Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fixed points in representations of categories

Authors: J. Adámek and J. Reiterman
Journal: Trans. Amer. Math. Soc. 211 (1975), 239-247
MSC: Primary 18A30
MathSciNet review: 0376799
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Fixed points of endomorphisms of representations, i.e. functors into the category of sets, are investigated. A necessary and sufficient condition on a category K is given for each of its indecomposable representations to have the fixed point property. The condition appears to be the same as that found by Isbell and Mitchell for Colim: $ {\text{Ab}^K} \to {\text{Ab}}$ to be exact. A well-known theorem on mappings of Katětov and Kenyon is extended to transformations of functors.

References [Enhancements On Off] (What's this?)

  • [1] J. Adámek and J. Reiterman, Fixed-point property of unary algebras, Algebra Universalis 4 (1975), 163-165. MR 0357278 (50:9746)
  • [2] -, Exactness of the set-valued colim (manuscript).
  • [3] J. R. Isbell, A note on exact colimits, Canad. Math. Bull. 11 (1968), 569-572. MR 39 #286. MR 0238926 (39:286)
  • [4] J. R. Isbell and B. Mitchell, Exact colimits, Bull. Amer. Math. Soc. 79 (1973), 994-996. MR 47 #6802. MR 0318255 (47:6802)
  • [5] M. Katětov, A theorem on mappings, Comment. Math. Univ. Carolinae 8 (1967), 431-433. MR 37 #4802. MR 0229228 (37:4802)
  • [6] H. Kenyon, Partition of a domain, Advanced problems..., Amer. Math. Monthly 71 (1964), 219. MR 1532554
  • [7] S. Mac Lane, Categories for the working mathematician, Springer-Verlag, New York, 1972. MR 0354798 (50:7275)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 18A30

Retrieve articles in all journals with MSC: 18A30

Additional Information

Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society