Graded Differential Categories and Graded Differential Linear Logic
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In Linear Logic (𝖫𝖫), the exponential modality ! brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic (𝖣𝗂𝖫𝖫) is an extension of Linear Logic which includes additional rules for ! which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic (𝖦𝖫𝖫) is a variation of Linear Logic in such a way that ! is now indexed over a semiring R. This R-grading allows for non-linear proofs of degree r ∈ R, such that the linear proofs are of degree 1 ∈ R. There has been recent interest in combining these two variations of 𝖫𝖫 together and developing Graded Differential Linear Logic (𝖦𝖣𝗂𝖫𝖫). In this paper we present a sequent calculus for 𝖦𝖣𝗂𝖫𝖫, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of 𝖦𝖣𝗂𝖫𝖫.
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