Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories

TL;DR

This paper demonstrates how categories of algebras of certain monads can be used to model the multiplicative and exponential fragments of linear logic, extending the structure from linear categories to their Eilenberg-Moore categories.

Contribution

It introduces the concept of $ extsf{MELL}$ lifting monads as Hopf monads with special distributive laws and shows how the linear category structure lifts to Eilenberg-Moore categories, enriching the modeling of linear logic.

Findings

Linear category structure lifts to Eilenberg-Moore categories of $ extsf{MELL}$ lifting monads.

Groups in coalgebra categories induce $ extsf{MELL}$ lifting monads.

Defined mixed distributive laws and lifted differential category structures.

Abstract

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and 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 $\mathsf{MELL}$. We define a $\mathsf{MELL}$ lifting monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and Virelizier sense -- with a special kind of 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 $\mathsf{MELL}$ lifting monads. We explain how groups in the category of coalgebras of the monoidal coalgebra modality induce $\mathsf{MELL}$ lifting monads and provide a source for such groups from enrichment over abelian groups. Along the way we also define mixed distributive laws of symmetric comonoidal monads over symmetric monoidal comonads and lifting differential category structure.