Abstract
Given an adjoint pair of functors F, G, the composite GF naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad? In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad. To demonstrate these methods, we explicitly compute the finitary homological presentations of the free A-module monad on the category of sets, for A a Dedekind domain. We also prove a criterion, reminiscent of Beck’s monadicity theorem, for when there is essentially (in a precise sense) only a single adjunction that gives rise to a given monad.
Similar content being viewed by others
References
Beck, J.: Triples, algebras, and cohomology. Reprints in Theory and Applications of Categories 2, 1–59 (2003)
Bénabou, J., Roubaud, J.: Monades et descente. C. R. Acad. Sci. Paris Sér. A-B 270, A96–A98 (1970)
Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, Modules, and Algebras in Stable Homotopy Theory, volume 47 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997). With an appendix by M. Cole
Grothendieck, A.: ÉLéments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222 (1961)
Mac Lane, S.: Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics, 2nd edn. Springer-Verlag, New York (1998)
Rada, J., Saorín, M., del Valle, A.: Reflective subcategories. Glasg. Math. J. 42(1), 97–113 (2000)
Tholen, W.: Reflective subcategories. In: Proceedings of the 8th International Conference on Categorical Topology (L,’Aquila, 1986), vol. 27, pp. 201–212 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salch, A. How many Adjunctions give Rise to the same Monad?. Appl Categor Struct 25, 875–891 (2017). https://doi.org/10.1007/s10485-016-9473-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9473-8