Cartesian Differential Categories as Skew Enriched Categories

Abstract We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind enriched category. The base for enrichment is category commutative monoids—or in straightforward generalisation, modules over rig k . However, tensor product on this not usual one, but rather warping it by certain monoidal comonad Q Thus sense, skew sense Szlachányi. Our first main result that are same with finite products base. involved is, fact, an example modality. Differential modalities symmetric -linear characteristic feature their co-Kleisli categories. Using our result, we able to prove second one: every small admits full, structure-preserving embedding into induced modality (in closed category—thus, model intuitionistic linear logic). This resolves important open question area.