  Differential lambda-categories were introduced by Bucciarelli et al. as
models for the simply typed version of the differential lambda-calculus of
Ehrhard and Regnier. A differential lambda-category is a cartesian closed
differential category of Blute et al. in which the differential operator is
compatible with the closed structure. We prove that any differential
lambda-category is equipped with a canonical strong commutative monad whose
construction resembles that of the tangent bundle in the category of smooth
manifolds. Most of the results of this note remain valid in an arbitrary
cartesian differential category. Our emphasis on differential lambda-categories
is motivated by the anticipated application of the theory developed in this
note to the design and semantics of a lambda-calculus extended by the
pushforward operator.

Category Theory