Abstract

The development of the formal theory of monads, begun by Street and later continued by Street and Lack, shows that large parts of the theory of monads can be developed within an arbitrary 2-category rather than in the 2-category of small categories, functors and natural transformations. I will describe some joint work with Tom Fiore and Joachim Kock in which we extend the basic concepts of the formal theory of monads from the setting of 2-categories to that of double categories. The motivation to do so derives from the desire to understand better the universal properties of the free category on a graph and of the free monad on a polynomial endofunctor. Our main result shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. After explaining this result, I will illustrate how it can be applied to obtain double adjunctions that extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.