Globalization for geometric partial comodules

Abstract: The study of partial symmetries (e.g. partial dynamical systems, (co)actions, (co)representations, comodule algebras) is a relatively recent research area in continuous expansion, whose origins can be traced back to the study of C*-algebras generated by partial isometries. One of the central questions in the field is the existence and uniqueness of a so-called globalization or enveloping (co)action.

In the framework of partial actions of groups, any global action of a group on a set induces a partial action of the group on any subset by restriction. The idea behind the concept of globalization of a given partial action is to find a (universal) global action such that the initial partial action can be realized as the restriction of this global one. The importance of this procedure is testified by the numerous globalization results already existing in the literature which, however, are based on some ad hoc constructions, depending on the nature of the objects carrying the partial action.

We propose here a unified approach to globalization in a categorical setting, explaining several of the existing results from the literature and, at the same time, providing a procedure to construct globalizations in concrete contexts of interest. Our approach relies on the notion of geometric partial comodules (recently introduced by Hu and Vercruysse in [HV]) which –unlike classical partial actions, that exist only for (topological) groups and Hopf algebras– can be defined over any coalgebra in an arbitrary monoidal category with pushouts.

[HV] J. Hu, J. Vercruysse, Geometrically partial actions. Trans. Amer. Math. Soc. 373 (2020), no. 6, 4085–4143.

[PJ] P. Saracco, J. Vercruysse, Globalization for geometric partial comodules. Preprint (2020).

category theory

Audience: researchers in the topic

---

ItaCa Fest 2021

Series comments: ItaCa Fest is an online webinar aimed to gather the community of ItaCa.

---