Comonadic base change for enriched categories

DOI10.1016/J.JPAA.2023.107357zbMATH Open1517.18002arXiv1809.02356MaRDI QIDQ6306374

Branko Nikolić, Ross H. Street

Publication date: 7 September 2018

Abstract: For our concepts of change of base and comonadicity, we work in the general context of the tricategory $mathrmCaten$ whose objects are bicategories $mathscrV$ and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad $G$ on a cocomplete closed monoidal category $mathscrC$, the forgetful functor $U:mathscr{C}^{G}omathscrC$ is comonadic when regarded as a morphism in $mathrmCaten$ between one-object bicategories. We show that the forgetful pseudofunctor $mathscrU:mathscr{V}^{m}athscrGightarrowmathscrV$ from the bicategory of Eilenberg-Moore coalgebras for a comonad $mathscrG$ on $mathscrV$ in $mathrmCaten$ induces a change of base pseudofunctor $widetildemathscrU:mathscr{V}^{m}athscrGext-mathrmModightarrowmathscrVext-mathrmMod$ which is comonadic in a bigger version of $mathrmCaten$. We define Hopfness for such a comonad $mathscrG$ and prove that having that property implies $mathscrU$ creates left (Kan) extensions in the bicategory $mathscr{V}^{m}athscrG$. We provide conditions under which Hopfness carries over from $mathscrG$ to the comonad $widetildemathscrG=widetildemathscrUcircwidetildemathscrR$ generated by the adjunction $widetildemathscrUdashvwidetildemathscrR$. This has implications for characterizing the absolute colimit completion of $mathscr{V}^{m}athscrG$-categories.

Mathematics Subject Classification ID

Module categories in associative algebras (16D90) Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) (18A40) Differential graded algebras and applications (associative algebraic aspects) (16E45)

This page was built for publication: Comonadic base change for enriched categories