# Tangent Categories and their Applications (Online) (21w5251)

## Organizers

Robin Cockett (University of Calgary)

(University of Calgary)

(Mount Allison University)

## Description

The Banff International Research Station will host a workshop on "Tangent Categories and their Applications": it will be on-line and be a joint workshop with the Foundational Methods in Computer Science (FMCS 2021) meeting. The workshop and the FMCS meeting will run from June 14th - June 18th, 2021.

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.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).