On the globalization of geometric partial (co)modules in the categories of topological spaces and algebras

Abstract

We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We show that this approach does not only allow to recover all known results in these settings, but it allows to treat new cases of interest, too.

Geometric partial comodules over flat coalgebras in Abelian categories are globalizable

“…As already mentioned in Remark 2.4, it is known that non-globalizable partial comodules may exist. This is the case, for instance, in the category (C)Alg k of (commutative) algebras over a field k (see [38,Corollary 3.7]) or in the opposite Top op of the category of topological spaces (see [39,Proposition 3.2]). The aim of the present paper is to show that (under mild assumptions on the coalgebra H or, more precisely, on its category of comodules) over an abelian monoidal category C we have gPCom H = gPCom H gl , obtaining in this way the optimum among the globalization results.…”

“…We also provided a genuine procedure to construct globalizations (whenever they exist) that can be applied to many concrete cases of interest. In [39], we already studied globalization of geometric partial comodules in the opposite of the categories of sets and topological spaces and in the category of algebras over a commutative ring (or, more precisely, we studied the globalization of geometric partial modules in Set, Top and Alg op k ). In the present paper we analyse the globalization problem in abelian categories, proving that in this setting globalization always exists under very mild conditions, and we apply this result in several concrete situations, recovering on the one hand some known globalization constructions, and providing on the other hand original ones, which have been never studied before.…”