Prof. Giuseppe Metere (University of Milano) will give a seminar on Wednesday 9 March 2025 at 11:30 in Sala Riunioni, DipMat.
Title: Towards a notion of unital action.
Abstract: A group action of a group $B$ on a group $X$ is usually described as a $B$-group $X$ , i.e., through a set-theoretic map:
$$B \times X \to X$$
satisfying some specified axioms. From a category-theoretic point of view, this approach is not fully satisfying, since it relies on a map which is not a group homomorphism, i.e. is not a morphism of the given category.
A solution to this issue was developed by Janelidze and others, leading to the definition of internal actions in a semi-abelian category.
Now, the notion of a semi-abelian category encompasses a broad spectrum of algebraic varieties but excludes those with more than one constant in their signature, as for instance: unital rings, unital non-associative algebras, MV-algebras.
In my talk, I will outline some ideas on how to extend the concept of actions to such non-pointed (protomodular) settings. If time permits, I will describe some examples of unital actions in different semi-abelian categories, such as unital non-associative algebras, bounded hoops, and the dual of the category of sets .
This is joint work (in progress) with Manuel Mancini and Federica Piazza.
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