Geometric partial comodules over flat coalgebras in Abelian categories are globalizable

TL;DR

This paper proves that geometric partial comodules over flat coalgebras in abelian categories are globalizable, introducing new globalization results and extending classical theorems to the partial setting.

Contribution

It establishes the globalizability of geometric partial comodules over flat coalgebras and introduces Hopf partial comodules, extending fundamental theorems to this new context.

Findings

Globalization of partial comodules is proven in new cases.

Partial corepresentations of Hopf algebras are globally realizable.

An analogue of the fundamental theorem for Hopf modules is established in the partial setting.

Abstract

The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literature. Finally, we introduce Hopf partial comodules over a bialgebra as geometric partial comodules in the monoidal category of (global) modules. By applying our globalization theorem we obtain an analogue of the fundamental theorem for Hopf modules in this partial setting.