Comonadic base change for enriched categories

TL;DR

This paper develops a general framework for change of base and comonadicity in enriched categories within a tricategory setting, introducing Hopfness for comonads and analyzing their impact on colimit constructions.

Contribution

It extends the theory of comonadicity and change of base to enriched categories in a tricategory context, defining Hopfness and exploring its implications for colimit creation.

Findings

The forgetful functor from Eilenberg-Moore coalgebras induces a comonadic change of base.

Hopfness of a comonad ensures the creation of left Kan extensions.

Conditions are provided for Hopfness to transfer between related comonads.

Abstract

For our concepts of change of base and comonadicity, we work in the general context of the tricategory $\mathrm{Caten}$ whose objects are bicategories $\mathscr{V}$ and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad $G$ on a cocomplete closed monoidal category $\mathscr{C}$, the forgetful functor $U : \mathscr{C}^G\to \mathscr{C}$ is comonadic when regarded as a morphism in $\mathrm{Caten}$ between one-object bicategories. We show that the forgetful pseudofunctor $\mathscr{U}:\mathscr{V}^\mathscr{G}\rightarrow \mathscr{V}$ from the bicategory of Eilenberg-Moore coalgebras for a comonad $\mathscr{G}$ on $\mathscr{V}$ in $\mathrm{Caten}$ induces a change of base pseudofunctor $\widetilde{\mathscr{U}}:\mathscr{V}^\mathscr{G}\text{-}\mathrm{Mod}\rightarrow \mathscr{V}\text{-}\mathrm{Mod}$ which is comonadic in a bigger version of $\mathrm{Caten}$. We define Hopfness for such a comonad $\mathscr{G}$ and prove that having that property implies $\mathscr{U}$ creates left (Kan) extensions in the bicategory $\mathscr{V}^\mathscr{G}$. We provide conditions under which Hopfness carries over from $\mathscr{G}$ to the comonad $\widetilde{\mathscr{G}}=\widetilde{\mathscr{U}}\circ \widetilde{\mathscr{R}}$ generated by the adjunction $\widetilde{\mathscr{U}}\dashv \widetilde{\mathscr{R}}$. This has implications for characterizing the absolute colimit completion of $\mathscr{V}^\mathscr{G}$-categories.