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

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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**(1974), 163–165. MR**0357278****[2]**-,*Exactness of the set-valued colim*(manuscript).**[3]**John R. Isbell,*A note on exact colimits*, Canad. Math. Bull.**11**(1968), 569–572. MR**0238926****[4]**John Isbell and Barry Mitchell,*Exact colimits*, Bull. Amer. Math. Soc.**79**(1973), 994–996. MR**0318255**, 10.1090/S0002-9904-1973-13296-3**[5]**M. Katětov,*A theorem on mappings*, Comment. Math. Univ. Carolinae**8**(1967), 431–433. MR**0229228****[6]**Hewitt Kenyon and I. N. Baker,*Advanced Problems and Solutions: Solutions: 5077*, Amer. Math. Monthly**71**(1964), no. 2, 219–220. MR**1532554**, 10.2307/2311775**[7]**Saunders MacLane,*Categories for the working mathematician*, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR**0354798**

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

Article copyright:
© Copyright 1975
American Mathematical Society