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Isbell, John; Mitchell, Barry

Exact colimits and fixed points. (English) Zbl 0358.18016

Trans. Am. Math. Soc. 220, 289-298 (1976).
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)
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Cited in 1 Review
Cited in 4 Documents

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

Keywords:

small categories; colimits of \(C\)-indexed diagrams; fixed-point property

Citations:

Zbl 0253.18014; Zbl 0318.18004
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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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