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**Exact colimits and fixed points.**
*(English)*
Zbl 0358.18016

Small categories \(C\) are characterized such that colimits of \(C\)-indexed diagrams of abelian groups commute with finite limits (i.e. such that \(\mathrm{colim}: Ab^c \to Ab\) is an exact functor). This result has been announced by the authors in [Bull. Am. Math. Soc. 79, 994–996 (1973; Zbl 0253.18014)]. The connection to the paper of the reviewer and J. Reiterman [Trans. Am. Math. Soc. 211, 239–247 (1975; Zbl 0318.18004)] is investigated. In that paper categories \(C\) are characterized with each indecomposable \(C\)-indexed diagram of sets having the fixed-point property. The two results are closely related; in the reviewed paper some results of Adámek and Reiterman are newly proved.

Reviewer: Jiří Adámek (Praha)

### MSC:

18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |

18G99 | Homological algebra in category theory, derived categories and functors |

18F05 | Local categories and functors |

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\textit{J. Isbell} and \textit{B. Mitchell}, Trans. Am. Math. Soc. 220, 289--298 (1976; Zbl 0358.18016)

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### References:

[1] | J. Adámek and J. Reiterman, Fixed points in representations of categories, Trans. Amer. Math. Soc. 211 (1975), 239 – 247. · Zbl 0318.18004 |

[2] | N. Bourbaki, Eléments de mathématiques: Algèbre commutative, Actualités Sci. Indust., nos. 1290, 1293, Hermann, Paris, 1961. MR 36 #146; 30 #2027. · Zbl 0108.04002 |

[3] | P. M. Cohn, On the free product of associative rings, Math. Z. 71 (1959), 380 – 398. · Zbl 0087.26303 |

[4] | John R. Isbell, A note on exact colimits, Canad. Math. Bull. 11 (1968), 569 – 572. · Zbl 0169.02302 |

[5] | John Isbell, Exact colimits. I, Ann. of Math. (2) 100 (1974), 633 – 637. · Zbl 0253.18014 |

[6] | John Isbell and Barry Mitchell, Exact colimits, Bull. Amer. Math. Soc. 79 (1973), 994 – 996. · Zbl 0253.18015 |

[7] | F. William Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869 – 872. · Zbl 0119.25901 |

[8] | Barry Mitchell, Rings with several objects, Advances in Math. 8 (1972), 1 – 161. · Zbl 0232.18009 |

[9] | Ulrich Oberst, Homology of categories and exactness of direct limits, Math. Z. 107 (1968), 87 – 115. · Zbl 0176.28902 |

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