Comonad cohomology of track categories
Simona Paoli, Department of Mathematics, University of Leicester, UK.
Abstract: Simplicial categories are one of the models of $(\infty,1)$-categories. They can be studied using the Postnikov decomposition, whose sections are categories enriched in simplicial n-types and whose k-invariants are defined in terms of the (S,O)-cohomology of Dwyer, Kan and Smith. The latter is defined topologically, while the understanding of the k-invariants calls for an algebraic description. In this talk I illustrate the first step of this program, for categories enriched in groupoids, also called track categories. We define a comonad cohomology of track categories and we show that, under mild hypothesis on the track category, its comonad cohomology coincides up to a dimension shift, with its (S,O)-cohomology, therefore obtaining an algebraic formulation of the latter. This is joint work with David Blanc.
A characterisation of Lie algebras via algebraic exponentiation.
Xabier Garcia Martinez, Departamento de Matemàticas, Universidade de Vigo, Spain.
Abstract: In this talk we will describe the variety of Lie algebras via algebraic exponentiation, a concept introduced by James Gray in his Ph.D. thesis. We will prove that the variety of Lie algebras over a field K of characteristic zero is the only non-abelian variety of non-associative algebras over K which is locally algebraically cartesian closed (LACC). We will also extend this result to varieties of n-algebras, and we will discuss what happens in prime characteristic.
Here you can find an extended abstract.
Approximations and torsion pairs in triangulated categories
Jorge Vitoria, Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Italy.
Abstract: Triangulated categories are fundamental objects in homological algebra, and many of those naturally occurring in algebra are generated by their small (compact) objects. Key examples range from stable module categories over group rings to derived categories of modules over a ring or sheaves over a scheme. For such a triangulated category T, it is then natural to consider the abelian category of (contravariant) functors on the compact objects. This abelian category is a locally coherent Grothendieck category and it naturally induces a pure-exact structure on T, controlling some of its structural aspects.In this talk we will discuss the interactions between the pure-exact structure mentioned above and the existence of approximations (precovers and preenvelopes) for subcategories of a compactly generated algebraic triangulated category. Moreover, we will focus on how to use the pure-exact structure to create torsion pairs, which are pairs of subcategories with special approximation-theoretic properties that allow for a useful decomposition of the underlying triangulated category. This is based on joint work with Rosanna Laking.