has a colax double functor left adjoint to a lax one. This cannot be viewed as an adjunction in some bicategory, because lax and colax morphisms do not compose well and do not form one. However, such adjunctions live within an interesting double category, formed of weak double categories, with lax and colax double functors as horizontal and vertical arrows, and suitable double cells.

BUNGE, FUNK, JIBLADZE & STREICHER, Definable completeness, 243-266.

The authors identify a completeness condition for geometric morphisms that they call definable completeness. They express the condition in the fibrational language associated with a geometric morphism. They prove that a geometric morphism is definably complete if and only if the pure factor of its comprehensive factorization is a surjection.

A. FRÖLICHER, Axioms for convenient calculus, 267-286.

In order to generalize and improve the traditional differential calculus, one tried to replace norms by other structures (locally convex spaces, bornologic, of convergence,…). To avoid an arbitrary choice, the author takes any class of structured vector spaces and supposes given for any E and F in it a set S(E,F) of maps called "smooth maps". If S(E,F) satisfies 3 axioms (valid e.g. for the class of Banach spaces with Cw (E,F)), he shows that any E in  has a single structure of convenient vector space so that the "smooth" applications are exactly the smooth maps in the terminology of the convenient calculus of Frölicher and Kriegl; thus (,S) is a category equivalent to a full sub-category of the category Convw of convenient spaces with their smooth maps. Conversely, any full sub-category of Convw which includes the object R satisfies the 3 axioms. Several remarks on the convenient calculus are given.

KRUML & RESENDE, On quantales that classify C*-algebras, 287-296.

The functor Max of Mulvey associates to each C*-algebra A the unitary and involutive quantale MaxA of the closed linear sub-spaces of A. The aim of this article is to prove that this functor allows to classify all the unitary C*-algebras modulo a *-isomorphism. In particular it is proven that for each isomorphism u from MaxA to MaxB there exists a *-isomorphism û from A to B such that Max û(a) = u(a) for any element a of L(MaxA). But it is also proven that generally there exists isomorphisms from MaxA to MaxB which are not of the form Max v for some v from A to B.

D. van der ZYPEN, Order convergence and compactness, 297-300.

Let (P,≤) be a partially ordered set and let T be a compact topology on P which is finer than the interval topology. The author proves that T is then included in the order convergence topology.

GUO, SOBRAL & THOLEN, Descent equivalence, 301-315.

For a C-indexed category, A, an A-descent equivalence is a morphism of bundles in C which induces an equivalence between the A-descent categories of its domain and codomain. In this note, properties of such morphisms are studied, and those morphisms which are A-descent equivalences for all C-indexed categories A are fully characterized.

7. Abstracts Volume XLVI (2005)

BOURN & PENON, Catégorification de structures définies par monade cartésienne, 2-52.

The authors give a construction of the categorification of structures defined by cartesian monads. Contrary to the usual ways, they focus on the iteration process from level n to level n+l. Their starting point is a pair of a functor U : E ? B and a cartesian monad (M,h,µ) on E satisfying some conditions. From that point, they construct a new pair of a functor U1 : (El ? E) and a cartesian monad (M1,h1,µ1) on E1 which satisfy the same conditions. This new pair is defined as the categorification of the initial pair. Starting from E = Ens, B = 1 and (M,h,µ) the identity monad, the n-th step of this iteration process gives rise to the category of weak n-categories. The limit E¥ of this iteration admits a comparison functor towards the category of weak ¥-categories in the sense of Leinster.

M. GRANDIS, Equilogical spaces, homology and non-commutative Geoemetry, 53-80.

After introducing singular homology for D. Scott equilogical spaces, the author shows how these structures can express 'formal' quotients of topological spaces which do not exist as ordinary spaces and are related with well-known noncommutative C*-algebras. This study also uses a wider notion of local maps between equilogical spaces, which might be of interest for the general theory of the latter.

DUBUC & ZILBER, Weil prolongations of Banach manifolds in an analytic model of SDG, 83-98.

André Weil's theory of the "points proches" for real differential manifolds generalizes the fundamental notion of jet of Ehresmann and, as the jets, encompasses all the higher order differential calculus. This paper generalizes and develops this theory in the case of complex Banach manifolds. Given a Weil algebra W and an open set B of a Banach space, the analyticity and the infinite dimension impose some modifications in the definition of the prolongation B[W] of species W of B for it to have the suitable properties. For a holomorphic function f, an explicit formula in terms of higher order derivatives is given for the function f[W] induced between the prolongations of species W. A second part gives an analytic model of SDG with an embedding of the category of open subsets of a Banach space in it, and it is proved that the usual differential calculus in this category corresponds with the intrinsic differential calculus of the model.

I. STUBBE, Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories, 99-121.

The author studies presheaves on semicategories enriched in a quantaloid: it leads to the notion of a regular presheaf. A semicategory is regular if all its representable presheaves are regular, and then its regular presheaves form an essential (co)localisation of the category of all its presheaves. The concept of a regular semidistributor is used to obtain the Morita equivalence of the regular semicategories. The continuous orders and the omega-sets give examples.

M. LAWSON, Constructing ordered groupoids, 123-138.

The author proves that every ordered groupoid is isomorphic to one constructed from a category acting in a suitable fashion on a groupoid arising from an equivalence relation. This construction is used, in a subsequent paper, to analyse Dehornoy's structural or geometric monoid associated to a balanced variety.

BRIDGES, ISHIHARA, SCHUSTER & VÎTA, Products in the category of apartness spaces, 139-153.

The apartness structure on the product of two apartness spaces is defined, and the role of local decomposability in the theory is investigated. All the work is constructive - that is, uses intuitionistic, rather than classical, logic.

RESUMES DU COLLOQUE INTERNATIONAL "Charles Ehresmann : 100 ans", 163-239.

The issue XLVI-3 of the "Cahiers" is devoted to the publication of the abstracts of the lectures given to the International Conference "Charles Ehresmann : 100 ans", organized in Amiens (7-9 October 2005) to commemorate the centenary of Charles' birth (1905-1979). Creator of the "Cahiers" in 1958, Charles has remained their Director up to his death. His works are reprinted with comments in the 7 volumes of "Charles Ehresmann : Œuvres complètes et commentées" (Amiens, 1980-83).
The abstracts of the Conference are divided in three parts:
First, the abstracts of a general and/or historical nature given the first day.
Second, those of the session : "Categories, Topology, Geometry" (organizers Elisabeth Vaugelade and Francis Borceux) which consisted in a session of the SIC (Séminaire Itinérant des Catégories), joint to the 82th session of the PSSL (Peripatetic Seminar on Sheaves and Logic).
Third, abstracts of the session "Multidisciplinary Applications" consisting in the Symposium ECHO V (Emergence, Complexity, Hierarchy, Organization; organizers George Farre, Andrée Ehresmann and Jean-Paul Vanbremeersch who has already organized ECHO I in Amiens in 1996).
Articles developing these abstracts are posted on the internet site dedicated to Charles Ehresmann:
http://perso.wanadoo.fr/vbm-ehr/ChEh

W.D. GARRAWAY, Sheaves for an involutive quantaloid, 243-274.

This paper studies Q-valued sets where Q is an involutive quantaloid. The category of presheaves for Q is defined as functors with values in sets, and from them sheaves are obtained by the unique amalgamation property for compatible families. Then it is proved that the category of Q-valued sets is equivalent to the category of sheaves if Q is pseudo-rightsided.

GIULI & SLAPAL, Raster convergence with respect to a closure operator, 275-300.

The authors introduce and study the concept of convergence on a concrete category K with respect to a closure operator c on K. First the neighbourhoods of sub-objects of a K-object are defined and analyzed. Then these neighbourhoods are used to introduce the convergence with the help of some generalized filters. Some basic properties are examined and the notions of separation and compactness are thoroughly studied. It is proved that the separation and compactness induced by the convergence have similar properties to those for topological spaces, and are more appropriate than the usual c-separation and c-compactness.

B. BANASCHEWSKI, Projective frames: a general view, 301-312.

This paper deals with projectivity in the category Frm of frames relative to onto homomorphisms whose right adjoint be-longs to a suitable subcategory K of the category of meet semilattices with unit. Applications to several familiar K then yield various known results which are thus brought under a natural unified scheme.

Abstracts Volume XLVII (2006)

KOCK & REYES, Distributions and heat equation in SDG, 2-28.
This article gives a synthetic theory of distributions (which are not necessarily of compact support). This theory is compared with the classical theory of Schwartz. This comparison is made by a full embedding of the category of Convenient Vector Spaces (and their smooth maps) in some large topos, models of the synthetic differential geometry.

R. ATTAL, Combinatorial stacks and the four-color Theorem, 29-49.
The author interprets the number of good four-colourings of the faces of a trivalent, spherical polyhedron as the 2-holonomy of the 2-connection of a fibered category, p, modeled on Repf(sl2) and defined over the dual triangulation T. He also builds an sl2-bundle with connection over T, that is a global, equivariant section of p, and he proves that the four-colour Theorem is equivalent to the fact that the connection of this sl2-bundle vanishes nowhere. This geometric interpretation shows the cohomological nature of the four-colour nroblern.

S. NIEFIELD, Homotopic pullbacks, lax pullbacks and exponentiability, 50-80.
This article proposes a unified approach of homotopic pullbacks and other generalized lax pullbacks, and it studies the corresponding notion of exponentiability.

DUBUC & STREET, A construction of 2-filtered bicolimits of categories, 83-106.
The authors define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if and only if it iis a filtered category, and the construction yields a category equivalent to the category resulting from the usual construction of filtered colimits of categories. Weaker axioms suffice, and the corresponding notion is called a pre 2-filtered 2-category. The full set of axioms is necessary to prove that 2-filtered bicolimits have the properties corresponding to the essential properties of filtered bicolimits.

M. GRANDIS, 107-128.
Directed Algebraic Topology is a recent field, where a 'directed space' X, e.g. an ordered topological space, hasdirected paths (which are generally not reversibla) and a fundamental category, replacing the funcdamental groupoid of the classical case. In dimension 2, the directed singular 2-cubes of X naturally produce a fundamental lax 2-category. This is a generalization of a bicategory, where the comparison cells are not assumed to be invertible, and some choice for their direction is needed. Our geometric guideline gives a choice which is different from the ones previously considered.

H. NISHIMURA, Synthetic Differential Geometry of higher-order differentials, 129-154 and 207-232.
Given microlinear spaces M and N, with x in M and y in N, the author has studied, in a preceding paper, a type of mappings from the totality TDnx of Dn-microcubes on M in x to TDny, which were called n-th order preconnections there and are called Dn-tangentials in this paper; they give a without-germ generalization of the total differentials of order n. This paper, after a deeper study of this notion, proposes another type of mappings from TDnx to TDny, called Dn-tangentials, which give another generalization. The relation between Dn-tangentials and Dn-tangentials is studied, firstly in case coordinates are not available (i.e., M and N are general microlinear spaces), and secondly in case coordinates are available (i.e., M and N are formal manifolds). In the first case, there exists a natural mapping from Dn-tangentials into Dn-tangentials, and in the second, this map is bijective. The ideas are presented in the frame of Synthetic Differential Geometry, but are readily applicable to smooth manifolds as differential spaces and suitable infinite-dimensional manifolds. The paper is to be looked upon as a microlinear generalization of Kock's considerations on Taylor series calculus.

JIBLADZE & PIRASHVILI, Quillen cohomology and Baues-Wirsching cohomology of algebraic structures, 163-205.
Algebraic theories can themselves be considered as a kind of algebraic structures, so that it is possible to examine their cohomology in the sense of Quillen. In this paper, it is shown that the Quillen cohomology of an algebraic theory is isomorphic to its Baues-Wirsching cohomology.

C. TOWNSEND, A categorical proof of the equivalence of local compactness and exponentiability in locale theory, 233-239.
A well known result in locale theory shows that a locale is locally compact if and only if it is exponentiable. A recent result of Vickers and Townsend represents dcpo homomorphism between the opens of locales in terms of natural transformations. Here we use this representation theorem to give a categorical proof that a locale is locally compact if and only if it is exponentiable.

A. BARKHUDARYAN, V. KOUBEK & V. TRNKOVA, Structural properties of endofunctors, 242-260.
A functor F from K to L is a DVO-functor if it is naturally equivalent to any functor G from K to L such that, for each K-object X, FX is isomorphic to GX. It is proved that each DVO-functor F from SET to SET is finitary, i.e. preserves directed colimits.

R. GARNER, Double clubs, 261-317.
The author develops a theory of the double clubs which extends Kelly's theory fo clubs to the pseudo-double categories of Paré and Grandis. He proves that the club for the strict symmetric monoidal categories on Cat extends into a double club on the pseudo-double category dCat of categories, functors, profunctors and transformations.

Abstracts Volume XLVIII (2007)

N. GANTER, Smash products of E(1)-local spectra at an odd prime, 3-54.
Let (M,?) be a stable monoidal category. The author analyzes the interaction of the monoidal structure with the structure maps of the system of triangulated diagram categories Ho(MC) defined by Franke in 1996. As an application, it is proved that an equivalence of categories defined by Franke maps the smash-product of E(1)-local spectra to a derived tensor product of cochain complexes.

GOLASINSKI, GONÇALVES & WONG, Equivariant evaluation subgroups and Rhodes groups, 55-69.
In this paper, the authors define equivariant evaluation sub-groups of superior Rhodes groups, and they study their relations to the Gottlieb-Fox groups.

F. MYNARD, Unified characterization of exponential objects in TOP, PRTOP and PARATOP, 70-80.
A unified internal characterization of exponential objects in the categories of topological, pretopological and paratopological spaces (with continuous maps) is presented as an application of a theorem on product of D-compact filters.

G.E. REYES, Embedding manifolds with boundary in smooth toposes, 83-103.
Improving a preceding paper, the author constructs a fully faithful embedding of the category of manifolds with boundary in some "smooth" toposes, in particular the "Cahiers topos" and the topos of closed ideals of Moerdijk and Reyes. He proves that this embedding preserves the products of a manifold with boundary with a manifold without boundary and the open coverings. It also maps prolongations of manifolds by Weil algebras to exponentials with infinitesimal structures as exponents. The main tool is the operation "doubling" a manifold with boundary to get a manifold without boundary.

BOURN & RODELO, Cohomology without projectives, 104-153.
A Yoneda's Ext long exact cohomology sequence is obtained for additive categories which are not strictly abelian, without any projectives and even without an object 0. This allows us to also add, among many others, the categories of topological or Hausdorff abelian groups as natural environments where such a long exact sequence holds and, mainly, to provide a unified background for the known, but unexplained, classical parallelism in the treatment of the cohomology of groups and of the cohomology of Lie algebras.

GRANDIS & PARE, Lax Kan extensions for double categories, 163-199.
Right Kan extensions for weak double categories extend double limits other constructions, called vertical companions and vertical adjoints, studied previous papers. We prove that these particular cases are sufficient to construct pointwise unitary lax right Kan extensions, along those lax double functors satisfy a Conduché type condition. Double categories 'based on profunctors' complete, i.e. have all such constructions, while the double category of commutative
squares on a complete category is not, in general.

A. SOLOMON, A category model proof of the Cogluing Theorem, 200-219.
This paper presents a model category proof of the Cogluing Theorem that generalizes the proof given by Brown and Heath for the category of topological spaces and continuous maps. The aim of this paper is to give conditions under which a map between pullbacks is a weak equivalence in a model category.

COQUAND, LOMBARDI & SCHUSTER, The projective spectrum as a distributive lattice, 220-227.
The authors construct a distributive lattice whose prime filtres correspond to the homogeneous prime ideals of a graduate commuative ring. It gives a characteristic example of a non affine scheme in topology without points, and of general construction of gluing of distributive lattices. They prove a projective form of the "Théorème des zéros" de Hilbert.

I. KOLAR, On special types of nonholonomic jets, 228-237.
The author discusses the concept of special type of nonholonomic r-jets and nonholonomic (k, r)-velocities from a general point of view. Special attention is paid to the composition of nonholonomic r-jets of the same type. The product preserving cases are characterized in terms of Weil algebras.

COLEBUNDERS & GERLO, Firm reflections generated by complete metric spaces, 243-260.
The authors study concrete categories in which each object is a sub-space pf a product of "metrizable spaces". If the category is equipped with a closure operator s, the class Us of dense immersions is considere and the following two questions are investigated: (1) are the completely metrizable objects Us-injective? (2) is the class of all closed subspaces of products of completely metrizable objects firmly Us-reflective? It is shown that in this setting these questions are equivalent and conditions are given for a positive answer. The main theorem is applied to a large collection of examples.

R. GUITART, An anabelian definition of abelian cohomology, 261-269.

A.C. EHRESMANN, Sur Paulette Liebermann (1919-2007), 270-274.

A.C. EHRESMANN, Fiftieth anniversary of the "Cahiers", 275-316.