4 papers
The functorial semantics of Lie theory
Benjamin MacAdam
Ehresmann's introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie alge…
Vector bundles and differential bundles in the category of smooth manifolds
Benjamin MacAdam
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 diffe…
Reverse derivative categories
Robin Cockett, Geoffrey Cruttwell, Jonathan Gallagher +4
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 derivativ…
Involution algebroids: a generalisation of Lie algebroids for tangent categories
Matthew Burke, Benjamin MacAdam
We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in…