Public Thesis defense - IRMP


17 mars 2021



will take place in the form of a video conference Teams

An approach to commutator theory via higher Mal'tsev operations by Cyrille Sandry SIMEU

Pour l’obtention du grade de Docteur en sciences

The aim of this thesis is to introduce a categorical description of higher-order commutators of equivalence relations on the same object in an exact Mal’tsev category with finite coproducts. We focus on the ternary case n = 3, because on higher levels (n > 3), similar definitions will work. We introduce two concepts of ternary commutator of equivalence relations in exact Mal’tsev categories with finite coproducts. A first one, which we have called the ternary Bulatov commutator, generalizes its universal-algebraic counterpart, which was earlier introduced by A. Bulatov and further studied in detail by E. Aichinger and N. Mudrinski, J. Opršal, A. Moorhead, and A. Wires. The other, that we have called the ternary Smith commutator, whose universal-algebraic counterpart will be the subject of our future work, enables us to characterize higher central extensions with respect to Ab(X), useful for the understanding of homological algebra in non-abelian environments. In particular, our higher-order Smith commutator is related to higher-order pregroupoids which are used for the interpretation of comonadic cohomology with non-trivial coefficients. The analysis and the geometrical interpretation of the smallest and the largest n-fold equivalence relations on a collection of n equivalence relations on the same object allow us to better understand the geometry of higher-pregroupoids and to derive some properties of our higher-order commutators.

In our study, we found a distributivity condition on the kernel pairs of the

initial morphisms of an n-fold regular epimorphism which characterizes when it is an n-fold extension. We obtain in some sense a higher-order generalization of the 3 X 3-Lemma and the denormalized 3 X 3-Lemma due to D. Bourn. We prove that our ternary Bulatov commutator has some of the convenient properties of its universal-algebraic version and when we restrict the context to algebraically coherent semi-abelian categories, we find for n = 3, the ternary Higgins commutator of M. Hartl and T. Van der Linden. This answers the question, what universal property characterizes that commutator.

Jury members :

  • Prof. Tim Van der Linden (UCLouvain), supervisor
  • Prof. Aldo Ursini (Université de Sienne, Italy), supervisor
  • Prof. Marino Gran (UCLouvain), chairperson
  • Prof. Enrico Vitale (UCLouvain), secretary
  • Prof. James Gray (Université de Stellenbosch, South Africa)
  • Prof. Andrea Montoli (Université de Milan, Italy)

Pay attention :

The public defense of Cyrille Sandry Simeu scheduled for Wednesday 17 March at 16:00 will take place in the form of a video conference Teams

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