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A categorical model of the multiplicative and exponential fragments ofintuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linearcategory}, is a symmetric monoidal closed category with a monoidal coalgebramodality (also known as a linear exponential comonad). Inspired by Blute andScott's work on categories of modules of Hopf algebras as models of linearlogic, we study categories of algebras of monads (also known as Eilenberg-Moorecategories) as models of $\mathsf{MELL}$. We define a $\mathsf{MELL}$ liftingmonad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, andVirelizier sense -- with a special kind of mixed distributive law over themonoidal coalgebra modality. As our main result, we show that the linearcategory structure lifts to the category of algebras of $\mathsf{MELL}$ liftingmonads. We explain how groups in the category of coalgebras of the monoidalcoalgebra modality induce $\mathsf{MELL}$ lifting monads and provide a sourcefor such groups from enrichment over abelian groups. Along the way we alsodefine mixed distributive laws of symmetric comonoidal monads over symmetricmonoidal comonads and lifting differential category structure.

Lifting Coalgebra Modalities and MELL Model Structure to Eilenberg-Moore Categories