Double Adjunctions and Free Monads

TL;DR

This paper explores the structure of double adjunctions and their relation to free monads and Eilenberg--Moore objects within double categories, providing new characterizations and extending previous results in category theory.

Contribution

It introduces new characterizations of double adjunctions using presheaves and universal squares, and extends the construction of free monads and Eilenberg--Moore objects to double categories.

Findings

Double adjunctions characterized via presheaves and universal squares

Construction of free monads in double categories extended from horizontal 2-category

Eilenberg--Moore objects characterized by representability of a parameterized presheaf

Abstract

We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg--Moore objects in double categories. We improve upon our earlier result in "Monads in Double Categories", JPAA 215:6, pages 1174-1197, 2011, to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category. We also prove that a double category admits Eilenberg--Moore objects if and only if a certain parameterized presheaf is representable. Along the way, we develop parameterized presheaves on double categories and prove a double-categorical Yoneda Lemma.