Comonadic base change for enriched categories

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

29 pages. The paper is totally restructured and includes more examples