Fixed points in representations of categories

Authors:
J. Adámek and J. Reiterman

Journal:
Trans. Amer. Math. Soc. **211** (1975), 239-247

MSC:
Primary 18A30

DOI:
https://doi.org/10.1090/S0002-9947-1975-0376799-X

MathSciNet review:
0376799

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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: to be exact. A well-known theorem on mappings of Katětov and Kenyon is extended to transformations of functors.

**[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)**

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0376799-X

Article copyright:
© Copyright 1975
American Mathematical Society