Vollständigkeitseigenschaften freier Colimes-Komplettierungen. (Completeness properties of free colimit completions)
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 \textit{A. Grothendieck} and \textit{J. L. Verdier} [Préfaisceaux (SGA 4, No.1), (Lect. Notes Math. 269) (1972; Zbl 0249.18021)] as well as \textit{E. Artin} and \textit{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 \textit{P. Gabriel} and \textit{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.
scientific article; zbMATH DE number 3931228