## A 2-category of weak mixed distributive laws

### Gabriella Böhm – 3 February 2010

A weak mixed distributive law in a 2-category consists of a monad t and a comonad c together with a 2-cell tc --> ct relating both structures. The axioms are obtained from those of a mixed distributive law by weakening the compatibility conditions with the (co)unit of the (co)monad. In contrast to a mixed distributive law, the weak generalization is not known to be a (co)monad in any 2-category. We show, however, that it can be described as a compatible pair of a comonad in a 2-category extending the 2-category of monads, and a monad in a 2-category extending the 2-category of comonads. This observation is used to construct a 2-category in which the objects are the weak mixed distributive laws. A mixed distributive law is shown to induce a weak lifting of the involved monad t to the Eilenberg-Moore category of the comonad c and also a weak lifting of c to the Eilenberg-Moore category of t. These weakly lifted monad and comonad are proven to possess equivalent Eilenberg-Moore objects.

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