Related papers: Pullbacks in tangent categories and tangent displa…

Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract…

Previous work has shown that reverse differential categories give an abstract setting for gradient-based learning of functions between Euclidean spaces. However, reverse differential categories are not suited to handle gradient-based…

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…

In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that…

Tangent categories offer a categorical context for differential geometry, by categorifying geometric notions like the tangent bundle functor, vector fields, Euclidean spaces, vector bundles, connections, etc. In the last decade, the theory…

Tangent category theory is a well-established categorical framework for differential geometry. A long list of fundamental geometric constructions, such as the tangent bundle functor, vector fields, Euclidean spaces, and vector bundles have…

A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of…

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…

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…

We show how the tangent functor extends from ordinary smooth maps to "microformal morphisms" (also called "thick morphisms") of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations…

Tangent category theory is a well-established categorical context for differential geometry. In a previous paper, a formal approach was adopted to provide a genuine Grothendieck construction in the context of tangent categories by…

Differential lambda-categories were introduced by Bucciarelli et al. as models for the simply typed version of the differential lambda-calculus of Ehrhard and Regnier. A differential lambda-category is a cartesian closed differential…

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having…

This paper explores a new perspective on the universality of the vertical lift in tangent categories by presenting a categorification of the dimension of smooth manifolds. The universality of the vertical lift is a key part of the axioms of…

Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and…

Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks, and…

In this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we…

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…

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…

In this paper, we introduce the notion of a super tangent bundle of a manifold, and extend the basic notions of differential geometry such as differential forms, exterior derivation, connection, metric and divergence on manifolds that…