×

Vollständigkeitseigenschaften freier Colimes-Komplettierungen. (Completeness properties of free colimit completions). (German) Zbl 0582.18003

If \(C\) is a category with finite limits, then Ind-\(C\), its completion with respect to filtered colimits, has finite limits, too. To prove this A. Grothendieck and J. L. Verdier [Préfaisceaux (SGA 4, No.1), (Lect. Notes Math. 269) (1972; Zbl 0249.18021)] as well as E. Artin and B. Mazur [Etale homotopy (Lect. Notes Math. 100) (1969; Zbl 0182.260)] constructed an equivalence between \([X,Ind-C]\) and Ind-\([X,C]\), provided X is a ”rigid” finite category, i.e. X has no endomorphisms but the identities.
For \(\Delta\) a regular class of colimits in the sense of P. Gabriel and F. Ulmer [Lokal präsentierbare Kategorien (Lect. Notes Math. 221) (1971; Zbl 0225.18004)], let \(L(\Delta)\) be the class of limits such that in Sets, L(\(\Delta)\)-limits and \(\Delta\)-colimits commute. The present paper generalizes the above result: If \(C\) is a category with \(L(\Delta)\)-limits, then \(K_{\Delta}(C)\), its completion with respect to \(\Delta\)-colimits, has L(\(\Delta)\)-limits, too. In particular, for \(\alpha\) any regular cardinal and C a category with \(\alpha\)-small limits, the completion with respect to \(\alpha\)-filtered colimits has \(\alpha\)-small limits; and for \(\Delta\) the class of arbitrary coproducts, \(K_{\Delta}(C)\) has connected limits provided C has them. In this latter case, however, for \(X=\cdot \begin{matrix} \to \\ \to \end{matrix} \cdot\) the categories \([X,K_{\Delta}(C)]\) and \(K_{\Delta}([X,C])\) fail to be equivalent.

MSC:

18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M.Artin, A.Grothendieck et J.L.Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA 4). LNM269, Berlin-Heidelberg-New York 1972.
[2] M.Artin and B.Mazur, Etale Homotopy. LNM100, Berlin-Heidelberg-New York 1969.
[3] P.Gabriel und F.Ulmer, Lokal präsentierbare Kategorien. LNM221, Berlin-Heidelberg-New York 1971. · Zbl 0225.18004
[4] S.Mac Lane, Categories for the Working Mathematician. New York-Heidelberg-Berlin 1971. · Zbl 0232.18001
[5] C. V. Meyer, Approximation filtrante de diagrammes finis par Pro-C. Ann. Sci. Math. Québec4, 35-57 (1980). · Zbl 0435.18006
[6] H.Schubert, Categories. Berlin-Heidelberg-New York 1972. · Zbl 0253.18002
[7] H. Weberpals, Vertauschbarkeit von Limites und Colimites. Manuscripta Math.49, 9-26 (1984). · Zbl 0554.18001 · doi:10.1007/BF01174869
[8] H. Weberpals, über einen Satz von Gabriel zur Charakterisierung regulärer Colimites. Math. Z.161, 47-67 (1978). · Zbl 0369.18004 · doi:10.1007/BF01175612
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.