Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular, derivations. In this paper, we show that the differential modality of a differential category lifts to a monad on the arrow category and, moreover, that the algebras of this monad are precisely derivations. Furthermore, in the presence of finite biproducts, the differential modality in fact lifts to a differential modality on the arrow category. In other words, the arrow category of a differential category is again a differential category. As a consequence, derivations also form a tangent category, and derivations on free algebras form a cartesian differential category.
Differential categories, introduced by Blute, Cockett, and Seely in [2], provide a powerful categorical framework for the algebraic foundations of differentiation, as well as for the categorical semantics of differential linear logic [10]. At their core, differential categories isolate and axiomatize the essential structural properties of differentiation in a way that is both conceptually clean and mathematically flexible. Concretely, a differential category (Def 2.3) is a symmetric monoidal category equipped with a differential modality: a monad S together with a deriving transformation d, whose axioms categorify fundamental identities of differential calculus, such as the chain rule and the Leibniz rule. This structure admits an intuitive insight: S(A) may be viewed as an algebra of differentiable functions with input in A, while d plays the role of an abstract differential operator sending functions to their derivatives. Differential categories encompass a rich collection of meaningful and well-studied examples which capture important models of differentiation. These include polynomial differentiation [2], where S = Sym is the free symmetric algebra monad (Ex 2.4); smooth functions [8], where S = S ∞ is the free C ∞ -ring monad (Ex 2.5); as well as more exotic examples from computer science, such as differentiation over finiteness spaces or Köthe spaces [10].
The theory of differential categories has been quite successful in formalizing various key differentiation related concept such as, in particular, derivations. In classical algebra, a derivation is the generalization of the differential operator which, recall, is a linear operator from an algebra to a module which satisfies the Leibniz rule. Derivations are a fundamental and ubiquitous concept, with deep applications across algebra, differential geometry, algebraic geometry, and beyond.
In [4], inspired by earlier work on Kähler differentials [1], Blute, Lucyshyn-Wright, and O’Neill introduced a categorical generalization of derivations within a differential category, defining the notion of an S-derivation relative to a differential modality S. An S-derivation (Def 3.1) is a map from an S-algebra (in the monad sense) to a module, axiomatized not by the Leibniz rule, but instead by the chain rule. Remarkably, every S-derivation automatically satisfies the Leibniz rule as a consequence, and hence recovers the classical notion of derivation. However, this is much more than just a reformulation: S-derivations are precisely those derivations that are compatible with the class of differentiable functions encoded by the differential modality S. For example, Sym-derivation correspond to ordinary derivations, since they satisfy a chain rule with polynomials (Ex 3.2), while S ∞ -derivations correspond to C ∞ -derivations [9,11], which satisfy a chain rule with real smooth functions (Ex 3.3).
A substantial body of classical results and concepts about derivations admits meaningful generalizations to S-derivations. For example, derivations can be characterized as algebra morphisms into a suitable semidirect product of an algebra and a module; an analogous statement holds for S-derivations [4,Prop 5.21]. Likewise, the classical construction of the universal derivation via Kähler differentials extends, under mild colimit assumptions, to yield universal S-derivations for S-algebras [4,Thm 5.23]. This in turn enables the development of de Rham cohomology internal to differential categories [15]. In another direction, S-algebras equipped with S-derivations to themselves generalize differential algebras in a differential category [13].
The main contribution of this paper is a new and conceptually striking characterization of S-derivations: we show that they are precisely the algebras of a monad on the arrow category. More specifically, we first show that any differential modality S lifts canonically to a monad S on the arrow category (Prop 4.5). The functor S is constructed directly from the deriving transformation d, and its monad structure is well defined by the chain rule axiom. We then prove that the algebras of this lifted monad S are exactly S-derivations.
This characterization is useful for several reasons. It holds in any differential category, without requiring additional structure such as biproducts or colimits, in contrast to other approaches [4]. Moreover, it offers a genuinely new perspective on the well established notion of derivations, which is only visible through the lens of differential categories. When biproducts are present, this perspective becomes even richer. We show that the arrow category admits a special monoidal structure (Lemma 6.2) which makes S into a differential modality (Thm 7.2), which is well-defined this time thanks to the Leibniz rule. As a consequence, the arrow category of a differential category with biproducts is itself a differential category, providing a new and systematic source of examples. In this setting, commutativ
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