Monads in Double Categories

TL;DR

This paper generalizes the theory of monads from 2-categories to double categories, introducing the double category of monads and conditions for free monad construction, with applications to double adjunctions.

Contribution

It extends Street's formal monad theory to double categories, defining the double category of monads and establishing conditions for free monad construction in framed bicategories.

Findings

Double categories of monads are introduced.

Free monad construction in double categories is characterized.

Applications include double adjunctions extending classical ones.

Abstract

We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads. Our main theorem 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. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.