Differential Categories Revisited

Abstract

Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these apparently distinct notions of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent. Thus, for linear logic settings, there is only one notion of differentiation.This paper also presents a number of separating examples for coalgebra modalities including examples which are and are not monoidal, as well as examples which do and do not support differential structure. Of particular interest is the observation that -somewhat counter-intuitively -differential algebras never induce a differential category although they provide a monoidal coalgebra modality. On the other hand, Rota-Baxter algebras -which are usually associated with integration -provide an example of a differential category which has a non-monoidal coalgebra modality.

Convenient Antiderivatives For Differential Linear Categories

“…In particular we also discuss the natural transformations K and J (Definition 2.6), both of which play fundamental roles for the notion of antiderivatives. For a more complete story on differential categories, we refer the reader to [2,4], while for more details on the story of integration and antiderivatives, see [8,9].…”

“…The problem is that arbitrary coalgebra modalities do not necessarily extend to the finite biproduct completion. On the other hand, monoidal coalgebra modalities induce monoidal coalgebra modalities on the finite biproduct completion (see [2,Section 7] for more details). However, finite biproducts do not play an important techinical role in this paper, so we will continue without them.…”

“…It should be noted that the interchange rule [d.5] was not part of the definition in [4] but was later added to ensure that the coKleisli category of a differential category was a Cartesian differential category [5]. Many examples of differential categories can be found in [2,4].…”

“…For example, Hyland and Schalk provide a nice list of examples in [16,Section 2.4]. Examples of coalgebra modalities which are not monoidal can be found in [2].…”

“…There are multiple equivalent ways of providing a monoidal coalgebra modality, some of which can be found in [2,26]. Of particular interest for this paper is via the Seely isomorphisms: Definition 3.4 A coalgebra modality (!, ρ, ε, ∆, e) on a symmetric monoidal category with finite products × and terminal object T is said to have the Seely isomorphisms [1,3,27] if the natural transformations χ A,B : !…”