ABSTRACTS


Dirk Hofmann Complete spaces:
Employing the analogy between ordered sets and topological spaces, over the past years we have investigated the notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set (= presheaf) monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual concept of completeness does not generalise easily; and it is the aim of this talk to explain some of the difficulties and how to overcome them. In particular, we construct the "copresheaf functor" on the category of tensor-exponentiable spaces, show that it is the part of a monad defined on representable (=Nachbin) spaces, and discuss the Isbell conjugation adjunction between the space of presheafs and the opposite (in an appropriate sense) of the space of copresheafs.

Nelson Martins-Ferreira On split extensions underlying a cartesian product:
We characterize those varieties where every split extension has underlying a cartesian product as middle object.
Joint work with James Gray.

Andrea Montoli Semidirect products and internal structures in Jonsson-Tarski varieties:
The aim of this talk is to characterize those split extensions corresponding to the classical semidirect product in Jonsson-Tarski varieties, with particular attention to the cases of monoids and unitary magmas, and to compare the classical notion of semidirect product with the categorical one, due to Bourn and Janelidze, in this context. This characterization allows to describe internal crossed modules in these varieties.
Joint work with Nelson Martins Ferreira and Manuela Sobral.

Jorge Picado On the uniform structure of localic groups: functoriality:
When the remarkable Closed Subgroup Theorem for localic groups, saying that any localic subgroup of a localic group is closed, made its first appearance (Isbell, KrÝz, Pultr & Rosickř [2]), it naturally raised the problem of the uniformisation of localic groups (see e.g. [1] where it is proved that any localic group is complete in its two-sided uniformity).
In fact, similarly like in the classical case, one has natural cover uniformities induced by the group structure and one has equally (if not even more) natural uniformities described by entourages. Due to the nature of product in the category of locales, the entourage uniformities in the localic context only mimic the classical Weil approach while the cover (Tukey type) ones can be viewed as an immediate extension. Nevertheless the resulting categories of uniform locales are concretely isomorphic. We apply this isomorphism to the natural uniformities of localic groups. In particular we present an extremely simple proof of the fact that localic group homomorphisms are uniform, thus providing a natural forgetful functor from the category of localic groups into the category of uniform locales.
It should be noted that this fact has so far, to our knowledge, not been proved in the literature by the cover methods, and even remaking our simple entourage proof to a cover one by translation seems to be rather complex. We see it as a nice example of the usefulness of the entourage approach.

References:
[1] B. Banaschewski and J. C. C. Vermeulen, On the completeness of localic groups, Comment. Math. Univ. Carolinae 40 (1999), 293-307.
[2] J. R. Isbell, I. KrÝz, A. Pultr and J. Rosickř, Remarks on localic groups, in Categorical Algebra and its Applications (Proc. Int. Conf. Louvain-La-Neuve 1987, ed. by F. Borceux), Lecture Notes in Math. 1348, pp. 154-172.
[3] J. Picado and A. Pultr, Entourages, covers and localic groups, Applied Categorical Structures, to appear.

Joint work with Ales Pultr.

Lurdes Sousa Moving from orthogonality to Kan-injectivity:
In a poset enriched category X an object A is said to be Kan-injective with respect to a morphism f : X --> Y provided that the map hom(f,A) has a reflective right adjoint in Pos. If X is an arbitrary category enriched with the trivial order, this simply means that A is orthogonal to f. In this talk we will show that many aspects of the good behaviour of orthogonality concerning limits, colimits and reflectivity still hold for Kan-injectivity in the more general setting of poset enriched categories.
Joint work with Margarida Carvalho.

Walter Tholen Considering Kleisli morphisms as bimodules:
For a monad S on a category K whose Kleisli category is a quantaloid, we introduce the notion of modularity, in such a way that morphisms in the Kleisli category may be regarded as V-(bi)modules, for some quantale V. The passage from S to V is shown to belong to a global adjunction which, in the opposite direction, associates with every (commutative unital) quantale V the prototypical example of a modular monad, namely the presheaf monad on V-Cat.

Enver Uslu Actor of a precrossed module in Lie algebras:
The notion of action between objects in a category was extensively studied from several general points of view, particularly, in the framework of semiabelian categories [1, 2] and categories of interest [3].
Our goal in the present talk is the construction of the actor (object which represents actions in the category) of any object in the category PXLie of precrossed modules in the category of Lie algebras. Following [3], there is a general construction of the actor object in the framework of categories of interest. Unfortunately, PXLie is not equivalent to a category of interest, so we can not apply this approach in order to obtain the construction of actor. Then our proposal considers an adequate set of triples whose components are different kind of derivations and we endow this set with the necessary structure.
From this construction we derive the notions of action, center, semi-direct product, derivation, commutator and abelian precrossed module in PXLie. We prove that the notion of action is equivalent to the one given in semi-abelian categories and our construction of the actor of a precrossed module is the split extension classifier for the precrossed module in the sense of [1].
If we apply our construction to the case of the category of crossed modules in Lie algebras these notions give in a certain sense the corresponding ones for crossed modules in [4].

References:
[1] Borceux, F., Janelidze, G., Kelly, G. M. (2005). Internal object actions. Comment. Math. Univ. Carolin. 46:235-255.
[2] Borceux, F., Janelidze, G., Kelly, G. M. (2005). On the representability of actions in a semi-abelian category. Theory Appl. Categories 14:244-286.
[3] Casas, J. M., Datuashvili, T., Ladra, M. (2010). Universal Strict General Actors and Actors in Categories of Interest. Appl. Categ. Structures 18:85-114.
[4] Casas, J. M., Ladra, M. (1998). The actor of a crossed module in Lie algebras. Comm. Algebra 26:2065-2089.

Joint work with JosÚ Manuel Casas.

Tim Van der Linden The geometry of higher central extensions:
In recent work we establish a Galois-theoretic interpretation of semi-abelian cohomology: we prove that cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This result depends on a geometric viewpoint of the concept of higher central extension, next to the algebraic one in terms of commutators. In this talk I will explain how to deal with higher central extensions geometrically and how this leads to the claimed result.
Joint work with Diana Rodelo.

Filippo Viviani Cohomology of categories fibered in groupoids:
We will review the definition of Chow (resp. singular) cohomology for a category fibered in groupoids over the category of algebraic varieties (resp. topological spaces). A remarkable example is provided by the quotients stacks, where the above definitions agree with the usual equivariant Chow (resp. singular) cohomology. As an application of the general theory, we present the computation, obtained in a joint work with D. Fulghesu, of the Chow ring of the stack of cyclic covers of the projective line.

JoŃo Xarez Galois theories and generalised connectedness:
There is a Galois theory associated to a reflection into a full subcategory provided certain pullback diagrams are preserved by the left adjoint functor. Previously, we have given a setting in which such a preservation holds if and only if the simpler condition connected components are connected holds. This setting unifies both topological and algebraic known examples. In the current talk we will show that a more general setting allows new geometrical examples to be added to the renewed frame. The same setting also provides simplified versions of other relevant properties in categorical Galois Theory, either weaker or stronger than the former limit preservation.

Ivan Yudin The nerve of a crossed module:
To define (co)homology groups for a group G, Eilenberg and MacLane gave in 1945 three different explicit descriptions of the classifying space BG: homogeneous, non-homogeneous, and matrix construction. The non-homogeneous description of BG was used by Hochschild in 1946 to define the bar construction for an arbitrary associative algebra A. In 1948 Blakers constructed the classifying space for an arbitrary crossed complex. In the case of the group his construction is related to the matrix construction of Eilenberg and MacLane. In general, the definition of cells in the Blakers construction depends on the existence of inverse elements at every term of a crossed complex. In my talk I will give an explicit description of the nerve for a crossed module, that generalizes non-homogeneous description of BG.
The talk is based on the preprint arxiv.org/abs/1103.6275