## Venerdì 1 Marzo 2019, ore 15:00 - Aula 2AB40 - Nicola Gambino + Marino Gran

**ARGOMENTI:** Seminars

Venerdì 1 Marzo 2019 in Aula 2AB40, nell'ambito del Seminario Padova-Verona MALGA - Moduli Algebre Anelli si terranno i seguenti seminari.

ore 15:00 Nicola Gambino (School of Mathematics, University of Leeds): “Bicategories of bimodules”.

Abstract

The notion of a bicategory is a generalisation of the notion of a category which is obtained by allowing the composition of morphisms to be associative and unital up to isomorphism, rather than strictly. This generalisation is useful to capture many naturally-occurring mathematical structures. For example, there is a bicategory with rings as objects and bimodules as morphisms, in which composition of morphisms is given by tensor product of bimodules. In this talk, after introducing bicategories, I will review the so-called “bimodule construction” for bicategories, present some examples of it and some new results, based on joint work with Andre' Joyal.

ore 16:30 Marino Gran (Université Catholique de Louvain): “Groupoids, commutators and cocommutative Hopf algebras”.

Abstract

Internal structures are useful to understand some fundamental constructions in commutator theory. In this talk we shall first explain the relationship between groupoids and commutators in the so-called Mal’tsev varieties [1], that are the varieties in the sense of universal algebra whose algebraic theory has a ternary operation $p(x,y,z)$ satisfying the identities $p(x,y,y)= x$ and $p(x,x,y)=y$. Typical examples of Mal’tsev varieties are those of groups, where such an operation is given by $p(x,y,z)=x-y+z$, quasi-groups, rings, Lie algebras, Boolean algebras and crossed modules.

We shall then explain how some of these results can be naturally extended to a categorical context [2,3], that also includes the categories of compact groups and of cocommutative Hopf algebras [4].

References

[1] J. D. Smith, Mal’cev varieties, Springer Lect. Notes in Math. 554 (1976).

[2] M.C. Pedicchio, A categorical approach to commutator theory, J. Algebra 177 (1995) 647-657.

[3] A. Duvieusart and M. Gran, Higher commutator conditions for extensions in Mal’tsev categories, J. Algebra 515 (2018) 298-327.

[4] M. Gran, F. Sterck, J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra (2019), pub.online.