Abstract
Kock (1970) defined the notion of a commutative monad in a symmetric monoidal closed category V and in Kock (1971) showed that the algebras for such a monad had a canonical structure as a closed category and that the monad had a canonical closed structure. In this paper we are concerned with the relationship between distributive laws and commutivity. In particular, the following question arises: given a distributive law between two monads on V when is the composite monad commutatice? To answer this question we define commutative distributive laws and show that if the composite is commutative then the distributiove law must be commutative. We also show that if Y and J are commutative monads in V with a commutative distributive law between them then the composite is commutative. So we get that if Y and J are commutative then the composite is commutative if and only if the distributive law is commutative. In addition we show that if the monads and the distributive law are commutative then the lifting of the monad Y to the category of J-algebras has a canonical structure as a closed monad (closed relative to the canonical closed category structure on the J algebras).
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