Abstracts

ABSTRACTS

Maria Manuel Clementino The categorical van Kampen Theorem:
We outline the proof of Brown-Janelidze van Kampen Theorem [1], that shows that the classical van Kampen Theorem -- as well as Magid's Theorem for fundamental groups of commutative rings -- follows from Descent Theory, and characterize, for topological spaces, the diagrams that fulfil van Kampen Theorem, in the global case.
References:
[1] R. Brown, G. Janelidze, Van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Algebra 119 (1997) 255-263.

Ignacio Lopez Franco Pseudomonoids and Hopf algebras:
In this talk we explain how pseudomonoids and autonomous pseudomonoids generalise respectively bialgebras and Hopf algebras. If time permits, we show how some fundamental results on Hopf algebras are in fact results on autonomous pseudomonoids. Most of the talk is based on [2,1].
References:
[1] B. Day, P. McCrudden, and R. Street. Dualizations and antipodes. Appl. Categ. Struct., 11(3):229-260, 2003.
[2] B. Day and R. Street. Monoidal bicategories and Hopf algebroids. Adv. Math., 129(1):99-157, 1997.

Carla Reis Co-completeness and exponentiation in V-categories:
For a completely distributive quantale V, we present a sufficient condition for a co-complete V-category to be exponentiable. We verify that co-complete (probabilistic) metric spaces satisfy this condition, and show that even a weaker form of co-completeness suffices.

Tim Van der Linden Actions, modules and the "Smith is Huq" problem:
I sketch the background of categorical commutator theory needed to understand the "Smith is Huq" problem: under which conditions does the Smith-Pedicchio commutator of equivalence relations correspond to the Huq-Borceux-Bourn commutator of normal subobjects? I give an overview of known questions and results in this context, and explain how information on the reflection of actions to modules might help solve the problem.

Maria Manuel Clementino	The categorical van Kampen Theorem: We outline the proof of Brown-Janelidze van Kampen Theorem [1], that shows that the classical van Kampen Theorem -- as well as Magid's Theorem for fundamental groups of commutative rings -- follows from Descent Theory, and characterize, for topological spaces, the diagrams that fulfil van Kampen Theorem, in the global case. References: [1] R. Brown, G. Janelidze, Van Kampen theorems for categories of covering morphisms in lextensive categories, J. Pure Appl. Algebra 119 (1997) 255-263.
Ignacio Lopez Franco	Pseudomonoids and Hopf algebras: In this talk we explain how pseudomonoids and autonomous pseudomonoids generalise respectively bialgebras and Hopf algebras. If time permits, we show how some fundamental results on Hopf algebras are in fact results on autonomous pseudomonoids. Most of the talk is based on [2,1]. References: [1] B. Day, P. McCrudden, and R. Street. Dualizations and antipodes. Appl. Categ. Struct., 11(3):229-260, 2003. [2] B. Day and R. Street. Monoidal bicategories and Hopf algebroids. Adv. Math., 129(1):99-157, 1997.
Carla Reis	Co-completeness and exponentiation in V-categories: For a completely distributive quantale V, we present a sufficient condition for a co-complete V-category to be exponentiable. We verify that co-complete (probabilistic) metric spaces satisfy this condition, and show that even a weaker form of co-completeness suffices.
Tim Van der Linden	Actions, modules and the "Smith is Huq" problem: I sketch the background of categorical commutator theory needed to understand the "Smith is Huq" problem: under which conditions does the Smith-Pedicchio commutator of equivalence relations correspond to the Huq-Borceux-Bourn commutator of normal subobjects? I give an overview of known questions and results in this context, and explain how information on the reflection of actions to modules might help solve the problem.