Diagrammatics for comodule monads (Q6607372)

The paper under review extends the Tannakian formalism of oplax monoidal monads, characterizing monoidal structures on a category \(\mathcal{D}\) with a strict monoidal functor \(\mathcal{D} \rightarrow \mathcal{C}\) in terms of oplax monoidal monads on \(\mathcal{C}\), to the setting of module categories. The case of monoidal structures, described by Moerdijk and separately by McCrudden, has led to the introduction and study of Hopf monads by Bruguières and Virelizier. The relation between the case of monoidal categories and that of their module categories is analogous to that of bialgebras and their (co)module algebras, the latter pair of structures being special cases of the former pair.\N\NGiven a monoidal category \(\mathcal{C}\), an oplax monoidal monad \(B\) on \(\mathcal{C}\), a \(\mathcal{C}\)-module category \(\mathcal{M}\) and a monad \(K\) on \(\mathcal{M}\), characterize module category structures on the Eilenberg-Moore category \(\mathcal{M}^{K}\) over the Eilenberg-Moore category \(\mathcal{C}^{B}\), such that \(U^{K}: \mathcal{M}^{K} \rightarrow \mathcal{M}\) is a \(\mathcal{C}^{B}\)-module functor, where \(\mathcal{M}\) is endowed with the \(\mathcal{C}^{B}\)-module structure obtained by pullback of the \(\mathcal{C}\)-module structure along the monadic functor \(U^{B}: \mathcal{C}^{B} \rightarrow \mathcal{C}\). Specifically, it is shown that such module category structures correspond bijectively to \textit{coactions} of \(B\) on \(K\), in the sense of comodule-monads of \textit{S. Chase} and \textit{M. Aguiar} [Theory Appl. Categ. 27, 263--326 (2013; Zbl 1275.18011)].\N\NWhile this is the main result of the article, in order to prove it, the authors prove a different theorem about lifting adjunctions of module categories to module adjunctions, along a monoidal adjunction. The result states that such lifts correspond precisely to lifts of the right adjoint to a strong module functor. This closely resembles the fact that lifts of an adjunction of monoidal categories to a monoidal adjunction correspond to lifts of the right adjoint to a strong monoidal functor.\N\NThe proofs presented in the article are particularly clear and elegant, the proof technique follows the idea of \textit{S. Willerton} [Arab. J. Sci. Eng., Sect. C, Theme Issues 33, No. 2, 561--585 (2008; Zbl 1185.18003)] of realizing monoidal and module-categorical structures, functors and transformations by a three-dimensional diagrammatic calculus for the monoidal \(2\)-category \(\mathbf{Cat}\).