Double Adjunctions and Free Monads
Category Theory
2014-07-15 v2
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.
Cite
@article{arxiv.1105.6206,
title = {Double Adjunctions and Free Monads},
author = {Thomas M. Fiore and Nicola Gambino and Joachim Kock},
journal= {arXiv preprint arXiv:1105.6206},
year = {2014}
}
Comments
52 pages