Monads in Double Categories
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.
Cite
@article{arxiv.1006.0797,
title = {Monads in Double Categories},
author = {Thomas M. Fiore and Nicola Gambino and Joachim Kock},
journal= {arXiv preprint arXiv:1006.0797},
year = {2014}
}
Comments
30 pages; v2: accepted for publication in the Journal of Pure and Applied Algebra; added hypothesis in Theorem 3.7 that source and target functors preserve equalizers; on page 18, bottom, in the statement concerning the existence of a left adjoint, "if and only if" was replaced by "a sufficient condition"; acknowledgements expanded