Abstract
Let 𝔽 be a monad in the category Comp of compact Hausdorff spaces and continuous maps. An abstract convexity was constructed by Radul for each 𝔽-algebra of the monad 𝔽 in the category Comp. It was proved that if the convexity of the monad 𝔽 with some additional properties is binary then 𝔽 has good topological properties, in particular, FX is an absolute extensor in the class of 0-dimensional spaces for each openly generated compactum X. We show in this paper that binarity is also a necessary condition.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ditor, S., Haydon, R: On absolute retracts, P(S) and complemented subspaces of \(C(D^{\omega _{1}})\). Studia Math. 56, 243–251 (1976)
Ditor, S., Eifler, L.: Some open mapping theorems for measures. Trans. Amer. Math. Soc. 164, 287–293 (1972)
Eilenberg, S., Moore, J.: Adjoint functors and triples. Ill. J. Math. 9, 381–389 (1965)
Engelking, R.: Dimension theory. Warszawa (1978)
Fedorchuk: On softness of maps of subset in locally convex spaces. Uspekhi Mat. Nauk 45, 221–222 (1990)
Fedorchuk, V.V.: Probability measures in topology. Uspekhi Mat. Nauk 46, 41–80 (1991)
de Groot, J.: Supercompactness and Superextension. Contributions to extension theory of topological structures. Deuche Verlag der Wissenschaften, eds., Berlin (1967)
Ivanov, A.V.: Solution of van Mill’s problem on the characterization of compacta, the superextensions of which are absolute retracts. Dokl. Akad. Nauk SSSR 262, 526–528 (1982)
Ivanov, A.V.: Superextensions of openly generated compacta. Dokl. Akad. Nauk SSSR 259, 275–278 (1981)
McLane, S.: Categories for Working Matematicians. Springer (1976)
Michael, E.: Some extension theorems for continuous functions. Pacific J. Math. 3, 789–806 (1953)
Radul, T., Zarichnyi, M.: Monads in the category of compacta. Uspekhi Mat. Nauk 50, 83–106 (1995)
Radul, T.: Convexities generated by L-monads. Appl. Categ. Struct. 19, 729–739 (2011)
Radul, T.: On functional representations of Lawson monads. Appl. Categ. Struct. 9, 457–463 (2001)
Radul, T.: On strongly Lawson and I-Lawson monads. Boletin de Matematicas 6, 69–75 (1999)
Radul, T.: Monad of hyperspaces of inclusion and its algebras. Ukr. Mat. Zh. 42, 806–811 (1990)
Swirszcz, T.: Monadic functors and convexity. Bull. Pol. Acad. Sci. 22, 39–42 (1970)
Teleiko, A., Zarichnyi, M.: Categorical Topology of Compact Hausdorff Spaces. VNTL Publishers, Lviv (1999)
Shchepin, E.V.: Functors and uncountable powers of compacta. Uspekhi Mat. Nauk 36, 3–62 (1981)
Shchepin, E.V.: κ-metrizable spaces. Izv. Akad. Nauk SSSR Ser. Mat. 43, 442–478 (1979)
van de Vel, M.: Theory of convex structures. North-Holland (1993)
Zarichnyi, M.: Spaces and mappings of idempotent measures. Izv. Ross. Akad. Nauk Ser. Mat. 74, 45–64 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Radul, T. Absolute Extensors and Binary Monads. Appl Categor Struct 25, 269–278 (2017). https://doi.org/10.1007/s10485-016-9433-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-016-9433-3