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

Author: Jean-Simon Lemay

Paper Information

Title:Lifting Coalgebra Modalities and IMELL Model Structure to Eilenberg-Moore Categories
Authors:Jean-Simon Lemay
Proceedings:FSCD Presented Papers
Editor: Helene Kirchner
Keywords:Linear Category, Coalgebra Modality, Monoidal Coalgebra Modality, Mixed Distributive Law, Bimonad, Hopf Monad, Eilenberg-Moore Category

ABSTRACT. 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 Eilenberg-Moore categories of monads 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 Eilenberg-Moore categories of MELL lifting monads. We explain how monoids in the co-Eilenberg-Moore 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 Eilenberg-Moore categories of exponential lifting monads.

Talk:Jul 09 12:00 (Session 47B: Linear Logic)