  Cartesian differential categories were introduced to provide an abstract
axiomatization of categories of differentiable functions. The fundamental
example is the category whose objects are Euclidean spaces and whose arrows are
smooth maps.
  Tensor differential categories provide the framework for categorical models
of differential linear logic. The coKleisli category of any tensor differential
category is always a Cartesian differential category. Cartesian differential
categories, besides arising in this manner as coKleisli categories, occur in
many different and quite independent ways. Thus, it was not obvious how to pass
from Cartesian differential categories back to tensor differential categories.
  This paper provides natural conditions under which the linear maps of a
Cartesian differential category form a tensor differential category. This is a
question of some practical importance as much of the machinery of modern
differential geometry is based on models which implicitly allow such a passage,
and thus the results and tools of the area tend to freely assume access to this
structure.
  The purpose of this paper is to make precise the connection between the two
types of differential categories. As a prelude to this, however, it is
convenient to have available a general theory which relates the behaviour of
"linear" maps in Cartesian categories to the structure of Seely categories. The
latter were developed to provide the categorical semantics for (fragments of)
linear logic which use a "storage" modality. The general theory of storage,
which underlies the results mentioned above, is developed in the opening
sections of the paper and is then applied to the case of differential
categories.
