Lifting Coalgebra Modalities and MELL Model Structure to Eilenberg-Moore Categories
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A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (MELL), known as a linear category, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by R. Blute and P. Scott's work on categories of modules of Hopf algebras as models of linear logic, we study categories of algebras of monads (also known as Eilenberg-Moore categories) as models of MELL. We define a MELL lifting monad on a linear category as a Hopf monad -- in the Bruguieres, Lack, and Virelizier sense -- with a mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to the category of algebras of MELL lifting monads. We explain how monoids in the category of coalgebras of the monoidal coalgebra modality can induce MELL lifting monads and provide sources for such monoids. Along the way, we also define mixed distributive laws of bimonads over coalgebra modalities and lifting differential category structure to categories of algebras of exponential lifting monads.
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