Title: Tangent categories & tangent monads
Tangent categories, first introduced by Rosicky [Ros84] and more recently generalized by Cockett and Cruttwell [Coc14], provide a categorical axiomatization for differential geometry. The core idea of this concept is to equip a category with an endofunctor which replicates the role of the tangent bundle functor, i.e. which associates to each object another object that looks like the fibre bundle of tangent spaces of the object. In this talk we explore the basic definitions and results of this theory, providing concrete examples of tangent categories.
In the second part, we are going to explore the theory of tangent monads, first introduced by Cockett, Lemay and Lucyshyn-Wright [Coc19]. Monads are often regarded as powerful tools to produce algebraic theories. In this regard, tangent monads produce geometrical theories. We will dig into the definition of tangent monads and the main results of the theory. We will also show that coCartesian differential monads induce tangent monads. This is a striking result, which will allow us to provide tangent categories for many types of algebras.
- Coc14: Cockett, Cruttwell; Differential Structure, Tangent Structure, and SDG (2014) (https://link.springer.com/article/10.1007/s10485-013-9312-0)
- Coc19: Cockett, Lemay, Lucyshyn-Wright; Tangent Categories from the Coalgebras of Differential Categories (2019) (https://arxiv.org/abs/1910.05617)
- Ros84: Rosicky; Abstract tangent functors (1984) (https://eudml.org/doc/91746)