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 d
ifferential 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.
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its domain, to be
thought of as the partial identity that is defined to just the same degree as the original map. In this paper, we show that restriction categories can be identified with emph{enriched categories} in the sense of Kelly for a suitable enrichment base. By varying that base appropriately, we are also able to capture the notions of join and range restriction category in terms of enriched category theory.
The word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as proce
sses which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting
