M. Messora : Torsion theories in dimension 2 (based on ongoing joint work with E.M. Vitale)
Torsion theories were introduced in the 1970s by S. Dickson in the context of abelian categories. More recently, many different authors have been interested in various generalisations of this notion, both for pointed and non-pointed categories.
In this talk we focus on studying a definition of torsion theory in a 2-dimensional setting, i.e., in the context of bicategories and 2-categories, relying on the usual notions of bizero object, bikernel and bicokernel.
After quickly reviewing the needed technical framework, we provide a possible definition of bitorsion theory in a bicategory equipped with a bizero object, and we explore some relevant properties and examples.
Enrico M. Vitale : The snail lemma and the long homology sequence (joint work with Julia Ramos Gonzalez)
A cornerstone in homological algebra is the fact that, starting from a short exact sequence of chain complexes in an abelian category, one can construct a long exact sequence relating the homology objects of the original complexes. Usually one proves first the snake lemma, and then gets the long exact sequence in homology pasting together infinitely many copies of the six-term snake sequence.
The snake lemma is a special case of a more general result, the snail lemma, introduced in order to unify some higher dimensional exact sequences, see [1,2,3,4]. The difference between the snake and the snail lemmas lies in the fact that the snake requires as starting point a short exact sequence in the category of arrows, whereas the snail works starting from any morphism in the category of arrows.
In this talk, we show that it is possible to use the snail lemma instead of the snake lemma in order to construct a long exact sequence in homology starting from any morphism of chain complexes, and not necessarily from a short exact sequence of complexes.
Even if the idea is quite simple, to state and prove it properly we have to introduce a new concept, that we call a sequentiable family of arrows.
The idea behind a sequentiable family is to focus our attention not on the homology objects associated with a chain complex, but on the homology arrows, that is, those arrows whose kernel and cokernel provide the homology objects. This allows us to formulate the snail lemma inside the category of sequentiable families, which is equipped with a structure of nullhomotopies more convenient than the one usually considered in the category of chain complexes.
