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Monads in double categories

By Thomas Fiore, Nicola Gambino, and Joachim Kock

J. Pure Appl. Alg. 215 (2011), 1174-1197. ArXiv:1006.0797

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.

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Double adjunctions and free monads

By Thomas Fiore, Nicola Gambino, and Joachim Kock

Cahiers Topol. Géom. Différ. Catég. 53 (2012), 242-307. ArXiv:1105.6206

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 [Fiore-Gambino-Kock] 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.

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Last updated: 2013-03-12 by Joachim Kock.