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Closed categories generated by commutative monads

Published online by Cambridge University Press:  09 April 2009

Anders Kock
Anders Kock
Affiliation:
The University of AarhusDenmark
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The notion of commutative monad was defined by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the front- and end-adjunctions are closed transformations. (The terms ‘Closed Category’ etc. are from the paper [2] by Eilenberg and Kelly). In particular, the monad itself is a ‘closed monad’; this fact was also proved in [4].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 4 , November 1971 , pp. 405 - 424
Copyright
Copyright © Australian Mathematical Society 1971

References

references

[1]Eilenberg, S. and Kelly, G. M., ‘A generalization of the functorial calculus’, J. Algebra 3 (1966), 366–375.Google Scholar
[2]Eilenberg, S. and Kelly, G. M., ‘Closed Categories’, Proc. of the Conference on Categorical Algebra, La Jolla, 1965 (Springer Verlag, 1966).Google Scholar
[3]Eilenberg, S. and Moore, J. C., ‘Adjoint functors and triples’, Illinois J. Math. 9 (1965), 381–398.CrossRefGoogle Scholar
[4]Kock, A., ‘Monads on symmetric monoidal closed categories’, Arch. Math. 21 (1970), 1–10.CrossRefGoogle Scholar
[5]Lambek, J., ‘Deductive systems and categories’, Math. Systems Theory 2 (1968), 287–318.CrossRefGoogle Scholar
[6]Linton, F. E. J., ‘Autonomous equational categories’, J. Math. Mech. 15 (1966), 637–642.Google Scholar