Neeman, Ammon Loop spaces of the \(Q\)-construction. (English) Zbl 0967.18006 Fundam. Math. 164, No. 1, 71-95 (2000). C. H. Giffen [“Loop spaces for the \(Q\)-construction”, J. Pure Appl. Algebra 52, No. 1/2, 1-30 (1988; Zbl 0655.18008)] and H. Gillet and D. R. Grayson [“The loop space of the \(Q\)-construction”, Ill. J. Math. 31, 574-597 (1987; Zbl 0628.55011)] found independently a simplicial model for the loop space on Quillen’s \(Q\)-construction. The present article shows that their results generalize to the \(K\)-theory of triangulated categories. The proof is original even when restricted to the case of exact categories. It relies heavily on the ideas developed by the author in his series of papers on the \(K\)-theory of triangulated categories [A. Neeman, “\(K\)-theory for triangulated categories”; “I(A): Homological functors”, Asian J. Math. 1, No. 2, 330-417 (1997; Zbl 0906.19002); “I(B): Homological functors”, Asian J. Math. 1, No. 3, 435-529 (1997; 906.19003); “II: The subtlety of the theory and potential pitfalls”, Asian J. Math. 2, No. 1, 1-125 (1998; Zbl 0923.19002); “III(A): The theorem of the heart”, Asian J. Math. 2, No. 3, 495-589 (1998; Zbl 0937.19001); “III(B): The theorem of the heart”, Asian J. Math. 3, No. 3, 555-606 (1999)]. The first section of the present article gives an account of the ideas in this series of papers that are necessary for its understanding, so that the reader not familiar with them should get an insight into the homotopy theoretic machinery underlying the construction of \(K\)-theory for triangulated categories. Reviewer: Frédéric Patras (Nice) MSC: 18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) 19D06 \(Q\)- and plus-constructions 18E30 Derived categories, triangulated categories (MSC2010) Keywords:\(Q\)-construction; \(K\)-theory for triangulated categories; loop space Citations:Zbl 0655.18008; Zbl 0628.55011; Zbl 0906.19002; Zbl 0906.19003; Zbl 0923.19002; Zbl 0937.19001 PDFBibTeX XMLCite \textit{A. Neeman}, Fundam. Math. 164, No. 1, 71--95 (2000; Zbl 0967.18006) Full Text: EuDML