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Commutative distributive laws

Published online by Cambridge University Press:  09 April 2009

Harvey Wolff
Harvey Wolff
Affiliation:
The University of TexasAustin, Texas 78712, U.S.A.
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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).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 2 , March 1975 , pp. 180 - 195
Copyright
Copyright © Australian Mathematical Society 1975

References

Beck, J. (1969), Distributive law, Lecture Notes in Mathematics, 80, (Springer-Verlag, 1969) 119–140.Google Scholar
Bunge, M. (1969), ‘Relative functor categories and categories of algebras’, Journal of Algebra 11, 64–101.CrossRefGoogle Scholar
Dubuc, E. J. (1970), Kan extensions in enriched category theory, Lecture Notes in Mathematics, 106, (Springer-Verlag, 1970).CrossRefGoogle Scholar
Eilenberg, S. and Kelly, G. M. (1966), Closed categories, in: Proc. of the Conf. on Categorical Algebra (Springer-Verlag, 1966) 421–562.Google Scholar
Kock, A. (1970), ‘Monads on symmetric monoidal closed categories’, Archiv. der Math. 21, 1–10.CrossRefGoogle Scholar
Kock, A. (1971), ‘Closed categories generated by commutative monads’, J. Austral. Math. Soc. 12, 405–424.CrossRefGoogle Scholar