The Tangent Functor Monad and Foliations

Abstract: In category theory, monads, which are monoid objects on endofunctors, play a central role closely related to adjunctions. Monads have been studied mostly in algebraic situations. In this dissertation, we study this concept in some categories of smooth manifolds. Namely, the tangent functor in the category of smooth manifolds is the functor part of a unique monad, which is the main character of this dissertation. After its construction and the study of uniqueness properties in related categories, we study its algebras, which are to this monad what representations are to a group. We give some examples of algebras, and general conditions that they should satisfy. We characterize them in the category of affine manifolds. We also study an analog of the tangent functor monad and its algebras in algebraic geometry. We then prove our main theorem: algebras over the tangent functor monad induce foliations on the manifold on which they are defined. This result links the study of these algebras to the study of foliations. A natural question is then to characterize foliations which arise this way. We give some restrictions in terms of the holonomy of such foliations. Finally, we study in greater detail algebras on surfaces, where they take a very simple form, and we nearly characterize them.