# BIRS workshop : Tangent Categories and their Applications

Mathematics

Banff International Research Station

Audience: | Researchers in the topic |

Conference dates: | 14-Jun-2021 to 18-Jun-2021 |

Curator: | BIRS Programme Coordinator* |

*contact for this listing |

One of the most fundamental notions when studying functions of a real variable is the rate of change of a function, as measured by its derivative. Geometrically, the derivative is the slope of the tangent line. Both the notion of the derivative and the tangent line (or more generally, the tangent bundle of a smooth manifold) can be defined purely axiomatically because of the underlying structure of the category of smooth functions. The derivative, for example, is determined by its properties (such as the sum and product formulae for differentiation, the chain rule, etc.). This structural approach to the derivative leads to the notion of a differential category. <\p>

Similarly, the tangent bundle of a manifold is determined by what one normally thinks of as properties - for example, properties of its sections. The abstraction of these properties to their most general setting is the notion of a tangent category. <\p> When working with models of these categories such as smooth functions or manifolds, the properties seem like natural consequence of the model. Differential and tangent categories, however, suggest a rather different perspective: namely, that the properties above determine the structure of the derivative or the tangent bundle and that one should look more broadly for instances of these structures in mathematics.

Category theory has proven to be a powerful way to organize mathematical structures and to show how these structures relate. The goal of this workshop is to utilize the cross-disciplinary language of tangent categories to identify and delineate general phenomena related to tangent structures in a wide variety of disciplines, including algebraic and differential geometry, algebraic topology and theoretical computer science.