1.
Abstracts Volume XL (1999) 

M. BARR
& H. KLEISLI, Topological balls, 320 (and Correction in XLII1,
227228).
This
paper shows how the use of the "construction of Chu" can simplify the
rather complicated construction of the *autonomous category of reflexive
zz*complete set up by the first author in the original papers and lecture
notes on *autonomous categories (these Cahiers XVII, 1976, 335342;
and Springer Lecture Notes in Math. 752, 1979).
M.
DOUPOVEC & I. KOLAR, On the jets of fibered manifold morphisms,
2130.
The
(r,s,q)jet of a morphism of fibred manifolds f is determined
by the rjet of the map f, by the sjet of the restriction
of f to the fibre and by the qjet of the base map induced
by f, for s > r < q.
This paper shows that the (r,s,q)jets are the only homomorphic
images of finite dimension of germs of morphisms of fibred manifolds satisfying
two natural conditions.
J.
PENON, Approche polygraphique des ¥ catégories non
strictes, 3180.
The
Author gives a new definition of a nonstrict ¥ category, called Prolixe.
The prolixes are algebras for some monad built on the category of reflexive
¥ graphs. That monad can be defined by a universal property which
expresses the universal relaxation of the axioms of ¥ categories. On
the other hand, the building of this monad uses a 'polygraphical' material,
obtained by logical technics.
M.
GARCIA ROMÁN, M. MÁRQUEZ HERNÁNDEZ, P. JARA &
A. VERSCHOREN, Uniform filters, 82126.
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, 127140.
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 JibladzeJohnstone and of BungeFunk. 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, 141157.
In
this paper, two categories Qual^{+} et Qual^{}
are introduced, the dual of each of which is algebraic (up to a natural
equivalence) on the other, thus lifting the classical algebraicity of
Ens^{op} over Ens. Moreover Qual^{+}
is cartesian closed. The calulus of inverse (resp. direct) images is presented
as the data of Qual^{} (resp. Qual^{+})
and of a monad on this category lifting the "monad of Stone" on Ens.
The concept of duality between categories is extended in the duality between
monads, and in this way the calculus of direct and inverse images are
dual. A consequence is the algebraicity on Qual^{+}
(up to a natural equivalence) of the dual of the category Top
of topological spaces and of the dual of the category of sets equipped
with an equivalence relation.
M.
GRANDIS & R. PARE, Limits in double categories, 162220.
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 2category,
it gives back the construction of Street for indexed limits; if I
has only vertical arrows, it gives back the construction of BastianiEhresmann
of limits relative to double categories.
H.
HERRLICH & L. SCHRÖDER, Composing special epimorphisms and retractions,
221226.
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 subcategory
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,
227231.
In
a paper published in these "Cahiers" (Volume XXI3, 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, 233240
Using
the definition of an ideal of a poset introduced by Doctor, the Author
shows that more posets are representable as compactopen 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, 242260.
The
Authors obtain different types of exponential laws for fibrations in a
quasitopos C, such as
C_{ABD}(X×Y,Z)
@ C_{ABD}(X,C_{BD}(Y,Z))
and M_{XD}(X×_{B}Y,Z) @ M_{B}(X,C_{BD}(Y,Z)),
with
isomorphisms in C for spaces on different bases. They prove
that there exists an isomorphism in C between the space
of fiber preserving maps from q to r and the space of transversal
sections to q · _{1 }r . They discuss the examples
of convergence spaces, of sequential convergence spaces and of simplicial
spaces, as well as the related cases of quasitopological spaces and of
compactly generated spaces.
A.
MUTLU & T. PORTER, Free crossed resolutions from simplicial resolutions
with given CWbasis, 261282.
In
this paper, the Authors study the relationship between a CWbase 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, Multibimodels, 284296.
The
paper studies the equivalences between multireflective subcategories
of categories of covariant presheaves. An appropriate notion of multibimodel
allows to generalize the classical theorems of EilenbergWatts and of
Morita for module categories. The motivating example is given by the multipresentable
categories, i.e., the categories which are sketchable by limitcoproducts
sketches.
D.
BOURN, Baer sums and fibered aspects of Malc'ev operations, 297316.
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 Qmodule 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,
274 et 156.
Using
the tensor product of Graycategories, the Author defines the concept
of a 4dimensional tas, which generalizes the tas of a Graycategory,
and he deduces ideas for generalizing to higher dimensions. The first
result is that the 4dimensional teisi with a unique object and a unique
arrow are the semistrict braided monoidal 2categories. Combining this
with the idea that a sylleptic 2category should be a 5dimensional tas
with a unique object, a unique arrow and a unique 2arrow, 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
2category should be a 6dimensional tas with a unique object, a unique
arrow, a unique 2arrow and a unique 3arrow, 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 2categories.
ALVAREZ
& VILABOA, On Galois Hobjects and invertible H*modules, 7579.
For
a cocommutative Hopf algbra H in a symmetric closed category C,
the Authors obtain, as a generalization of a theorem of Childs, a homomorphism
between the group Gal_{C}(H) of isomorphism classes
of Hobjects 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 Hmodules of Azumaya with an internal action coincides
with the group of Brauer of triple Hmodules 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, 82100.
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, 101119 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, 121142.
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 homsets (A,B). In this way, C becomes a Hauscategory.
Moreover there is a naturally associated Loccategory C*
of which C is the spectrum; in C*, a frame C*(A,B)
can be nontrivial even if C(A,B) is void.
L.
SCHRÖDER, Isomorphisms and splitting of idempotents in semicategories,
143154.
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,
162206.
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,
207224.
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 Gholomorphic. 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 subobject 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,
225238.
The
representation of "frames" by the duality of Priestley gives a simple
spatiality criteria (in the sense to be isomorphic to a topology). This
criteria allows to easily deduce the spatiality of G_{d}absolute
frames (Isbell), or of continuous ditributive lattices (Hofmann &
Lawson, Banaschewski).
M.
BARR, On *autonomous categories of topological vector spaces, 243254.
The
Author proves that two (isomorphic) full subcategories 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, 255304.
This
paper studies categorical coherence in the setting of a 2dimensional
generalization of the functorial semantics of Lawvere. It introduced the
2theories which are a syntactic manner for describing categories with
structure. With this approach, several results on coherence become simple
assertions on the quasiYoneda Lemma and the morphisms of 2theories.
Given two 2theories and a morphism between them, the Author analyzes
the relation induced between the corresponding 2categories of algebras.
The strength of the induced quasiadjoints is classified by the strength
of the 2theories morphisms. These quasiadjoints reflect how one of the
structures can be replaced by the other. A 2dimensional 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, 305313.
The
Authors use bigroupoids to analyze the exact sequence connecting the group
of Picard and the group of Brauer, and to give a Ktheoretical 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,
250.
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
Amodule 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 , 5180.
This
paper proposes a new solution to the problem of orthogonal subcategories
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 omegaorthogonal classes in locally finitely
presentable categories, that is the full subcategories 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 subcategories
of accessible categories. Finally, a description of fraction categories
in small finitely complete categories is given.
M.
GOLASINSKI & D. LIMA GONÇALVES, Equivariant Gottlieb groups,
83100.
The
Authors study the diagram of Gottlieb groups G_{n}(X)
et G_{n}(X), for n ³ 1,
where X is a space of diagrams and X an equivariant space
respectively. They give several properties, extending those in the nonequivariant
case. Then, using the universal Gfibration p¥ : E¥ ®
B¥ , they obtain a relationship between G_{n}(F) and the
connection homomorphisms determined by a Gfibration E ® B with fibre
F.
M.
GRANDIS, Higher fundamental functors for simplicial sets, 101136.
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 ncategory functor
left adjoint to the nnerve. Similar constructions can be given
in several categories of presheaves.
S.
PICCARRETA, Rational nilpotent groups as subgroups of selfhomotopy equivalences,
137153.
Let
X be a CWcomplexe, E(X) the group whose elements are the
homotopy classes of selfhomotopy equivalences of X, and E_{#}(X)
and E_{*}(X) its subgroups 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 subgroup has a rank equal to 1, are realized
as E_{#}(X) and E_{*}(X) where
X is the rationalization of a finite CWcomplexe.
W.
RUMP, Almost abelian categories, 163225.
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 CohenMacaulay
rings of dimension less than or equal to 2.
P.
AGERON, Esquisses inductives et presque inductives, 229240.
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 resketch it. As a special
case, the category of models of a colimit sketch can always be resketched
by some limit sketch. Specific examples are investigated further.
E.
LOWENCOLEBUNDERS, R. LOWEN & M. NAUWELAERTS, The cartesian closed
hull of the category of approach spaces, 242260.
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 subcategory of the category of pseudoapproach
spaces which two ot the authors had proved to be the topological quasitopos
hull of AP.
KLAUS,
Cochain operations and higher cohomology operations, 261284.
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, 285316.
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 preapproach 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, 218.
Given
a morphism between two finite commutative Hopf algebras G and H
in a symmetric closed category C with projective basic object,
the Authors construct an homomorphism of abelian groups between Gal_{C}(H)
and Gal_{C}(G) (groups of isomorphism classes of
Galois Hobjects and Gobjects, respectively). Its restriction
gives a homomorphism between the groups of isomorphism classes of Galois
Hobjects and Gobjects with a normal basis N_{C}(H)
et N_{C}(G), thanks to two exact sequences relating
these groups with G(H*) and G(G*). Finally,
a commutative diagram is constructed which links the preceding morphisms
to other seqences, such as a derivative of the Decomposition Theorem of
Beattie.
J.
PASEKA, A Note on nuclei of quantale modules, 1934.
The
aim of this paper is to give factorization theorems for Qmodules (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, 3548.
Barr
has shown that the category of locally convex spaces admits full subcategories
A with the following properties: A is complete and cocomplete; A admits
bifunctors L and Ä satisfying the usual proprieties of a closed category,
in particular
L(E,
L(F, G)) @ L(E Ä F, G) et E Ä F @ F Ä E ;
moreover
D
:= L(,R) : E ® E' := L(E,R)
is
an involutive functor, that is D _{°} D @ Id_{A}.
Hence any object E is reflective in the sense E @ E". This is remarkable
since generally dim E = ¥ . Explicit descriptions and proofs are given.
Finally an involutive duality functor is constructed for a category of
projective geometries of any dimension.
V.
LYUBASHENKO, Tensor product of categories of equivariant perverse sheaves,
4980.
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 Gvariety X and to a Hvariety Y is the category associated
to the G x Hvariety X x Y  product of the constructible spaces.
J.
ADAMEK, H. HERRLICH, J. ROSICKY & W. THOLEN, On a generalized smallobject
argument for the injective subcategory problem, 83106.
The
Authors prove a generalization of the Small Object Argument wellknown
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 subcategory of Hinjective
objects is weakly reflective, and moreover that the weak reflections are
Hcellular.
P.
GAUCHER, About the globular homology of higher dimensional automata,
107156.
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,
BERNICANANI & BORCEUX, On regular presheaves and regular semicategories,
163190.
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 semicategory. They work in the context of enriched categories. The
regularity axiom on a presheaf canonically corresponds to a colimit of
representable presheaves and the semicategory 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 semicategories
is obtained. Several examples are given related to the theory of matrices,
of HilbertSchmidt operators and of sets.
R.
LECLERCQ, Symétries de Hecke à déterminant associé
central, 191212.
This
paper gives an explicit construction of a family of quantum Rmatrices
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 Rmatrix 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,
213220.
It
is proved that, in a nonvoid collection of at most d2 squares
of a hypercube Q_{d} of dimension d there exists
a 3cube subgraph of Q_{d} which contains exactly one
of these squares. It follows that a diagram of isomorphisms on the scheme
of the ddimensional hypercube which has strictly less than d1
noncommutative squares must actually be commutative. Statistical implications
to verify the commutativity are deduced.
T.
PIRASHVILI, On the PROP corresponding to bialgebras, 221239.
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, 242256.
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, 257312
Given
any CW complex X, and x Î X, it is well known
that p_{1}(X,x) @ Aut(w_{x}^{0}),
where w_{x}^{0} 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 1Segal category of locally constant
(¥ )stacks on X, and he proves that the H¥ space of endomorphisms
of its fibre functor at x is equivalent to the loop space W_{x}X.
Y.T.
RHINEGHOST, The Boolean Prime Ideal Theorem holds iff maximal open filters
exist, 313315.
This
paper proves that the following properties are equivalent in ZF Theory
of Sets:
(a)
Any nontrivial Boolean algebra has a prime ideal.
(b)
Any nonvoid topological space has an open filter.

5. Abstracts
Volume XLIV (2003) 

B.
JOHNSON & R. MCCARTHY, A classification of degree n functors,
I, 238.
Using
the theory of calculus for functors from pointed categories to abelian
categories developed by the authors in a preceding paper, they prove in
Part II that degree n functors can be classified in terms of modules
over a particular DGA P_{nxn}(C). In Part I, they develop
the calculus structures needed to prove this and related results. They
also construct a filtration by rank for functors from pointed categories
to abelian categories and compare rank n functors with degree n
functors.
M.
MACKAAY, Note on the holonorny of connections in twisted bundles,
3962.
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 & GARCÍAMUÑOZ, Homotopy colimits and cohomology
with local coefficients, 6380.
The
authors describe the structure of the generalized EilenbergMac 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, 85104.
The
Author describes a precise 2dimensional sense in which the stack of principal
Gbundles 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 nfold monoidal categories model all loop spaces,
105148.
In
a preceding paper, the Authors proved that the classifying space of an
nfold monoidal category is equivalent to a C_{n}space,
where C_{n} is the little ncubes operad. Here they
show a partial converse: any C_{n}space is up to weak
equivalence the classifying space of a simplicial nfold 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, 149160.
A general
rth order jet functor on fibered manifolds is defined as a fiber
product preserving subfunctor of the rth nonholonomic prolongation
containing the rth 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, 153216.
Using
the theory of calculus for functors from pointed categories to abelian
categories developed in a preceding paper, the Authors prove that degree
n functors can be classified in terms of modules over a particular
DGA P_{nxn}(C). They further show that homogeneous
degree n functors have natural classifications in terms of three
different module categories. They use the structures developed for these
classification theorems to show that ail degree n functors factor
through a certain category P_{n}C, extending a result of
Pirashvili. This paper depends on results established in Part I (above).
BARKHUDARYAN,
EL BASHIR & TRNKOVÁ, Endofunctors of Set and cardinalities,
217239.
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 settheoretical assumption (EUCE) (relatively
consistent with (ZFC+GCH)) is introduced and, under (GCH+EUCE),
the classes W of cardinal numbers which have the form W
= {X; FX = X} for some F:
Set ® Set,
are characterized. The presented results solve several problems raised
by Rhineghost and by Zmrzlina.
BIOGRAPHIE
de René LAVENDHOMME, 242246.
GARZON
& del RIO, Lowdimensional cohomology for categorical groups, 247280.
In
this article, the authors define the cohomology categorical groups H^{i}(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) Gmodules.
They show that to any short exact sequence of symmetric Gmodules one
can associate a sixterm exact sequence connecting H^{0}
and H^{1}. Wellknown 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 H^{1}.
M.
GRANDIS, Directed homotopy Theory, I, 281316.
Directed
Algebraic Topology is emerging, from several applications. The basic structure
that the author studies in this paper, called a directed space or dspace,
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 dspaces,
as well as the usual homotopical constructions. The autheor introduces
the fundamental category of a dspace, computable with the help of a van
Kampentype Theorem; its homotopic invariance is brought back to the directed
homotopy of categories.
It
should be noted taht this study reveals new 'shapes' for dspaces 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 spacetime, or a concurrent system.

6. Abstracts
Volume XLV (2004) 

N.
BALL & A. PULTR, Forbidden forests in Priestley spaces, 222.
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, 2334.
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,
3580.
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, 82108.
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.
DC.
CISINSKI, Le localisateur fondamental minimal, 109140.
In
"Pursuing stacks", Grothendieck defines basic localizors as
classes of weak equivalences in the category of small categories satisfying
good descent properties (closely related to Quillen's Theorem A). For
example, the usual weak equivalences (defined by the nerve functor) form
a basic localizor W¥ . More generally, every cohomological theory
on small categories defines canonically a basic localizor. In this paper,
the author gives a proof of Grothendieck's conjecture that states that
W¥ is the smallest basic localizor. Furthermore, we
get a second characterization of W¥ involving Quiiien's
Theorem B. This gives an elementary and axïomatic way to define the
classical homotopy theory of CWcomplexes.
CENTAZZO,
ROSICKY & VITALE, A characterization of locally Dpresentable categories,
141146.
In
a preceding paper, Adamak, Borceux Lack and Rosicky have generalized the
locally finitely presentable categories to locally Dpresentable
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 Dpresentable categories are characterized
as the cocomplete categories with a strong generator consisting of Dpresentable
categories. This unifies results known for locally finitely presentable
categories, varieties and categories of presheaves.
EBRAHIMI,
TABATABAEE & MAHMOUDI, Metrizability of frames, 147156.
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 MFrm of metric frames with
uniform frame maps. Then they prove, among other things, the
counterparts for frames of the pointfree metrization theorems
proved by Barrasrhewski and Pultr. Finally, they characterize the category
MFrm as the intersection of the categories of metric Lindelöf
frames, and of regular frarnes.
Y.
KOPYLOV, Exact couples in a Raikov semiabelian category, 162178.
The
author studies exact couples in semiabelian categories of Raikov, a class
of additive categories which includes many nonabelian 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 semiabelian 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,
179192.
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, 193240.
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, 243266.
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, 267286.
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 subcategory
of the category Convw of convenient spaces with their smooth maps.
Conversely, any full subcategory 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, 287296.
The
functor Max of Mulvey associates to each C*algebra A the unitary
and involutive quantale MaxA of the closed linear subspaces 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, 297300.
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, 301315.
For
a Cindexed category, A, an Adescent equivalence is a morphism of bundles
in C which induces an equivalence between the Adescent categories of
its domain and codomain. In this note, properties of such morphisms are
studied, and those morphisms which are Adescent equivalences for all
Cindexed categories A are fully characterized.

7. Abstracts
Volume XLVI (2005) 

BOURN
& PENON, Catégorification de structures définies par
monade cartésienne, 252.
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 nth step of this iteration process gives rise to
the category of weak ncategories. 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 noncommutative Geoemetry, 5380.
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 wellknown
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, 8398.
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, 99121.
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 omegasets give examples.
M. LAWSON,
Constructing ordered groupoids, 123138.
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, 139153.
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",
163239.
The issue XLVI3 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 (79 October
2005) to commemorate the centenary of Charles' birth (19051979). 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, 198083).
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 JeanPaul
Vanbremeersch who has already organized ECHO I in Amiens in 1996).
Articles developing these abstracts are posted on the internet site dedicated
to Charles Ehresmann:
http://perso.wanadoo.fr/vbmehr/ChEh
W.D. GARRAWAY,
Sheaves for an involutive quantaloid, 243274.
This paper studies Qvalued 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 Qvalued
sets is equivalent to the category of sheaves if Q is pseudorightsided.
GIULI & SLAPAL, Raster convergence with respect to a closure operator,
275300.
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 subobjects of a Kobject 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
cseparation and ccompactness.
B. BANASCHEWSKI,
Projective frames: a general view, 301312.
This paper deals with projectivity in the category Frm of frames relative
to onto homomorphisms whose right adjoint belongs 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, 228.
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 fourcolor Theorem, 2949.
The author interprets the number of good fourcolourings of the faces
of a trivalent, spherical polyhedron as the 2holonomy of the 2connection
of a fibered category, p, modeled on Repf(sl2) and defined over the dual
triangulation T. He also builds an sl2bundle with connection over T,
that is a global, equivariant section of p, and he proves that the fourcolour
Theorem is equivalent to the fact that the connection of this sl2bundle
vanishes nowhere. This geometric interpretation shows the cohomological
nature of the fourcolour nroblern.
S. NIEFIELD,
Homotopic pullbacks, lax pullbacks and exponentiability, 5080.
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 2filtered bicolimits of categories,
83106.
The authors define the notion of 2filtered 2category and give an explicit
construction of the bicolimit of a category valued 2functor. A category
considered as a trivial 2category is 2filtered 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 2filtered 2category. The full set of axioms is necessary
to prove that 2filtered bicolimits have the properties corresponding
to the essential properties of filtered bicolimits.
M. GRANDIS,
107128.
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
2cubes of X naturally produce a fundamental lax 2category. 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 higherorder differentials, 129154
and 207232.
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 Dnmicrocubes on M in x to TDny, which were called nth order preconnections
there and are called Dntangentials in this paper; they give a withoutgerm
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 Dntangentials, which give another generalization.
The relation between Dntangentials and Dntangentials 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 Dntangentials into Dntangentials, 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 infinitedimensional 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 BauesWirsching cohomology of
algebraic structures, 163205.
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 BauesWirsching cohomology.
C. TOWNSEND,
A categorical proof of the equivalence of local compactness and exponentiability
in locale theory, 233239.
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,
242260.
A functor F from K to L is a DVOfunctor if it is naturally equivalent
to any functor G from K to L such that, for each Kobject X, FX is isomorphic
to GX. It is proved that each DVOfunctor F from SET to SET is finitary,
i.e. preserves directed colimits.
R. GARNER,
Double clubs, 261317.
The author develops a theory of the double clubs which extends Kelly's
theory fo clubs to the pseudodouble 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 pseudodouble 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, 354.
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 smashproduct of E(1)local spectra to a derived tensor product of
cochain complexes.
GOLASINSKI,
GONÇALVES & WONG, Equivariant evaluation subgroups and Rhodes
groups, 5569.
In this paper, the authors define equivariant evaluation subgroups of
superior Rhodes groups, and they study their relations to the GottliebFox
groups.
F. MYNARD,
Unified characterization of exponential objects in TOP, PRTOP and PARATOP,
7080.
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 Dcompact
filters.
G.E. REYES,
Embedding manifolds with boundary in smooth toposes, 83103.
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, 104153.
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, 163199.
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, 200219.
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,
220227.
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, 228237.
The author discusses the concept of special type of nonholonomic rjets
and nonholonomic (k, r)velocities from a general point of view. Special
attention is paid to the composition of nonholonomic rjets of the same
type. The product preserving cases are characterized in terms of Weil
algebras.
COLEBUNDERS
& GERLO, Firm reflections generated by complete metric spaces,
243260.
The authors study concrete categories in which each object is a subspace
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 Usinjective? (2) is the class of all closed subspaces
of products of completely metrizable objects firmly Usreflective? 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, 261269.
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 (19192007), 270274.
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 Gstructures, Lie pseudogroups and higher order geometry.
A.C. EHRESMANN,
Fiftieth anniversary of the "Cahiers", 275316.
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, 168.
The
goal of this article is to propose precise definitions, for the higher
and nonstrict version, of the concepts of natural transformations and
analogues (modifications, etc.), and all that within the framework of
the EilenbergMoore algebras. Thus the author obtains objects with algebraic
nature, and a good candidate for the graphic part of the nonstrict 1category
of nonstrict 1categories. 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, 6980.
In
this paper, the authors give a characterization of shape and strong shape
equivalences in the general setting of a tensored and cotensored Topcategory
C. The case of equivariant shape equivalences is also considered.
GUITART
R., Toute théorie est algébrique et topologique, 83128.
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 byproduct 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, 129141.
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, 142153.
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, 163175.
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, 176195.
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 SchreierMac Lane. It naturally appears as a pushout along an internal
Mal'tsev operation.
GRAN
& RODELO, A universal construction in Goursat categories, 196208.
The
authors prove that the category of internal groupoids Grd(E) is a reflective
subcategory 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, 209227.
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, 228239.
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, 243266.
For
a topos C of presheaves which is locally finitely presentable in the enriched
context, the authors prove that the notion of filtered weighted Ccolimit
reduces to the usual pointwise filtered colimits.
HERRLICH,
KEREMEDIS & TACHTSIS, Tychonoff products of super second countable and
super separable metric spaces, 267279.
The
authors prove that, in ZF, i.e. in ZermeloFraenkel 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 nonempty
subsets of the reals. 2. The Tychonoff product of a finite number of metric
spaces with superenumerable bases has also a superenumerable 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 Cantorcomplete super
separable (resp. hereditarily separable) sets is super separable (resp.
hereditarily separable)" is provable in ZF.
JANELIDZE
G. & SOBRAL, Profinite relational structures, 280288.
In
this paper it is proved that a preordered topological space (on a Stone
space) is profinite if, and only if, il is interclopen, i.e. if it can
be presented as the intersection of closedandopen 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 modeltheoretic
meaning.
KOUBEK
& SICHLER, On synchronized relatively full embeddings and Quniversality,
289306.
M.
E. Adams et W. Dziobiak have proved that each ffalgebraic universal quasivariety
of algebraic systems of finite signature is Quniversal. This paper introduces
the notion of a synchronized relatively full embedding which is used to
modify their result for algebraic quasivarieties.
A.H.
ROQUE, Protomodular quasivarieties of universal algebras, 307314.
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 quasivarieties of universal algebras and
he obtains a sufficient condition for the full subcategories 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, 322.
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,
2366.
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 smallorthogonality classes, 6779.
The
author gives a characterization of locally bounded categories and a criterion
to identify the ?orthogonal subcategories in these categories (for a
given regular cardinal ?).
CHENG
E. & MAKKAI M., A note on the Penon definition of ncategory, 83101.
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 nonreflexive 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), 108143.
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, 144155.
Initially
inspired by the case of the standard borromean link, the author introduces
the notion of a borromean object in a category. He provides examples
in groups, Boolean algebras, semirings, 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, 171188.
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
semiabelian, and moreover, grouptheoretic, involving semidirect products
of cyclic groups. They also briefly compare their new results with what
is known for the socalled locally semisimple coverings and for generalized
central extensions.
BÖRGER
R. & KEMPER R., Infinitary linear combinations in reduced cotorsion modules,
189210.
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 EilenbergMoore 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 subcategory; 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, 211232.
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?
233238.
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 settheoretical status of the original Raptis'
result is open.
GRANDIS M., Limits in weak cubical categories (On weak cubical categories,
II), 242272.
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., Quasiequations in locally presentable categories,
273297.
Following
the tradition of Hatcher and BanaschewskiHerrlich, the authors introduce
quasiequations as parallel pairs of unitary morphisms. An object satisfies
the quasiequation if its contravariant homfunctor merges the parallel
pair. The subcategories of a locally finitely presentable category which
can be presented by quasiequations 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,
298319.
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 quasiequations on locally presentable categories II: A logic,
328.
Quasiequations,
given by parallel pairs of finitary morphisms, represent properties of
objects: an object satisfies the property if its contravariant homfunctor
merges the parallel pair. Recently Adamek and Hébert characterized subcategories
of locally finitely presentable categories specified by quasiequations.
The authors now present a logic of quasiequations 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? 2950.
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) YangBaxter operators,
weak YangBaxter 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 YangBaxter operator is the image of
a genuine YangBaxter operator under a separable Frobenius monoidal functor.
Prebimonoidl functors are also defined and discussed.
I. STUBBE, "Hausdorff distance" via conical cocompletion, 5176.
In
the context of quantaloidenriched categories, the author explains how
each saturated class of weights defines, and is defined by, an essentially
unique full subKZdoctrine of the free cocompletion KZdoctrine. The
KZdoctrines which arise as full subKZdoctrines of the free cocompletion,
are characterised by two simple "fully faithfulness" conditions. Conical
weights form a saturated class, and the corresponding KZdoctrine is precisely
(the generalisation to quantaloidenriched categories of) the Hausdorff
doctrine of Akhvlediani et al (2009).
E.
VITALE, Bipullbacks and calculus of fractions, 83113.
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 2algebras are fractions of internal functors
with respect to weak equivalences.
JANELIDZE,
MARKI, THOLEN & URSINI, Ideal determined categories, 115125.
The
authors clarify the role of Hofmann's Axiom in the oldstyle definition
of a semiabelian 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 counterexamples 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 semiabelian.
M.M.
CLEMENTINO & GUTIERRES, On regular and homological closure operators,
127142.
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, 143153.
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, 162181.
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, 182204.
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 ncategory, 205223.
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 nonreflexive rather than reflexive globular sets, and show that
the problem with doubly degenerate tricategories does not arise.
A.
KOCK, Abstract projective lines, 224240.
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 1dimensional subspaces of k2).
W.
RUMP, Objective categories and schemes, 243271.
Quasicoherent
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 (preobjective 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 Sposets,
272281.
A poset
with an action of a pomonoid S on it is called an Sposet. There are two
different notions of completeness for an Sposet. 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, 282315.
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 wellknown BrunnDebrunnerKanenobu 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 ngroupoids,
244
In
this work the authors study exactness in the sesquicategory of ngroupoids.
Using homotopy pullbacks, they construct a six term sequence of (n1)groupoids
from an nfunctor between pointed ngroupoids. 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), 4576.
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 Ckweights
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
semiabelian categories, 77 80.
Semiabelian
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 ncategories II: Degenerate tricategories,
82125.
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, 126160.
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, 163187.
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 orderadjoint. The EilenbergMoore 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 omegagroupoids
are closed under tensor product, 188208.
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 omegagroupoids 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, 209240.
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 wellknown equivalence between Rel, and a
full subcategory of the category Sup, of complete lattices and suppreserving
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 2cofiltered bilimits of topoi, 242251.
The
authors show the existence of bilimits of 2cofiltered diagrams of topoi,
generalizing the construction of cofiltered bilimits developed before.
For any given such diagram represented by any 2cofiltered 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 2filtered
bicolimit of the underlying categories and inverse image functors. They
use the construction of this bicolimit 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, 253268.
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 deriving transformation
satisfying typical differentiation axioms, expressed algebraically. The
traditional notion of Kähler differentials defines the notion of a module
of Adifferential forms with respect to A, where A is a commutative kalgebra.
This module is equipped with a universal Aderivation. 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 nonstrict cells, 269231.
In
a preceding paper the author has pursued Penon's work in higher dimensional
categories by defining weak omegafunctors, weak natural omegatransformations,
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 nétirements
catégoriques". Here he is pursuing Batanin's work in higher dimensional
categories by defining weak omegafunctors, natural transformations, and
so on, using Batanin's framework by extending his universal contractible
omegaoperad K, by building the adapted globular colored contractible.

14.
Abstracts
Volume LIII (2012) 

MALTSINIOTIS
Georges, Carrés exacts homotopiques et dérivateurs, 363.
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, 6480.
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, 82114.
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,
115160.
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 laxunital. 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, 162210.
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, finitedimensional
manifolds, these results extend previous work of J. W. Barrett.
BLUTE,
EHRHARD & TASSON, A convenient differential category, 211232.
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 coKleisli 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 (19472012), 233240.
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 powerset functor
on Ens.
FIORE,
GAMBINO & J. KOCK, Double adjunctions and free monads, 242307.
The
authors characterize double adjunctions in terms of presheaves and universal
squares, and then apply these characterizations to free monads and EilenbergMoore
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 2category, then it also admits the construction
of free monads as a double category. They also prove that a double category
admits EilenbergMoore objects if and only if a certain parameterized
presheaf is representable. Along the way, they develop parameterized presheaves
on double categories and prove a doublecategorical Yoneda Lemma.
BRODOLONI
& STRAMACCIA, Saturation for classes of morphisms, 308315.
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 FreydKelly. 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 3computads is not cartesian
closed, 312.
The
author proves by counterexample that the category of 3computads is not
cartesian closed, a result originally proved by Makkai and Zawadowski.
She gives a 3computad 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, 1327.
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 biadjoint 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, 2855
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,
5674
Motivated
by the case of Klein's group GL_3(F_2), the author introduces the new
notions of ncyclable groups and nbrunnian 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 nbrunnian 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 nonstrict
cells" (2011), 7580.
This
short note proposes a new notion of contractibility for colored omegaoperad
defined in the paper cited in the title. It also proposes an alternative
direction to build the monad for free contractible colored omegaoperads,
FOREWORD,
8290.
Foreword
to the special volume (consisting of Volumes LIV2, 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), 91136
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, 137160
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,
MARTINSFERREIRA & VAN DER LINDEN, A characterisation of the "Smith is
Huq" condition in the pointed Mal'tsev setting, 163183
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,
184210
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, 211220
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, 221239
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, 243263
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 wellknown 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, 264291
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, 292297
The
author explicitly describes a projective sketch of diffeological spaces.
A.C.
EHRESMANN, Parcours de René Guitart, un mathématicien aux multiples facettes,
298316
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 2ProObjects, 335.
Grothendieck
has developed the theory of proobjects 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, precomposition 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 2dimensional theory of proobjects. Given
a 2category C, it defines the 2category 2Pro(C) whose objects are called
2proobjects. It is proved that 2Pro(C) has all the expected basic properties
adequately relativized to the 2categorical setting, including the universal
property corresponding to the one described above. The authors have at
hand the results of Catenriched category theory, but their theory
goes beyond the Catenriched cases since they consider the nonstrict
notion of pseudolimit, which is usually that of practical interest.
BEZHANISHVILI
& HARDING, Stable compactifications of frames, 3664.
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 T0space and described them in terms of
quasiproximities on the space. The author provides an alternate description
of stable compactifications of T0spaces 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 T0space 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 T0space, 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 T0space.
R.
GUITART, Autocategories I. A common setting for knots and 2categories,
6580.
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 2graphs. 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, 2categories, double categories (stratified case),
which so live in the same context, the category of autocategories.
MARTINSFERREIRA
& VAN DER LINDEN, Categories vs. groupoids via generalised Mal'tsev properties,
83112.
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 npermutability 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, 113149.
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 pullingback 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, 151160.
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 2generated
graphic topos, categories, autocategories, associative autographs, are
autographic algebras.
D.
GARRAWAY, QValued Sets and RelationalSheaves, 161204.
The
authors show that a sheaf for a quantaloid is an idempotent supremapreserving
laxsemifunctor (a relationalsheaf). This implies that for a Grothendieck
topos E a sheaf is a relationalsheaf on the category of relations of
E and thus E is equivalent to the category of relationalsheaves and functionaltransformations.
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,
205240.
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 submatrix
of order less or equal to 4.
EHRESMANN,
GRAN & GUITART, New Editorial Board of the "Cahiers'', 242243.
Announce
of the enlargement of the Editorial Board of the "Cahiers", starting from
January 2015.
LACK
& STREET, On monads and warpings, 244266.
The
authors explain the sense in which a warping on a monoidal category is
the same as a pseudomonad on the corresponding oneobject 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, 267313.
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 ("Ktheory
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, 214.
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 2catégories, 1575.
This
text develops a homotopy theory of 2categories analogous to Grothendieck's
homotopy theory of categories in Pursuing Stacks. We define the notion
of basic localizer of 2Cat, a 2categorical generalization of Grothendieck's
notion of basic localizer, and we show that the homotopy theories of Cat
and 2Cat are equivalent in a remarkably strong sense: there is an isomorphism,
compatible with localization, between the ordered classes of basic localizers
of Cat and 2Cat. It follows that weak homotopy equivalences in 2Cat
can be internally characterized, without mentioning topological spaces
or simplicial sets.
A.
EHRESMANN, Parcours d'un topologuecatégoricien : JeanMarc Cordier (19462014),
7680.
This
Note contains the List of Publications of JeanMarc 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 2catégories strictes, 83108.
This
paper is a complement to J. Chiche's paper in Volume LVI1 (cf. above)
which studies homotopy theories on 2Cat, given by classes of weak equivalences
called basic localizers of 2Cat (a 2categorical 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 2Cat, there is a model category structure à la Thomason on 2Cat
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
systems. As a consequence, we arrive at a restatement of the sobrietyspatiality
equivalence for affine sets, patterned after the equivalence between the
categories of sober topological spaces and spatial locales.
E.
MEHDINEZHAD, Abstract annihilation graphs, 133145.
We
propose a new, widely generalized context for the study of the zerodivisor/annihilatingideal
graphs, where the vertices of graphs are not elements/ideals of a commutative
ring, but elements of an abstract ordered set (imitating the lattice of
ideals), equipped with a binary operation (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, 147159.
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
prohomotopy category deûned by Lisica and Mardesic when C is the category
of topological spaces.
BARR, KENNISON & RAPHAEL, On reflective and coreflective hulls, 163208.
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 quantaloidenriched categories and their
distributors, 209231.
It
is shown that, for a small quantaloid Q, the category of small Qcategories
and Qfunctors is total and cototal, and so is the category of Qdistributors
and QChu transforms.
CARLETTI
& GRANDIS, Generalised pushouts, connected colimits and codiscrete groupoids,
232240.
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, 242300.
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, 301317.
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, C1rings, 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 diffeological spaces,
350.
The authors
study how the notion of tangent space can be extended to diffeological
spaces, which are generalizations of smooth manifolds that include singular
spaces and infinitedimensional 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 establish 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, infinitedimensional vector spaces and diffeological
groups, and spaces of smooth maps between smooth manifolds.
HOSSEINI
& QASEMI NEZHAD, Equalizers in Kleisli categories, 5176.
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. JANELIDZEGRAY,
Calculus of ERelations in incomplete relatively regular categories,
83102.
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 relations 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 EGoursat axiom is equivalent to the relative 3x3Lemma.
GRANDIS
& PARE, An introduction to multiple categories on weak and lax multiple
categories, I) 103159.
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 3dimensional 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 infinitedimensional 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 respected 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 mentioned above can still be defined, but a general
theory of multiple limits is not developed here.
W. RUMP,
The completion of a quantum Balgebra, 203228.
It is shown
that quantales are the injective objects in the category of quantum Balgebras,
and that every quantum Balgebra has an injective envelope. By an explicit
construction, the injective envelope is revealed as a completion more
general than the DedekindMacNeille completion. A recent result of Lambek
et al., where residual structures unexpectedly arise, is explained in
the light of quantum Balgebras, which gives another Instance for their
ubiquity. Connections to promonoidal structures and multicategories are
indicated.
N. GILL,
On a conjecture of Degos, 229237.
This note
givea a proof of a conjecture of Degos concerning groups generated by
companion matrices in GLn(q).
BLUTE,
LUCYSHYNWRIGHT & O'NEILL Derivations in codifferential categories, 243279.
We define
and study a novel, general notion of derivation in the setting of the
codifferential categories of BluteCockettSeely, generalizing the notion
of Klinear derivation from commutative algebra. Given a codifferential
category (C;T;d), a Tderivation del from A to M on an algebra A of the
monad T is defined as a morphism in C into an Amodule M satisfying a
form of the chain rule expressed in terms of the deriving transformation
d. We show that such Tderivations correspond to Thomomorphisms from
A to W(A;M) over A valued in an associated Talgebra. We establish the
existence of universal Tderivations from A to OmegaTA valued in an associated
Amodule of Kählertype differentials OmegaTA. Where as previous work
of BluteCockettPorterSelly on Kähler categories employed a notion of
derivation expressible without reference to the monad T, we show that
the use of the above Tbased 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 omegagroupoids from types,
topological spaces..., 281294.
A short introduction
to Grothendieck weak omegagroupoids is given. The aim is to give evidence
that, in certain contexts, this simple language is a convenient one for
constructing globular weak omegagroupoids. To this end, we give a short
reworking of van den Berg and Garner's construction of a Batanin weak
omegagroupoid from a type using the language of Grothendieck weak omegagroupoids.
This construction also applies to topological spaces and Kan complexes.
W. THOLEN,
A mathematical tribute to Reinhard Börger, 295314.
A synopsis
of the life and work of Reinhard Börger (19542014) is presented, withan
emphasis on his early or unpublished works.
