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Register a new userTangent Categories from the Coalgebras of Differential Categories
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Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science. This is an extended version of a conference paper for CSL2020.
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We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids -- or in a straightforward generalisation, the category of modules over a commutative rig $k$. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad $Q$. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachanyi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad $Q$ involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal $k$-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category -- thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.
We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of $R^n$ in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.
We make precise the analogy between Goodwillies calculus of functors in homotopy theory and the differential calculus of smooth manifolds by introducing a higher-categorical framework of which both theories are examples. That framework is an extension to infinity-categories of the tangent categories of Cockett and Cruttwell (introduced originally by Rosicky). A tangent structure on an infinity-category X consists of an endofunctor on X, which plays the role of the tangent bundle construction, together with various natural transformations that mimic structure possessed by the ordinary tangent bundles of smooth manifolds and that satisfy similar conditions. The tangent bundle functor in Goodwillie calculus is Luries tangent bundle for infinity-categories, introduced to generalize the cotangent complexes of Andre, Quillen and Illusie. We show that Luries construction admits the additional structure maps and satisfies the conditions needed to form a tangent infinity-category, which we refer to as the Goodwillie tangent structure on the infinity-category of infinity-categories. Cockett and Cruttwell (and others) have started to develop various aspects of differential geometry in the abstract context of tangent categories, and we begin to apply those ideas to Goodwillie calculus. For example, we show that the role of Euclidean spaces in the calculus of manifolds is played in Goodwillie calculus by the stable infinity-categories. We also show that Goodwillies n-excisive functors are the direct analogues of n-jets of smooth maps between manifolds; to state that connection precisely, we develop a notion of tangent (infinity, 2)-category and show that Goodwillie calculus is best understood in that context.
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over Cartesian categories, so that every Cartesian category has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.
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.
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