Abstract
We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.
Highlights
This paper brings together two active strands of research in current category theory
With the refinement just noted, the embedding theorem can be stated as follows: Theorem Any small cartesian differential category has a full, structure-preserving embedding into the co-Kleisli category Kl(!) of the monoidal differential modality associated to a model of intuitionistic differential linear logic
As explained in the introduction, a cartesian differential category is a category endowed with an abstract notion of differentiation; the motivating example is the category whose objects are Euclidean spaces Rn and whose maps are smooth functions, but there are other examples coming from algebraic geometry and linear logic, which we will recall below
Summary
This paper brings together two active strands of research in current category theory. With the refinement just noted, the embedding theorem can be stated as follows: Theorem Any small cartesian differential category has a full, structure-preserving embedding into the co-Kleisli category Kl(!) of the monoidal differential modality associated to a model of intuitionistic differential linear logic. We will obtain this using our other main result, the enrichment theorem, and to describe that we must turn to the other side of our story: skew monoidal categories.
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