Monads in Double Categories

Details Monads in Double Categories Monads in Double Categories

2010-06-04

arxiv.org/abs/1006.0797v2

Mathematics

Category Theory

30 pages; v2: accepted for publication in the Journal of Pure and Applied Algebra; added hypothesis in Theorem 3.7 that source

Scientific paper

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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