Abstracts

ABSTRACTS

F. Borceux Variations on the notion of filtered colimit:
It is well-known how to compute a filtered colimit in the category of sets; it is well-known as well that filtered colimits commute in SET with finite limits.
A well-known generalization of filtered colimits is the case of sifted colimits: those which commute in SET with finite products. This property of sifted colimits carries over to algebraic theories. In the case of semi-abelian theories, sifted colimits commute also with right exact sequences, but generally not with short exact sequences.
The well-known construction of filtered colimits in SET holds already under a weaker axiom: every span can be completed in a commutative square. Call "pro-filtered" a connected colimit with this property. In the case of pointed sets, pro-filtered colimits commute with finite products and kernels, but not with equalizers. In the semi-abelian case, pro-filtered sifted colimits commute again with finite products and kernels ... but the case of equalizers remains an open problem.

C. Caleiro Quantum institutions (Joint work with P. Mateus, A. Sernadas and C. Sernadas):
The exogenous approach to enriching any given base logic for probabilistic and quantum reasoning is brought into the realm of institutions. The theory of institutions helps in capturing the precise relationships between the logics that are obtained, and, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract.

M. Gran Relative commutator associated with varieties of nilpotent and of solvable groups:
In this joint work with Tomas Everaert we determine explicit formulas for the relative commutator of groups with respect to the subvarieties of n-nilpotent groups and of n-solvable groups, for each natural number n. In particular, a characterization of the extensions of groups that are central relatively to these subvarieties is deduced.

G. Gutierres Sequential spaces in topological structures:
Topological sequential spaces are the fixed points of a Galois connection between collections of open sets and sequential convergence structures. The same procedure can be done replacing open sets by other topological concepts, e.g. closure operators or (ultra)filter convergences. The fixed points of these other Galois connections are not topological spaces in general, but they can be embedded into larger topological categories, such as pretopological, pseudotopological or convergence spaces.
We characterize the sequential convergences which are fixed points of these connections as well as their restrictions to topological spaces.

D. Hofmann Categorical notions in T-categories:
In this talk we introduce topological theories as a possible general framework for the study of lax algebras. We recall that lax algebras (=models of such a theory) can be seen as a generalisation of Eilenberg--Moore algebras and, at the same time, as a generalisation of the concept of a category. Here we focus on the latter point of view and present topics like distributors, weighted (co)limits, completeness, duality theory and Morita-equivalence in the realm of lax algebras.

M. Mackaay Categorification and knot homologies:
I will first explain the idea of categorification as first formulated by Crane and Frenkel in 1993 and discuss Khovanov's knot homology, which is a non-trivial example of categorification. Then I will discuss generalizations of Khovanov's original work. The idea is to give a sort of review with the main ideas without getting (too) technical.

D. Rodelo How to avoid crossed modules in group cohomology (Joint work with D. Bourn):
The classical interpretation for the cohomology groups of a group is given by equivalence classes of crossed n-fold extensions equipped with the Baer sum ([4], [2], [3]). The search for an adequate categorical context to generalize non-abelian cohomology leads us to a simpler interpretation where crossed modules are avoided ([1]).
References:
[1] D. Bourn and D. Rodelo, Cohomology without projectives, submitted for publication.
[2] D. Holt, An interpretation of the cohomology groups Hn(G;M), J. Algebra 60 (1979) 307-318.
[3] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980) 302-314.
[4] S. Mac Lane, Homology, Springer, 1963.

A.H. Roque About descent in quasi-varieties:
The category of structures for a (one sorted) first order language is regular but not necessarily exact. A quasi-variety is a full subcategory of such, closed under equalizers, products and filtered colimits. We present characterizations of descent and effective descent morphisms in some quasi-varieties.

A. Sernadas Towards a universal theory of fibring (Joint ongoing work with Marcelo Coniglio, Cristina Sernadas and Luca Viganò):
A new notion of deductive system is proposed in order to allow the development of a truly universal theory of fibring. After a brief review of the notions of 2-category, multicategory and polycategory, a deductive system is defined as a 3-multipolymulticategory. Objects are sorts, multimorphisms are language constructors, 2-polycells are judgments, and 3-multicells are derivations. The universality of the proposed concept is discussed. Fibring is introduced as a cocartesian lifting. Along the way, some examples are provided. Finally, pending problems are discussed.

C. Van Olmen Eilenberg-Moore algebras and categories for pointfree topology:
It is well-known that the category of frames can be described as the Eilenberg-Moore algebras associated with the Hom-functor monad of the initially dense object of Top. When looking at approach spaces and approach frames an analogous result is true. This result and questions following from it gives us an algebraic description of the collection of lower semi-continuous functions on a topological space and different ways of looking at this. If time permits, we will also look at the case of metric spaces.

F. Borceux	Variations on the notion of filtered colimit: It is well-known how to compute a filtered colimit in the category of sets; it is well-known as well that filtered colimits commute in SET with finite limits. A well-known generalization of filtered colimits is the case of sifted colimits: those which commute in SET with finite products. This property of sifted colimits carries over to algebraic theories. In the case of semi-abelian theories, sifted colimits commute also with right exact sequences, but generally not with short exact sequences. The well-known construction of filtered colimits in SET holds already under a weaker axiom: every span can be completed in a commutative square. Call "pro-filtered" a connected colimit with this property. In the case of pointed sets, pro-filtered colimits commute with finite products and kernels, but not with equalizers. In the semi-abelian case, pro-filtered sifted colimits commute again with finite products and kernels ... but the case of equalizers remains an open problem.
C. Caleiro	Quantum institutions (Joint work with P. Mateus, A. Sernadas and C. Sernadas): The exogenous approach to enriching any given base logic for probabilistic and quantum reasoning is brought into the realm of institutions. The theory of institutions helps in capturing the precise relationships between the logics that are obtained, and, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract.
M. Gran	Relative commutator associated with varieties of nilpotent and of solvable groups: In this joint work with Tomas Everaert we determine explicit formulas for the relative commutator of groups with respect to the subvarieties of n-nilpotent groups and of n-solvable groups, for each natural number n. In particular, a characterization of the extensions of groups that are central relatively to these subvarieties is deduced.
G. Gutierres	Sequential spaces in topological structures: Topological sequential spaces are the fixed points of a Galois connection between collections of open sets and sequential convergence structures. The same procedure can be done replacing open sets by other topological concepts, e.g. closure operators or (ultra)filter convergences. The fixed points of these other Galois connections are not topological spaces in general, but they can be embedded into larger topological categories, such as pretopological, pseudotopological or convergence spaces. We characterize the sequential convergences which are fixed points of these connections as well as their restrictions to topological spaces.
D. Hofmann	Categorical notions in T-categories: In this talk we introduce topological theories as a possible general framework for the study of lax algebras. We recall that lax algebras (=models of such a theory) can be seen as a generalisation of Eilenberg--Moore algebras and, at the same time, as a generalisation of the concept of a category. Here we focus on the latter point of view and present topics like distributors, weighted (co)limits, completeness, duality theory and Morita-equivalence in the realm of lax algebras.
M. Mackaay	Categorification and knot homologies: I will first explain the idea of categorification as first formulated by Crane and Frenkel in 1993 and discuss Khovanov's knot homology, which is a non-trivial example of categorification. Then I will discuss generalizations of Khovanov's original work. The idea is to give a sort of review with the main ideas without getting (too) technical.
D. Rodelo	How to avoid crossed modules in group cohomology (Joint work with D. Bourn): The classical interpretation for the cohomology groups of a group is given by equivalence classes of crossed n-fold extensions equipped with the Baer sum ([4], [2], [3]). The search for an adequate categorical context to generalize non-abelian cohomology leads us to a simpler interpretation where crossed modules are avoided ([1]). References: [1] D. Bourn and D. Rodelo, Cohomology without projectives, submitted for publication. [2] D. Holt, An interpretation of the cohomology groups Hn(G;M), J. Algebra 60 (1979) 307-318. [3] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980) 302-314. [4] S. Mac Lane, Homology, Springer, 1963.
A.H. Roque	About descent in quasi-varieties: The category of structures for a (one sorted) first order language is regular but not necessarily exact. A quasi-variety is a full subcategory of such, closed under equalizers, products and filtered colimits. We present characterizations of descent and effective descent morphisms in some quasi-varieties.
A. Sernadas	Towards a universal theory of fibring (Joint ongoing work with Marcelo Coniglio, Cristina Sernadas and Luca Viganò): A new notion of deductive system is proposed in order to allow the development of a truly universal theory of fibring. After a brief review of the notions of 2-category, multicategory and polycategory, a deductive system is defined as a 3-multipolymulticategory. Objects are sorts, multimorphisms are language constructors, 2-polycells are judgments, and 3-multicells are derivations. The universality of the proposed concept is discussed. Fibring is introduced as a cocartesian lifting. Along the way, some examples are provided. Finally, pending problems are discussed.
C. Van Olmen	Eilenberg-Moore algebras and categories for pointfree topology: It is well-known that the category of frames can be described as the Eilenberg-Moore algebras associated with the Hom-functor monad of the initially dense object of Top. When looking at approach spaces and approach frames an analogous result is true. This result and questions following from it gives us an algebraic description of the collection of lower semi-continuous functions on a topological space and different ways of looking at this. If time permits, we will also look at the case of metric spaces.