Related papers: Cartesian Difference Categories
We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the…
In this paper we provide an abstract model theory for the untyped differential lambda-calculus and the resource calculus. In particular we propose a general definition of model of these calculi, namely the notion of linear reflexive object…
A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced…
Interventional causal models describe several joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between the different joint distributions and…
The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program…
We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
Disentangling the factors of variation in data is a fundamental concept in machine learning and has been studied in various ways by different researchers, leading to a multitude of definitions. Despite the numerous empirical studies, more…
In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. When pullbacks are required, often submersions are employed as an ideal class of maps which…
Category theory provides a compact method of encoding mathematical structures in a uniform way, thereby enabling the use of general theorems on, for example, equivalence and universal constructions. In this article we develop the method of…
Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work…
A tangent category is a category with an endofunctor, called the tangent bundle functor, which is equipped with various natural transformations that capture essential properties of the classical tangent bundle of smooth manifolds. In this…
In this paper we show that if $\mathscr{C}$ is a tangent category then the Ind-category $\operatorname{Ind}(\mathscr{C})$ is a tangent category as well with a tangent structure which locally looks like the tangent structure on…
The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian…
A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in…
Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
This is the fourth (and last) prepublication version of a book on derived categories, that will be published by Cambridge University Press. The purpose of the book is to provide solid foundations for the theory of derived categories, and to…
A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…
We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it…