A note on Frobenius-Eilenberg-Moore objects in dagger 2-categories

TL;DR

This paper introduces Frobenius-Eilenberg-Moore objects within dagger 2-categories, extending universal properties and free completions to the dagger context, and explores dagger lax functors and limits.

Contribution

It defines Frobenius-Eilenberg-Moore objects for dagger Frobenius monads and extends universal properties and free completions to the dagger setting.

Findings

Frobenius-Eilenberg-Moore objects are well-defined in dagger 2-categories.

The free completion under Eilenberg-Moore objects extends to the dagger context.

Frobenius-Eilenberg-Moore objects are examples of dagger lax limits.

Abstract

We define Frobenius-Eilenberg-Moore objects for a dagger Frobenius monad in an arbitrary dagger 2-category, and extend to the dagger context a well-known universal property of the formal theory of monads. We show that the free completion of a 2-category under Eilenberg-Moore objects extends to the dagger context, provided one is willing to work with such dagger Frobenius monads whose endofunctor part suitably commutes with their unit. Finally, we define dagger lax functors and dagger lax-limits of such functors, and show that Frobenius-Eilenberg-Moore objects are examples of such limits.