Abstract
For a topological category ϰ over Set we prove that if a functor T: ϰ → ϰ has a fixed cardinal α (i.e. for each object K with card (UK)=α we have card (UTK)≤α), then T has a least fixed point, and if T has a successive pair of fixed cardinals α and α+, then T has a greatest fixed point. This extends results of Adámek and Koubek.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adámek, J.: Free algebras and automata realizations in the language of categories, Comment. Math. Univ. Carolinae 15 (1974), 589–602.
Adámek, J., Herrlich, H. and Strecker, G. E.: Abstracts and Concrete Categories, Wiley, New York, 1991.
Adámek, J. and Koubek, V.: On the greatest fixed point of a set functors, Theoret. Comput. Sci., to appear.
Adámek, J. and Trnková, V.: Automata and Algebras in Category, Kluwer, Dordrecht, 1990.
Lambek, J.: A fixpoint theorem in complete categories, Math. Z. 103 (1968), 151–161.
Reiterman, J.: A more categorical model of universal algebra, Lecture Notes in Comput. Sci. 56 (1977), 308–313.
Trnková, V., Adámek, J., Koubek, V., and Reiterman, J.: Free algebras, input processes and free monads, Comment. Math. Univ. Carolinae 16 (1975), 339–351.
Author information
Authors and Affiliations
Additional information
Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Adámek, J. A remark on fixed points of functors in topological categories. Appl Categor Struct 4, 121–126 (1996). https://doi.org/10.1007/BF00124120
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00124120