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arXiv · 1009.3454

On the 2-categories of weak distributive laws

Gabriella Böhm·Stephen Lack·Ross Street
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Abstract

A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category K which admits Eilenberg-Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to K^{2 x 2}.

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Gabriella Böhm, Stephen Lack, Ross Street. 2011-11-14. On the 2-categories of weak distributive laws. https://doi.org/10.1080/00927872.2011.616436

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