## Cartesian Differential Storage Categories

**by R. Blute, J.R.B. Cockett, and R.A.G. Seely**

### Abstract

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.
Monoidal differential categories provide the framework for categorical models
of differential linear logic. The coKleisli category of any monoidal 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 monoidal differential categories.

This paper provides natural conditions under which the linear maps of a Cartesian differential
category form a monoidal 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.