Firm Frobenius monads and firm Frobenius algebras

TL;DR

This paper explores the categorical properties of firm Frobenius algebras, establishing conditions under which their comodules and modules categories are equivalent, with implications for algebras with local units and coseparable coalgebras.

Contribution

It introduces a categorical framework for firm Frobenius algebras and characterizes when their comodules and modules categories are isomorphic, based on adjunctions and functor factorizations.

Findings

Categories of comodules and firm modules are isomorphic under certain conditions.

Equivalence occurs if the algebra has local units or the comultiplication splits the multiplication.

Provides categorical characterizations for firm Frobenius algebras arising from co-Frobenius coalgebras.

Abstract

Firm Frobenius algebras are firm algebras and counital coalgebras such that the comultiplication is a bimodule map. They are investigated by categorical methods based on a study of adjunctions and lifted functors. Their categories of comodules and of firm modules are shown to be isomorphic if and only if a canonical comparison functor from the category of comodules to the category of non-unital modules factorizes through the category of firm modules. This happens for example if the underlying algebra possesses local units, e.g. the firm Frobenius algebra arises from a co-Frobenius coalgebra over a base field; or if the comultiplication splits the multiplication (hence the underlying coalgebra is coseparable).