TY - JOUR

T1 - On the 2-categories of weak distributive laws

AU - Böhm, Gabriella

AU - Lack, Stephen

AU - Street, Ross

PY - 2011/12

Y1 - 2011/12

N2 - A weak mixed distributive law (also called weak entwining structure [8]) 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 in [13]. 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 2×2.

AB - A weak mixed distributive law (also called weak entwining structure [8]) 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 in [13]. 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 2×2.

UR - http://www.scopus.com/inward/record.url?scp=84857016331&partnerID=8YFLogxK

U2 - 10.1080/00927872.2011.616436

DO - 10.1080/00927872.2011.616436

M3 - Article

AN - SCOPUS:84857016331

VL - 39

SP - 4567

EP - 4583

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 12

ER -