1. Abstracts Volume XL (1999)


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 > 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.

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 Ensop 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

CABD(X×Y,Z) @ CABD(X,CBD(Y,Z)) and MXD(X×BY,Z) @ MB(X,CBD(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.

2. Abstracts Volume XLI (2000)


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 GalC(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 Gd-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.


3. Abstracts Volume XLII (2001)



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 Gn(X) et Gn(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 : E¥ ® B¥ , they obtain a relationship between Gn(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 GalC(H) and GalC(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 NC(H) et NC(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 ;


D := L(-,R) : E |® E' := L(E,R)

is an involutive functor, that is D ° D @ IdA.

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 Qd of dimension d there exists a 3-cube sub-graph of Qd 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(wx0), where wx0 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 -space of endomorphisms of its fibre functor at x is equivalent to the loop space WxX.

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:

(a) Any non-trivial Boolean algebra has a prime ideal.
(b) Any non-void topological space has an open filter.

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 Pnxn(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 Cn-space, where Cn is the little n-cubes operad. Here they show a partial converse: any Cn-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 Pnxn(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 PnC, 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.


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 is the smallest basic localizor. Furthermore, we get a second characterization of 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:

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,

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.


8. 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.


9. 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.

The author proposes a general algebraic definition of homology and cohomology which completely divides the concept and the computations, and which is valid without any abelian hypothesis nor completeness. He proves that, in the case of abelian, complete and cocomplete computations, this homology gives back the usual abelian definition.

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

P.Libermann,who died in July, had been one of the first research students of Charles Ehresmann in Strasbourg, and she has developed and extended his work on differential geometry. Some indications are given on some of her results, in particular on G-structures, Lie pseudogroups and higher order geometry.

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

Brief story of the "Cahiers" since their creation by Charles Ehresmann in 1957, followed by the index of all the papers published up to now.


10. Abstracts Volume XLIX (2008)

KACHOUR K., Définition algébrique des cellules non strictes, 1-68.
The goal of this article is to propose precise definitions, for the higher and non-strict version, of the concepts of natural transformations and analogues (modifications, etc.), and all that within the framework of the Eilenberg-Moore algebras. Thus the author obtains objects with algebraic nature, and a good candidate for the graphic part of the non-strict 1-category of non-strict 1-categories. But to complete the formalism of the categorification is also a motivation of this work.

GOLASINSKI & STRAMACCIA, Weak homotopy equivalences of mapping spaces and Vogt's Lemma, 69-80.
In this paper, the authors give a characterization of shape and strong shape equivalences in the general setting of a tensored and cotensored Top-category C. The case of equivariant shape equivalences is also considered.

GUITART R., Toute théorie est algébrique et topologique, 83-128.
The author wants to explain how any mathematical theory is committed to a pulsation between algebra and topology. 1) His main aim is to show on one hand how every theory is an algebraic one, according to various notions of an algebraic theory, through sketches and up to figurative algebras, that is to say a question of equations between laws of composition of figures; and on the other hand how every theory is topological or toposical, that is to say an expression of facts on continuity and a geometrical organization of these facts (but this approach leads to some ambiguity). So he gets insights into a possible construction of an algebraic theory similar to the algebraic geometry of Grothendieck. On the way he proves two new facts which are essential for his analysis, one of a general nature, and the other which is a rather peculiar observation: 2.1) He stresses the role of figurative algebras, and as a by-product gets that every category of models is the full subcategory of the category of fractions of a category EnsEnsC generated by the representable objects. 2.2) He introduces axioms for an "end" operation B on sequences on [0, 1] and he proves that the continuity of a fonction f: [0, 1] ? [0, 1] is a strict algebraic fact, it is equivalent to the commutation B°fN = f°B, for B: [0, 1]N ? [0, 1] an everywhere defined end operation.

SOLOMON, A note on Kieboom's Pullback Theorem for cofibrations, 129-141.
The aim of this paper is to show that Kieboom's Pullback Theorem for Cofibrations has many important applications and generalizations of some well known classical results of homotopy theory. Kieboom has shown that Strøm's Pullback Theorem is a special case of his theorem and has given a number of applications of his theorem on locally equiconnected spaces. In this paper, the author presents more important applications of his theorem and even shows that a version of the main theorem of another Kieboom's paper is in fact a consequence of his Pullback Theorem thereby demonstrating that most of the well classical results of Strøm are in fact special cases of this Pullback Theorem

EHRESMANN A.C., Fifty years of research, 142-153.
The article contains the list of publications of the author, whose first papers have been published exactly 50 years ago. It is followed by a large diagram which indicates the main connections between the various notions she has introduced and/or studied, and points out some essential threads running through her work along the years.

BORCEUX, JUNG, ROSICKY & SOUSA, Foreword, 163-175.
Presentation of the 1st of 3 special issues of the "Cahiers" dedicated to J.Adámek on the occasion of his 60th birthday and in appreciation of his contribution to category theory and its applications to theoretical computer science. It is followed by the list of his publications.

BOURN D., The Mal'tsev operation on extensions, 176-195.
Given a pointed protomodular category C with split extension classifiers, the author gives an explicit description of the Mal'tsev operation associated to the simply transitive action on extensions determined by the theorem of Schreier-Mac Lane. It naturally appears as a pushout along an internal Mal'tsev operation.

GRAN & RODELO, A universal construction in Goursat categories, 196-208.
The authors prove that the category of internal groupoids Grd(E) is a reflective sub-category of the category RGph(E) of internal reflexive graphs in a regular Goursat category E with coequalizers. They deduce from this fact that the category Grd(E) is also a regular Goursat category.

SICHLER & TRNKOVÁ, On clones determined by their initial segments, 209-227.
The answers to the questions to know when the existence of isomorphisms of local clones implies the existence of an isomorphism of global clones are different for terminal clones, polynomial clones and centralizer clones of unitary universal algebras. In each of these 3 cases, the answer is strongly connected to the similarity type of the corresponding algebras.

THOLEN W., Injectivity versus exponentiability, 228-239.
At the morphisms level, the exponentiability implies the existence of some injective hulls. The author proves a converse statement, thus exhibiting a strong link between the concepts of injectivity and exponentiability.

BORCEUX & ROSICKY, On filtered weighted colimits of presheaves, 243-266.
For a topos C of presheaves which is locally finitely presentable in the enriched context, the authors prove that the notion of filtered weighted C-colimit reduces to the usual pointwise filtered colimits.

HERRLICH, KEREMEDIS & TACHTSIS, Tychonoff products of super second countable and super separable metric spaces, 267-279.
The authors prove that, in ZF, i.e. in Zermelo-Fraenkel set theory without the axiom of choice, the following conditions are equivalent : 1. CAC(R), i.e. the axiom of choice restricted to countable families of non-empty subsets of the reals. 2. The Tychonoff product of a finite number of metric spaces with super-enumerable bases has also a super-enumerable base Hence condition 2 is not a theorem of the ZF set theory. They prove also that the statement: "The Tychonoff product of two metric Cantor-complete super separable (resp. hereditarily separable) sets is super separable (resp. hereditarily separable)" is provable in ZF.

JANELIDZE G. & SOBRAL, Profinite relational structures, 280-288.
In this paper it is proved that a preordered topological space (on a Stone space) is profinite if, and only if, il is inter-clopen, i.e. if it can be presented as the intersection of closed-and-open preorders on the same space. In particular, it gives a new characterization of the spaces which are called Priestley spaces. Then this result is extended from preordered spaces to the models of a first order language verifying some condition. The paper also gives a stronger condition which has a clear model-theoretic meaning.

KOUBEK & SICHLER, On synchronized relatively full embeddings and Q-universality, 289-306.
M. E. Adams et W. Dziobiak have proved that each ff-algebraic universal quasi-variety of algebraic systems of finite signature is Q-universal. This paper introduces the notion of a synchronized relatively full embedding which is used to modify their result for algebraic quasi-varieties.

A.H. ROQUE, Protomodular quasivarieties of universal algebras, 307-314.
Protomodular varieties of universal algebras have been syntactically characterized by Bourn and Janelidze. In this paper, the author proves that the same characterization is valid for quasi-varieties of universal algebras and he obtains a sufficient condition for the full sub-categories of a category of structures closed under subobjects and products to be protomodular.

11. Abstracts Volume L (2009)

KARAZERIS P. & VELEBIL J., Representability relative to a doctrine, 3-22.
The authors propose the notion of doctrine to give a uniform environment to study weak representability notions. Since (co)limits are representability notions, this allows defining and studying weakened (co)limit concepts. For example when the doctrine is that of free cocompletions under a given class of colimits, the existence of weakened limits in the given category is strongly linked to honest limits in its free completion. Similarly, some weak promonoidal structures on a category can be linked to real monoidal structures on a free cocompletion.

MILIUS S. & MOSS L.S., Equational properties of recursive program scheme solutions, 23-66.
In preceding papers, the authors have proposed a general thery of recursive program schemes and of their solutions. These works generalized older approaches using ordered spaces or metric spaces, thanks to a theory using the notions of final coalgebra, Elgot algebra and what is known about them. The theory proved the existence and unicity of very general uninterpreted recusrsive program schemes. They also gave a theory of the interpreted solutions. The theory is developed in this paper. It gives general principles used to prove that two such recursive scheme programs have identical or linked uninterpreted solutions, or have identical or linked solutions well connected to the interpretation.

SOUSA L., On boundedness and small-orthogonality classes, 67-79.
The author gives a characterization of locally bounded categories and a criterion to identify the ?-orthogonal sub-categories in these categories (for a given regular cardinal ?).

CHENG E. & MAKKAI M., A note on the Penon definition of n-category, 83-101.
The authors show that doubly degenerate Penon tricategories give symmetric rather than braided monoïdal categories. They prove that Penon tricategories cannot give all tricategories, but show how to modify the definitions lightly in order to rectify the situation. They give the modified definition, using non-reflexive rather than reflexive globular sets, and show that the problem with doubly degenerate tricategories does not arise.

GRANDIS M., The role of symmetries in cubical sets and cublical categories (On weak cubical categories, I), 108-143.
Symmetric weak cubical categories were introduced by the author as a basis for the study of cubical cospans and higher cobordism. Such cubical structures are equipped with an action of the symmetric groups, which simplifies the coherence conditions. He gives now a deeper study of the role of symmetries. While ordinary cubical sets have a tensor product which is non symmetric and biclosed, the symmetric ones have a symmetric monoidal closed structure (and one internal hom). Similar facts hold for cubical categories and the symmetric ones, and should play a relevant role in the sequel, the study of cubical limits and adjunctions. Weak double categories are a cubical truncation of the present structures.

GUITART R., Klein's group as a Borromean object, 144-155.
Initially inspired by the case of the standard borromean link, the author introduces the notion of a borromean object in a category. He provides ex-amples in groups, Boolean algebras, semi-rings, rings, fields. But the notion is introduced mainly for the case of the famous Klein's group G168 = GL3(F2), that he describes as a borromean object in groups.

GRAN M. & JANELIDZE G., Covering morphisms and normal extensions in Galois structures associated with torsion theories, 171-188.
The authors study covering morphisms and normal extensions with respect to Galois structures equipped with what they call test functors. These test functors naturally occur in Galois structures associated with torsion theories in homological categories. Under suitable additional conditions, every morphism with a torsion free kernel is a covering, and every covering is a normal extension whenever it is an effective descent morphism. Their counterexamples showing the relevance of those additional conditions are semi-abelian, and moreover, group-theoretic, involving semi-direct products of cyclic groups. They also briefly compare their new results with what is known for the so-called locally semi-simple coverings and for generalized central extensions.

BÖRGER R. & KEMPER R., Infinitary linear combinations in reduced cotorsion modules, 189-210.
The authors investigate sets within unitary linear combinations subject to the usual axioms with coefficients in a suitable ring, e.g. a complete valuation ring. They are Eilenberg-Moore algebras for a monad of countable arity. Moreover, they are always modules; surprisingly infinitary linear combinations yield a property. This is quite different from the real or the complex case studied by Pumplün and Röhrl. These modules were called cotorsion modules and defined by a cohomological property by Matlis. They form a reflective sub-category; the reflection also has a cohomological description. This yields some insight, particularly if the first Ulm functor does not vanish.

BOURN D. & JANELIDZE G., Centralizers in action accessible categories, 211-232.
This article introduces and studies action accessible categories. They provide a wide class of protomodular categories, including all varieties of groups, rings, associative and Lie algebras, in which it is possible to calculate centralizers of equivalence relations and subobjects. It is shown that, in those categories, the equivalence relation and subobject commutators agree with each other.

ROSICKY J., Are all cofibrantly generated model categories combinatorial? 233-238.
G.Raptis has recently proved that, assuming Vop?nka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial one. His result remains true for a slightly more general concept of a cofibrantly generated model category. The author shows that Vopenka's principle is equivalent to this claim. The set-theoretical status of the original Raptis' result is open.

GRANDIS M., Limits in weak cubical categories
(On weak cubical categories, II), 242-272.
A weak symmetric cubical category is equipped with an action of the symmetric groups. This action, besides simplifying the coherence conditions, yields a symmetric monoidal closed structure and one path functor - a crucial fact for defining cubical transformations. Here the author deals with symmetric cubical limits, showing that they can be constructed from symmetric cubical products, equalisers and tabulators. Weak double categories are a cubical truncation of the present structures, so that double limits can be compared with the cubical ones.

ADAMEK J. & HEBERT M., Quasi-equations in locally presentable categories, 273-297.
Following the tradition of Hatcher and Banaschewski-Herrlich, the authors introduce quasi-equations as parallel pairs of unitary morphisms. An object satisfies the quasi-equation if its contravariant hom-functor merges the parallel pair. The sub-categories of a locally finitely presentable category which can be presented by quasi-equations are precisely those closed under products, subobjects and filtered colimits. They characterize the corresponding theory morphisms in the style of Makkai and Pitts as precisely the strong quotient morphisms. These results can be seen as ananalogue of the classical Birkhoff Theorem for locally finitely presentable categories. On the way, they show the rather surprising fact that in locally finitely presentable categories, every fnitary strong epimorphism is a composite of finitely many regular epimorphisms.

JANELIDZE Z., Closedness properties of internal relations, VI: Approximate operations, 298-319.
The method of translating universal algebraic term conditions into purely categorical conditions, which was presented in the first paper from this series, is now revisited with a new insight that is based on the idea of considering so called approximate operations, which is due to D. Bourn and the present author. In some sense, these approximate operations arise as categorical counterparts of terms of an algebraic theory of a variety.


12. Abstracts Volume LI (2010)

ADAMEK & SOUSA. On quasi-equations on locally presentable categories II: A logic, 3-28.
Quasi-equations, given by parallel pairs of finitary morphisms, represent properties of objects: an object satisfies the property if its contravariant hom-functor merges the parallel pair. Recently Adamek and Hébert characterized subcategories of locally finitely presentable categories specified by quasi-equations. The authors now present a logic of quasi-equations close to Birkhoff's classical equational logic. They prove that it is sound and complete in all locally finitely presentable categories with effective equivalence relations.

MCCURDY & STREET, What separable Frobenius monoidal functors preserve? 29-50.
Separable Frobenius monoidal functors were defined and studied under that name by Kornél Szlachányi and by Brian Day and Craig Pastro. They are a special case of the linearly distributive functors of Robin Cockett and Robert Seely. The purpose of the authors is to develop the theory of such functors in a very precise sense. They characterize geometrically which monoidal expressions are preserved by these functors (or rather, are stable under conjugation in an obvious sense). They show, by way of corollaries, that they preserve lax (meaning not necessarily invertible) Yang-Baxter operators, weak Yang-Baxter operators (in the sense of Alvarez, Vilaboa and González Rodríguez), and (in the braided case) weak bimonoids in the sense of Pasro and Street. Actually, every weak Yang-Baxter operator is the image of a genuine Yang-Baxter operator under a separable Frobenius monoidal functor. Prebimonoidl functors are also defined and discussed.

I. STUBBE, "Hausdorff distance" via conical cocompletion, 51-76.
In the context of quantaloid-enriched categories, the author explains how each saturated class of weights defines, and is defined by, an essentially unique full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines which arise as full sub-KZ-doctrines of the free cocompletion, are characterised by two simple "fully faithfulness" conditions. Conical weights form a saturated class, and the corresponding KZ-doctrine is precisely (the generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of Akhvlediani et al (2009).

E. VITALE, Bipullbacks and calculus of fractions, 83-113.
In this article it is proved that the class of weak equivalences between internal groupoids in a regular protomodular category is a bipullback congruence and, therefore, has a right calculus of fractions. As an application, it is shown that monoidal functors between internal groupoids in groups and homomorphisms of strict Lie 2-algebras are fractions of internal functors with respect to weak equivalences.

JANELIDZE, MARKI, THOLEN & URSINI, Ideal determined categories, 115-125.
The authors clarify the role of Hofmann's Axiom in the old-style definition of a semi-abelian category. By removing this axiom they obtain the categorical counterpart of the notion of an ideal determined variety of universal algebras - which they therefore call an ideal determined category. Using known counter-examples from universal algebra they conclude that there are ideal determined categories which fail to be Mal'tsev. They also show that there are ideal determined Mal'tsev categories which fail to be semi-abelian.

M.M. CLEMENTINO & GUTIERRES, On regular and homological closure operators, 127-142.
Observing that weak heredity of regular closure operators in Top and of homological closure operators in homological categories identifies torsion theories, the authors study these closure operators in parallel, showing that regular closure operators play the same role in topology as homological closure operators do algebraically.

EVERAERT & VAN DER LINDEN, A note on double central extensions in exact Mal'tsev categories, 143-153.
The characterisation of double central extensions in terms of commutators due to Janelidze (in the case of groups) and Gran and Rossi (in the case of Mal'tsev varieties) is shown to be still valid in the context of exact Mal'tsev categories with coequalisers.

J.R.A. GRAY, Representability of the strict extension functor for categories of generalized Lie algebras, 162-181.
For an additive symmetric closed monoidal category C with equalizers, suppose M is a monoid defined with respect to the monoidal structure. In this setting we can define a Lie algebra with respect to M and the monoidal structure. For the category Lie(M; C) of Lie algebras the author shows that the functor SplExt( -, X) from Lie(M, C) to Set is representable by constructing a representation.

E. BURRONI & PENON, Representation of metric jets, 182-204.
Guided by the heuristic example of the tangential Tfa of a map f differentiable at a which can be canonically represented by the unique continuous affine map it contains, the authors extend this property of representation of a metric jet, into a specific metric context. This yields a lot of relevant examples of such representations.

CHENG & MAKKAI, A Note on the Penon definition of n-category, 205-223.
The authors show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. They prove that Penon tricategories cannot give all tricategories, but they show how to modify the definition slightly in order to rectify the situation. They give the modified definition, using non-reflexive rather than reflexive globular sets, and show that the problem with doubly degenerate tricategories does not arise.

A. KOCK, Abstract projective lines, 224-240.
The article describes a notion of projective line (over a fixed field k): a groupoid with a certain structure. A morphism of projective lines is then a functor preserving the structure. The author proves a structure theorem, namely: such projective lines are isomorphic to the coordinate projective line (= set of 1-dimensional subspaces of k2).

W. RUMP, Objective categories and schemes, 243-271.
Quasi-coherent sheaves over a scheme are regarded as modules over an objective category. The category Obj of objective categories is shown to be dual to the category of schemes. The author exhibits Obj as a reflective full subcategory of a category POb (pre-objective categories) whose objects are contravariant functors from a poset to the category of commutative rings while the morphisms of POb take care of the structure responsible for the generation of schemes. In this context, morphisms of schemes just turn into functors between objective categories preserving the relevant structure. The main result gives a more explicit version of Rosenberg's reconstruction of schemes (1998).

EBRAHIMI, MAHMOUDI & RASOULI, Characterizing pomonoids S by complete S-posets, 272-281.
A poset with an action of a pomonoid S on it is called an S-poset. There are two different notions of completeness for an S-poset. One just as a poset, and the other as a poset taking also account of the actions which are distributive over the joins. In this paper, comparing these two notions with each other, the authors find characterizations for some pomonoids.

S. DUGOWSON, On connectivity spaces, 282-315.
This paper presents some basic facts about connectivity spaces. In particular, it explains how to generate connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the tensor product of integral connectivity spaces; it defines homotopy for connectivity spaces (mentioning some related difficulties) and the smash product of pointed integral connectivity spaces, showing that this operation results in a closed monoidal category with such spaces as objects. Then, it studies finite connectivity spaces, associating a directed acyclic graph with each such space and then defining a new numerical invariant for links: the connectivity order. Finally, it mentions the not very well-known Brunn-Debrunner-Kanenobu Theorem which asserts that every finite integral connectivity space can be represented by a link


13. Abstracts Volume LII (2011)

KASANGIAN, METERE and VITALE, The Ziqqurath of exact sequences of n-groupoids, 2-44
In this work the authors study exactness in the sesquicategory of n-groupoids. Using homotopy pullbacks, they construct a six term sequence of (n-1)-groupoids from an n-functor between pointed n-groupoids. They show that the sequence is exact in a suitable sense, which generalizes the usual notions of exactness for groups and categorical groups. Moreover, iterating the process, they get a ziqqurath of exact sequences of increasing length and decreasing dimension. For n = 1; they recover a classical result due to R. Brown and, for n = 2; its generalizations due to Hardie, Kamps and Kieboom and to Duskin, Kieboom and Vitale.

M. GRANDIS, Singularities and regular paths (an elementary introduction to smooth homotopy), 45-76.
This article is a basic introduction to a particular approach within smooth algebraic topology: our aim is to study 'smooth spaces with singularities', by methods of homotopy theory adapted to this task. Here it explores euclidean regions by paths of (variable) class Ck, counting their stops. The fundamental groupoid of the space acquires thus a sequence of integral Ck-weights that depend on a smoothness index; a sequence that can distinguish 'linear' singularities and their order. These methods can be applied to the theory of networks.

HARTL & LOISEAU, A characterization of finite cocomplete homological and of semi-abelian categories, 77- 80.
Semi-abelian and finitely cocomplete homological categories are characterized in terms of four resp. three simple axioms, in terms of the basic categorical notions introduced in the first few chapters of MacLane's classical book.

CHENG & GURSKI, The periodic table of n-categories II: Degenerate tricategories, 82-125.
This article continues the project begun in a preceding article by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories it is exhibited a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case the authors explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between BrMonCat and Tricat. They indicate how to iterate the icon construction to produce an equivalence, but leave the details to a sequel. Finally they study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure MonBicat.

BARILE, BARONE & TULCZYJEW, The total exterior differential, 126-160.
The definition of mixed jets includes the finite sequences of vertical vectors tangent to jet bundles. This allows to define differential operators on vertical forms on jet bundles by using mixed jets prolongations. The total exterior differential is a special case.

G. SEAL, On the monadic nature of categories of ordered sets, 163-187.
If S is an ordered joint monad, that is, a monad on Set that factors through the category of ordered sets with left adjoint maps, then any monad morphism between S and T makes T order-adjoint. The Eilenberg-Moore category of T is then monadic over the category of monoids in the Kleisli category of S.

BROWN & STREET, Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product, 188-208.
The aim is the proof of the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoïds is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.

KENNEY & PARE, Categories as monoids in Span, Rel and Sup, 209-240.
The authors study the representation of small categories as monoids in three closely related monoidal bicategories. Categories can be expressed as special types of monoids in the category Span. In fact, these monoids also live in Rel. There is a well-known equivalence between Rel, and a full subcategory of the category Sup, of complete lattices and sup-preserving morphisms. This allows to represent categories as a special kind of monoid in Sup. Monoids in Sup are called quantales, and are of interest in a number of different areas. They also study the appropriate ways to express other categorical structures such as functors, natural transformations and profunctors in these categories.

DUBUC & YUHJTMAN, A construction of 2-cofiltered bilimits of topoi, 242-251.
The authors show the existence of bilimits of 2-cofiltered dia-grams of topoi, generalizing the construction of cofiltered bilimits developed before. For any given such diagram represented by any 2-cofiltered diagram of small sites with finite limits, we construct a small site for the bilimit topos (there is no loss of generality since we also prove that any such diagram can be so represented). This is done by taking the 2-filtered bicolimit of the underlying categories and inverse image functors. They use the construction of this bicol-imit developed precedently,where it is proved that if the categories in the diagram have finite limits and the transition functors are exact, then the bicolimit category has finite limits and the pseudocone functors are exact. An application of the result gives back the fact that every Galois topos has points.

BLUTE, COCKETT, PORTER & SEELY, Kähler categories, 253-268.
This paper establishes a relation between the notion of a codifferential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad T, which is furthermore an algebra modality, i.e .a natural assignment of an associative algebra structure to each object of the form T(C). Finally, a codifferential category comes equipped with a deriv-ing transformation satisfying typical differentiation axioms, expressed algebraically. The traditional notion of Kähler differentials defines the notion of a module of A-differential forms with respect to A, where A is a commutative k-algebra. This module is equipped with a universal A-derivation. A Kähler category is an additive monoidal category with an algebra modality and an object of differential forms associated to every object. Under the assumption that the free algebra monad exists and that the canonical map to T is epimorphic, codifferential categories areKähler.

C. KACHOUR, Operadic definition of non-strict cells, 269-231.
In a preceding paper the author has pursued Penon's work in higher dimensional categories by defining weak omega-functors, weak natural omega-transformations, and so on, all that with Penon's framework, i.e. the "étirements catégoriques", where he has used an extension of this object, namely the n-étirements catégoriques". Here he is pursuing Batanin's work in higher dimensional categories by defining weak omega-functors, natural transformations, and so on, using Batanin's framework by extending his universal contractible omega-operad K, by building the adapted globular colored contractible.


14. Abstracts Volume LIII (2012)

MALTSINIOTIS Georges, Carrés exacts homotopiques et dérivateurs, 3-63.
The aim of this paper is to generalize in a homotopical framework the notion of exact square introduced by René Guitart, and explain the relationship between this generalization and the theory of derivators.

MANES Ernie, Varieties generated by compact metric spaces, 64-80.
A nonprincipal ultrafilter r on omega chooses a convergent point for each sequence in a compact metric space. The class of algebras produced by this operation is a full subcategory of countably tight topological spaces and continuous maps which contains all compact metrizable spaces. The equations which determine this class are precisely those satisfied by the characteristic function of r.

M.L. del HOYO, On the homotopy type of a (co)fibred category, 82-114.
This paper describes two ways on which (co)fibred categories give rise to bisimplicial sets. The fibred nerve is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of the homotopy type of the total category. If the fibration is splitting, then one can construct the cleaved nerve, a smaller variant which emerges from a closed cleavage. Some classical theorems by Thomason and Quillen are interpreted in terms of these constructions, and the fibred and cleaved nerve are used to establish new results on homotopy and homology of small categories.

M. GRANDIS, A lax symmetric cubical category associated to a directed space, 115-160.
The recent domain of directed algebraic topology studies 'directed spaces', where paths and homotopies need not be reversible. The main applications are concerned with concurrency. The author introduces here, for a directed space, an infinite dimensional fundamental category, of a lax cubical type: the singular cubes of the space have a cubical structure, where concatenations are associative up to invertible reparametrisation while degeneracies are only lax-unital. Moreover, this structure is symmetric, by permuting the variables of singular cubes; this simplifies the coherence properties. By a similar construction, the 'Moore cubes' of the space give a strict symmetric cubical category.

K. WALDORF, Transgression to loop spaces and its inverse, I: Diffeological bundles and fusion maps, 162-210.
The author proves that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. The bijections are established by explicit group isomorphisms: transgression and regression. Restricted to smooth, finite-dimensional manifolds, these results extend previous work of J. W. Barrett.

BLUTE, EHRHARD & TASSON, A convenient differential category, 211-232.
It is shown in this paper that the category of convenient vector spaces in the sense of Frölicher and Kriegl is a differential category. Differential categories were introduced by Blute, Cockett and Seely as the categorical models of the differential linear logic of Ehrhard and Regnier. Indeed it is claimed that this category fully captures the intuition of this logic. It was already evident in the monograph of Frölicher and Kriegl that the category of convenient vector spaces has a remarkable structure. Here much of this structure is given a logical interpretation. For example, this category supports a comonad for which the co-Kleisli category is the category of smooth maps on convenient vector spaces.; it is shown that this comonad models the exponential modality of linear logic. It is also shown that the logical system suggests new structure, in particular, the existence of a codereliction map. Such a map allows for the differentiation of arbitrary maps by simple precomposition.

R. GUITART, Pierre Damphousse, mathématicien (1947-2012), 233-240.
Pierre Damphousse was an active member in the community of categoricians. The author surveys his mathematical itinerary, with mainly three subjects : purity in modules, cellular maps, fixob functors and the power-set functor on Ens.

FIORE, GAMBINO & J. KOCK, Double adjunctions and free monads, 242-307.
The authors characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg-Moore objects in double categories. They improve upon their earlier results to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category. They also prove that a double category admits Eilenberg-Moore objects if and only if a certain parameterized presheaf is representable. Along the way, they develop parameterized presheaves on double categories and prove a double-categorical Yoneda Lemma.

BRODOLONI & STRAMACCIA, Saturation for classes of morphisms, 308-315.
An externally saturated class S of morphisms in a category C is a class of morphisms that are inverted by some functor F from C to D. On the other hand, S is internally saturated if it coincides with its double orthogonal in the sense of Freyd-Kelly. In this short note the authors prove that S is an internally saturated class if and only if it is externally saturated and admits a calculus of left fractions.


15. Abstracts Volume LIV (2013)

E. CHENG, A direct proof that the category of 3-computads is not cartesian closed, 3-12.
The author proves by counterexample that the category of 3-computads is not cartesian closed, a result originally proved by Makkai and Zawadowski. She gives a 3-computad B and shows that the functor × B does not have a right adjoint, by giving a coequaliser that is not preserved by it.

R. GUITART, Trijunctions and triadic Galois connections, 13-27.
This paper introduces the notion of a trijunction, which is related to a triadic Galois connection just as an adjunction is to a Galois connection. It constructs the trifibered tripod associated to a trijunction, the trijunction between toposes of presheaves associated to a discrete trifibration, and the generation of any trijunction by a bi-adjoint functor. While some examples are related to triadic Galois connections, to ternary relations, others are associated to some symmetric tensors, to toposes and algebraic universes.

J. E. BERGNER, Derived Hall algebras for stable homotopy theories, 28-55
In this paper the author extends Toën's derived Hall algebra construction, in which he obtains unital associative algebras from certain stable model categories, to one in which such algebras are obtained from more general stable homotopy theories, in particular stable complete Segal spaces satisfying appropriate finiteness assumptions.

J.-Y. DEGOS, Linear groups and primitive polynomials over F_p, 56-74
Motivated by the case of Klein's group GL_3(F_2), the author introduces the new notions of n-cyclable groups and n-brunnian groups of type I and II. He then proves that the groups SL_n(F_p) and GL_n(F_p) enjoy a structure of n-brunnian groups of type I for p prime and n equal or more than 3. Finally he states two conjectures about primitive polynomials over Fp, and he gives some partial results on them.

C. KACHOUR, Correction to the paper "Operadic definition of the non-strict cells" (2011), 75-80.
This short note proposes a new notion of contractibility for colored omega-operad defined in the paper cited in the title. It also proposes an alternative direction to build the monad for free contractible colored omega-operads,

FOREWORD, 82-90.
Foreword to the special volume (consisting of Volumes LIV-2, 3 and 4) of the "Cahiers" dedicated to Professor René Guitart on the occasion of his 65th birthday. It is followed by the list of publications of René Guitart.

M. GRANDIS, Adjoints for symmetric cubical categories (on weak cubical categories, III), 91-136
Extending a previous article (with R. Paré) on adjoints for double categories, the author deals now with weak symmetric cubical categories (of infinite dimension). Also here, a general 'cubical adjunction' has a colax cubical functor left adjoint to a lax one. This cannot be viewed as an adjunction in some bicategory, because composing lax and colax morphisms destroys all comparisons. However, as in the case of double adjunctions, cubical adjunctions live in an interesting double category; this now consists of weak symmetric cubical categories, with lax and colax cubical functors as horizontal and vertical arrows, linked by suitable double cells.

S. DUGOWSON, Espaces connectifs : représentations, feuilletages, ordres et difféologies, 137-160
This article is a continuation of a former article of the author on connectivity spaces. After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case is now generalized to all connectivity spaces, and so to connective foliations. Finally, the author starts the study of some functorial relations between connectivity and diffeological spaces, and gives a characterization of diffeologisable connectivity spaces.

BOURN, MARTINS-FERREIRA & VAN DER LINDEN, A characterisation of the "Smith is Huq" condition in the pointed Mal'tsev setting, 163-183
The authors give a characterisation of the "SmithisHuq" condition for a pointed Mal'tsev category C by means of a property of the fibration of points from Pt(C) to C, namely: any change of base functor h* from Pt_Y(C) to Pt_X(C) reflects commuting of normal subobjects.

EVERAERT, GOEDECKE & VAN DER LINDEN, The fundamental group functor as a Kan extension, 184-210
The authors prove that the fundamental group functor from categorical Galois theory may be computed as a Kan extension.

J.-Y. DEGOS, Borroméanité du groupe pulsatif, 211-220
Section 1 defines the pulsative group Pul as the group of symetries of the integral sphere of radius V5, and gives a description of this group. Section 2 gives a borromean presentation of it, and proposes a link with the special orthogonal group. The focus of section 3 relates to different ways of representing this group graphically, using Newton's method. At last, an interpretation and a generalization are given.

ABBAD & VITALE, Faithful calculus of fractions, 221-239
The aim of this note is to develop a simple argument on bicategories of fractions showing that, if Sis the class of weak equivalences between groupoids internal to a regular category A with enough regular projective objects, then the description of Grpd(A)[S^-1] can be considerably simplified.

D. BOURN, Normality, commutation and suprema in the regular Mal'tsev and protomodular settings, 243-263
The author develops, in the contexts of regular Mal'tsev and protomodular categories, the consequences of a characterization, obtained in a mere unital category without any cocompleteness assumption, of the fact that the supremum of two commuting subobjects is their common codomain. In this way some well-known results in the category Gp of groups are recovered with conceptual proofs,

BARANOV & SOLOVIEV, Equality in Lambda Calculus. Weak universality in Category Theory and reversible computations, 264-291
In this paper the authors consider how the type of equality (extensional or intensional) in lambda calculus with inductive types and recursion influences universal constructions in certain categories based on this calculus (weak universality is connected with intensionale quality). Then they establish the link between weak universality and conditional reversibility in the theory of reversible computations.

J.-P. LAFFINEUR, Esquissabilité projective des espaces difféologiques, 292-297
The author explicitly describes a projective sketch of diffeological spaces.

A.C. EHRESMANN, Parcours de René Guitart, un mathématicien aux mul-tiples facettes, 298-316
Brief recall of René Guitart's works, extracted from the presentation done during the "Journée en l'honneur de René Guitart" (Paris, 2012) for his 65th birthday.


16. Abstracts Volume LV (2014)

DESCOTTE & DUBUC, A theory of 2-Pro-Objects, 3-35.
Grothendieck has developed the theory of pro-objects over a category C . The fundamental property of the category Pro(C) is that there is an embedding c from C to Pro(C), the category Pro(C) is closed under small cofiltered limits, and these limits are free in the sense that for any category E closed under small cofiltered limits, pre-composition with c determines an equivalence of categories between Cat(Pro(C); E) and Cat(C; E) (where the "+" indicates the full subcategory of the functors preserving cofiltered limits). This paper develops a 2-dimensional theory of pro-objects. Given a 2-category C, it defines the 2-category 2-Pro(C) whose objects are called 2-pro-objects. It is proved that 2-Pro(C) has all the expected basic properties adequately relativized to the 2-categorical setting, including the universal property corresponding to the one described above. The authors have at hand the results of Cat-enriched category theory, but their theory goes beyond the Cat-enriched cases since they consider the nonstrict notion of pseudo-limit, which is usually that of practical interest.

BEZHANISHVILI & HARDING, Stable compactifications of frames, 36-64.
In a classic paper, Smirnov characterized the poset of compactifications of a completely regular space in terms of the proximities on the space. Banaschewski formulated Smirnov's results in the point free setting, defining a compactification of a completely regular frame, and characterizing these in terms of the strong inclusions on the frame. Smyth generalized the concept of a compactification of a completely regular space to that of a stable compactification of a T0-space and described them in terms of quasi-proximities on the space. The author provides an alternate description of stable compactifications of T0-spaces as embeddings into stably compact spaces that are dense with respect to the patch topology, and relates such stable compactifications to ordered spaces. Each stable compactification of a T0-space induces a companion topology on the space, and he shows the companion topology induced by the largest stable compactification is the topology studied by Salbani. In the point free setting, he introduces a notion of a stable compactification of a frame that extends Smyth's stable compactification of a T0-space, and Banaschewski's compactification of a frame. He characterizes the poset of stable compactifications of a frame in terms of proximities on the frame, and in terms of stably compact subframes of its ideal frame. These results are then specialized to coherent compactifications of frames, and related to Smyth's spectral compactifications of a T0-space.

R. GUITART, Autocategories I. A common setting for knots and 2-categories, 65-80.
An autograph is an action of the free monoid with 2 generators; it could be drawn with no use of objects, by arrows drawn between arrows. As examples the author gets knot diagrams as well as 2-graphs. The notion of an autocategory is analoguous to the notion of a category, by replacing the underlying graph by an autograph. Examples are knots or links diagrams (unstratified case), categories, 2-categories, double categories (stratified case), which so live in the same context, the category of autocategories.

MARTINS-FERREIRA & VAN DER LINDEN, Categories vs. groupoids via generalised Mal'tsev properties, 83-112.
The authors study the difference between internal categories and internal groupoids in terms of generalised Mal'tsev properties - the weak Mal'tsev property on the one hand, and n-permutability on the other. In the first part of the article they give conditions on internal categorical structures which detect whether the surrounding category is naturally Mal'tsev, Mal'tsev or weakly Mal'tsev. They show that these do not depend on the existence of binary products. In the second part they focus on varieties of algebras.

M. MENNI, Sufficient cohesion over atomic toposes, 113-149.
Let (D;Jat) be an atomic site and j from Sh(D;Jat) to D^ be the associated sheaf topos. Any functor w from C to D induces a geometric morphism from C^ to D^ and, by pulling-back along j, a geometric morphism q to Sh(D;Jat). The author gives a sufficient condition on wfor q to satisfy the Nullstellensatz and Sufficient Cohesion in the sense of Axiomatic Cohesion. This is motivated by the examples where the opposite of D is a category of finite field extensions.

R. GUITART, Autocategories: II. Autographic algebras, 151-160.
An autograph is a set A with an action of the free monoid with 2 generators; it could be drawn as arrows between arrows. The category of autographs is a topos, and an autographic algebra will be the algebra of a monad on this topos. This paper compares autographic algebras with graphic algebras of Burroni, via graphic monoids of Lawvere. For that the author uses monadicity criteria of Lair and of Coppey. The point is that when it is possible to replace graphic algebras by autographic algebras, the situation with 2 types of arities is changed into a situation with only 1 type, the type "object" being avoided. So graphs, basic graphic algebras, autographs in a category of algebras of a Lawvere theory, elements of any 2-generated graphic topos, categories, autocategories, associative autographs, are auto-graphic algebras.

D. GARRAWAY, Q-Valued Sets and Relational-Sheaves, 161-204.
The authors show that a sheaf for a quantaloid is an idempotent suprema-preserving lax-semifunctor (a relational-sheaf). This implies that for a Grothendieck topos E a sheaf is a relational-sheaf on the category of relations of E and thus E is equivalent to the category of relational-sheaves and functional-transformations. The theory is developed in the context of enriched taxons, which are enriched semicategories with an added structural requirement.

ALLOUCH & Simpson, Classification des matrices associées aux catégories finies, 205-240.
In this paper, the authors find necessary and sufficient conditions for a positive square matrix to have at least one corresponding category. A corollary is that it suffices to verify this condition for every sub-matrix of order less or equal to 4.

EHRESMANN, GRAN & GUITART, New Editorial Board of the "Cahiers'', 242-243.
Announce of the enlargement of the Editorial Board of the "Cahiers", starting from January 2015.

LACK & STREET, On monads and warpings, 244-266.
The authors explain the sense in which a warping on a monoidal category is the same as a pseudomonad on the corresponding one-object bicategory, and they describe extensions of this to the setting of skew monoidal categories: these are a generalization of monoidal categories in which the associativity and unit maps are not required to be invertible. The analysis leads to describe a normalization process for skew monoidal categories, which produces a universal skew monoidal category for which the right unit map is invertible.

BERTRAM & SOUVAY, A general construction of Weil functors, 267-313.
The authors construct the Weil functor T^A corresponding to a general Weil algebra A; this is a functor from the category of manifolds over a general topological base field or ring K (of arbitrary characteristic) to the category of manifolds over A. This result simultaneously generalizes results known for ordinary, real manifolds, and results obtained for the case of the higher order tangent functors and for the case of jet rings. Some algebraic aspects of these general Weil functors are investigated ("K-theory of Weil functors", action of the "Galois group" Aut_K(A)), which will be of importance for subsequent applications to general differential geometry.



17. Abstracts Volume LVI (2015)


































A. KOCK, Duality for generic algebras, 2-14.
Duality theorems often assert that the canonical map d of an object into its double dual (possibly a "restricted" double dual) is an isomorphism. Both dualizations used in the formation of the double dual are with respect to some basic object R. An example is Gelfand duality. In this note, we prove that the generic algebra R for an algebraic theory serves as a basic object for such a duality theorem: the (suitably restricted) double dual of any representable object y(C), in the presheaf topos E in which R lives, is isomorphic, via d to y(C) itself. The proof goes via a "complete pairing" - a notion which we abstract from the proof. Among the immediate corollaries of this is that the generic commutative ring R is a model for synthetic differential geometry.

CHICHE, Théories homotopiques des 2-catégories, 15-75.
This text develops a homotopy theory of 2-categories analogous to Grothendieck's homotopy theory of categories in Pursuing Stacks. We define the notion of basic localizer of 2-Cat, a 2-categorical generalization of Grothen-dieck's notion of basic localizer, and we show that the homotopy theories of Cat and 2-Cat are equivalent in a remarkably strong sense: there is an isomorphism, compatible with localization, between the ordered classes of basic localizers of Cat and 2-Cat. It follows that weak homotopy equivalences in 2-Cat can be inter-nally characterized, without mentioning topological spaces or simplicial sets.

A. EHRESMANN, Parcours d'un topologue-catégoricien : Jean-Marc Cordier (1946-2014), 76-80.
This Note contains the List of Publications of Jean-Marc Cordier and an outline of his works (some in collaboration with Bourn or Porter), from Topology, to Shape Theory and Coherent Homotopy.

D. ARA, Structures de catégorie de modèles à la Thomason sur la catégorie des 2-catégories strictes, 83-108.
This paper is a complement to J. Chiche's paper in Volume LVI-1 (cf. above) which studies homotopy theories on 2-Cat, given by classes of weak equivalences called basic localizers of 2-Cat (a 2-categorical generalization of Grothendieck's notion). The author deduces, from Chiche's results and a result he has obtained with G. Maltsiniotis, that for essentially every basic localizer W of 2-Cat, there is a model category structure à la Thomason on 2-Cat whose weak equivalences are given by W; such structures model exactly combinatorial left Bousfield localization of the classical homotopy theory of simplicial sets.

S. A. SOLOVYOV. Localification procedure for affine systems, 109-.131.
Motivated by the concept of affine set of Y. Diers, this paper studies the notion of affine system, extending topological systems of S. Vickers. The category of affine sets is isomorphic to a full coreflective subcategory of the category of affine systems. We find the necessary and sufficient condition for the dual category of the variety of algebras, underlying affine sets, to be isomorphic to a full reflective subcategory of the category of affine sys-tems. As a consequence, we arrive at a restatement of the sobriety-spatiality equivalence for affine sets, patterned after the equivalence between the categories of sober topological spaces and spatial locales.

E. MEHDI-NEZHAD, Abstract annihilation graphs, 133-145.
We propose a new, widely generalized context for the study of the zero-divisor/annihilating-ideal graphs, where the vertices of graphs are not ele-ments/ideals of a commutative ring, but elements of an abstract ordered set (imitating the lattice of ideals), equipped with a binary opera-tion (imitating products of ideals). The intermediate level of congruences of any algebraic structure admitting a 'good' theory of commutators is also considered.

L. STRAMACCIA, The coherent category of inverse systems, 147-159.
For every model category C enriched over the category Gpd of groupoids a new category ProC is defined, with objects the inverse systems in C, which is isomorphic to the Steenrod homotopy category Ho(ProC) and to the coherent pro-homotopy category deûned by Lisica and Mardesic when C is the category of topological spaces.

BARR, KENNISON & RAPHAEL, On reflective and coreflective hulls, 163-208.
This paper explores the reflective hull (smallest full reflective subcategory) generated by a full subcategory of a complete category. We apply this to obtain the coreflective hull of the full subcategory of pointed cubes inside the category of pointed topological spaces. We also find the reflective hull of the category of metric spaces inside the category of uniform spaces as well as certain subcategories.

SHEN & THOLEN, Limits and colimits of quantaloid-enriched categories and their distributors, 209-231.
It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.

CARLETTI & GRANDIS, Generalised pushouts, connected colimits and codiscrete groupoids, 232-240.
This is a brief study of a particular kind of colimit, called a 'generalised pushout'. We prove that, in any category, generalised pushouts amount to connected colimits; in the finite case the latter are known to amount to ordinary pushouts and coequalisers (R. Paré, 1993). This study was motivated by remarking that every groupoid is, up to equivalence, a generalised pushout of codiscrete subgroupoids. For the fundamental groupoids of suitable spaces we get finer results concerning finite generalised pushouts.

GAUCHER, The geometry of cubical and regular transition systems, 242-300.
There exist cubical transition systems containing cubes having an arbitrarily large number of faces. A regular transition system is a cubical transition system such that each cube has the good number of faces. The categorical and homotopical results of regular transition systems are very similar to the ones of cubical ones. A complete combinatorial description of fibrant cubical and regular transition systems is given. One of the two appendices contains a general lemma of independent interest about the restriction of an adjunction to a full reflective subcategory.

COCKETT & CRUTTWELL. The Jacobi identity for tangent categories, 301-317.
A tangent category is a category equipped with an endofunctor with abstract properties modelling those of the tangent bundle functor on the category of smooth manifolds. Examples include many settings for differential geometry; for example, convenient manifolds, C1-rings, and models of synthetic differential geometry all give rise to tangent categories. Rosicky showed that in this abstract setting, one can define a Lie bracket operation for the resulting vector fields. He also provided a proof of the Jacobi identity for this bracket operation; however, his proof was unpublished, quite complex, and made additional assumptions on the tangent category. We provide a much shorter proof of the Jacobi identity in this setting that does not make any additional assumptions. Moreover, the techniques developed for the proof, namely the use of a graphical calculus, may be of use in proving other results for tangent categories.


18. Abstracts Volume LVII (2016)


CHRISTENSEN & ENXIN WU, Tangent spaces and tangent bundles for dif-feological spaces, 3-50.

The authors study how the notion of tangent space can be extended to diffeologi-cal spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. They focus on the internal tangent space, defined using smooth curves, and the external tangent space, defined using smooth derivations. They prove fundamental results about these tangent spaces, compute them in many examples, and observe that while they agree for many of the examples, they do not agree in general. Next, they recall Hector's definition of the tangent bundle, and show that both scalar multiplication and addition can fail to be smooth. Then they give an improved definition of the tangent bundle, using what they call the dvs diffeology, which fixes these deficiencies. They es-tablish basic facts about these tangent bundles and study the question of whether the fibres of tangent bundles are fine diffeological vector spaces. The examples include singular spaces, irrational tori, infinite-dimensional vector spaces and diffeological groups, and spaces of smooth maps between smooth manifolds.

HOSSEINI & QASEMI NEZHAD, Equalizers in Kleisli categories, 51-76.

This article gives necessary and sufficient conditions for a pair of morphisms in a Kleisli category, corresponding to a general monad, to have an equalizer. It also presents a better criterion for equalizers in a number of cases of interesting monads, and in all these cases it is explained what is an equalizer (if it exists) of a pair of morphisms.

T. JANELIDZE-GRAY, Calculus of E-Relations in incomplete relatively regu-lar categories, 83-102.

The author defines an incomplete relative regular category as a pair (C; E), where C is an arbitrary category and E is a class of regular epimorphisms in C satisfying certain conditions. She then develops what she calls a relative calculus of rela-tions in such categories; it applies to relations (R; r1; r2) from A to B in C having the morphisms r1 and r2 in E. This generalizes previous results, including the recent work with J.Goedecke on relative Goursat categories. She defines incomplete relative regular Goursat categories, and: (a) proves the incomplete relative versions of the equivalent conditions defining relative regular Goursat categories; (b) shows that in this setting the E-Goursat axiom is equivalent to the relative 3x3-Lemma.

GRANDIS & PARE, An introduction to multiple categories on weak and lax multiple categories, I) 103-159.

We introduce here infinite dimensional weak multiple categories, an extension of double and triplecategories. We also consider a partially lax, 'chiral' form with directed interchangers and a laxer form already studied in two previous papers for the 3-dimensional case, under the name of intercategory. In these settings we also begin a study of tabulators, the basic higher limits, that is concluded in the following:

GRANDIS & PARE, Limits in multiple categories (On weak and lax multiple categories, II) 163- 202.

Continuing the preceding paper,we study multiple limits in infinite-dimensional chiral multiple categories - aweak, partially lax form with directed interchangers. After defining multiple limits, we prove that all of them can be constructed from (multiple) products, equalisers and tabulators - all of them assumed to be re-spected by faces and degeneracies. Tabulators appear thus to be the basic higher limits, as was already the case for double categories. Intercategories, a laxer form of multiple category already studied in two previous papers, are also considered. In this more general setting the basic limits men-tioned above can still be defined, but a general theory of multiple limits is not developed here.

W. RUMP, The completion of a quantum B-algebra, 203-228.

It is shown that quantales are the injective objects in the category of quantum B-algebras, and that every quantum B-algebra has an injective envelope. By an ex-plicit construction, the injective envelope is revealed as a completion more gen-eral than the Dedekind-MacNeille completion. A recent result of Lambek et al., where residual structures unexpectedly arise, is explained in the light of quantum B-algebras, which gives another Instance for their ubiquity. Connections to promonoidal structures and multi-categories are indicated.

N. GILL, On a conjecture of Degos, 229-237.

This note givea a proof of a conjecture of Degos concerning groups generated by companion matrices in GLn(q).

BLUTE, LUCYSHYN-WRIGHT & O'NEILL Derivations in codifferential categories, 243-279.

We define and study a novel, general notion of derivation in the setting of the codifferential categories of Blute-Cockett-Seely, generalizing the notion of K-linear derivation from commutative algebra. Given a codifferential category (C;T;d), a T-derivation del from A to M on an algebra A of the monad T is defined as a morphism in C into an A-module M satisfying a form of the chain rule expressed in terms of the deriving transformation d. We show that such T-derivations corre-spond to T-homomorphisms from A to W(A;M) over A valued in an associated T-algebra. We establish the existence of universal T-derivations from A to OmegaTA valued in an associated A-module of Kähler-type differentials OmegaTA. Where as previous work of Blute-Cockett-Porter-Selly on Kähler categories employed a notion of derivation expressible without reference to the monad T, we show that the use of the above T-based notion of derivation resolves an open problem concerning Kähler categories, showing that Property K for codifferential categories is unnecessary. Along the way, we establish a succinct equivalent definition of codifferential categories in terms of a given monad morphism on the symmetric algebra monad S to T and a compatible transformation d satisfying the chain rule.

J. BOURKE, Note on the construction of globular weak omega-groupoids from types, topological spaces..., 281-294.

A short introduction to Grothendieck weak omega-groupoids is given. The aim is to give evidence that, in certain contexts, this simple language is a convenient one for constructing globular weak omega-groupoids. To this end, we give a short reworking of van den Berg and Garner's construction of a Batanin weak omega-groupoid from a type using the language of Grothendieck weak omega-groupoids. This con-struction also applies to topological spaces and Kan complexes.

W. THOLEN, A mathematical tribute to Reinhard Börger, 295-314.

A synopsis of the life and work of Reinhard Börger (1954-2014) is presented, withan emphasis on his early or unpublished works.